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Dissertation

AdvancedTechniquesfor Resonance

FrequencyMonitoringin

Nanomechanical Sensing: Integrative

Approaches andApplications

Submittedinpartial fulﬁllmentofthe requirements forthe

academic degree Doktor derTechnischenWissenschaften (Dr.

Techn.), at

InstituteofSensorand Actuator Systems

Technische UniversitätWien

By:

Dipl.-Ing. Hajrudin Bešić

01226946

Supervisedby:

Univ.Prof. Dr.sc. Silvan Schmid

Vienna,November19, 2024

We cannot solve problems with the same thinkingweusedtocreate them.

-AlbertEinstein

Contents
Contents
Contents I
Abstract III
Acknowledgment V
ListofPublications VII
ListofSymbols IX
Contributions to Original Knowledge 1
Introduction 3
1TheoreticalBackground andMethods 7
1.1Nanomechanical Resonators ............................. 7
1.1.1DampedLinear Resonator .......................... 7
1.1.2FreeUndampedVibration ..........................9
1.1.3FreeDampedVibration ........................... 9
1.1.4DrivenDampedVibration .......................... 9
1.1.5QualityFactor .................................12
1.2Noise in Nanomechanical Resonators ........................ 14
1.2.1Noise Fundamentals .............................. 14
1.2.2Relationbetween Amplitude,Phase andFrequency Noise ....... 18
1.2.3Thermomechanical Noise ........................... 32
1.2.4Detection (Transduction)Noise ....................... 33
1.2.5Temperature FluctuationNoise .......................36
1.3AllanVariance ..................................... 37
1.4Noise in Closed-LoopSystems ............................ 40
1.5Transduction andActuation .............................42
1.5.1Magneto-motive Transduction andActuation ..............43
1.5.2Optical Transduction WithLaser-DopplerVibrometer ......... 44
2Schemes forTrackingResonance Frequency 45
3Adaptable FrequencyCounter with PhaseFiltering 51
I

Contents

4ThermalResponseModellingand Kalman Filtering 57

5Conclusionand Outlook 63

5.1Conclusion ........................................ 63

5.2Outlook ......................................... 64

Abstract

This dissertation exploresadvanced techniquesfor resonancefrequency monitoring in

nanomechanical sensing,withafocusonintegrative approachesand theirapplications.The

researchpresentsacomprehensivestudy on variousfrequency tracking schemesfor NEMS

(nanoelectromechanical systems) resonators,comparing open-loopand closed-loopmethods.

The ﬁndings demonstrate that alltrackingschemes canachieve comparableperformance in

termsofaccuracy andspeed,suggesting that theperformance limitsare intrinsic to the

NEMS resonatoritself. This allows thechoiceoftracking scheme to be basedonpractical

considerationssuchascostand ease of implementation.

Additionally,anadaptable frequency counterarchitectureisalsodeveloped,oﬀering

enhanced usabilityfor frequencymonitoring taskswithinthe NEMS community.

Further contributionsinclude thedevelopment of athermal modelfor NEMS infrared spec-

troscopy, accurately capturing thetransient response of NEMS to infraredlight absorption.

This modelaccountsfor both theintrinsic response of theresonator andthe frame-induced

response. Theintroduction of an adaptable Kalman ﬁlter, integratingthe NEMS response

modelwithinherent noiseproﬁles, enablesfasterand more precisemeasurementsthrough

optimaladaptiveﬁltering.

These advancements contributetothe ﬁeld of nanomechanical sensing,providing new

toolsand methodologies for more accurate andeﬃcient monitoring of resonancefrequencies

in NEMS devices.

III

Acknowledgment

Iwould liketoexpress my deepestgratitude to thosewho have supportedmethroughout

thejourney of completing this dissertation.

First andforemost, Iwanttothank my parents, Mujo and Elbira,for theirunconditional

love,support, andencouragement.You have always believedinme, andyourguidancehas

been invaluable in helping me reachthispoint.Tomybrother Haris,thank youfor always

beingthere andfor your unwavering support.

Iamimmenselygratefultomywife, Andrea,for herendless patience,love, and

understanding throughout this process. Your supporthas been my pillarofstrength.To

my belovedchildren, Nadia and Hana,you have been my inspirationand motivation to

pushthrough thetoughest moments.

Iwould also liketoextend my sincerethankstomyprofessorand mentor,Univ.Prof. Dr.sc.

Silvan Schmid.Yourguidance, expertise,and encouragement have been instrumental in

shaping this work. Iamdeeply thankful foryourmentorship andfor theopportunitiesyou

have providedme.

IamalsogratefultoProf. AlperDemir forhis valuablecontributions as acoauthor

andtoJohannesSteurer,whose expertise in electronics hasgreatly enriched my research.

Your insightsand assistance have been crucialtothe success of this work.

Special thanks go to my colleagues Kostas,Robert, Niklas,and Paolo.Your

collaboration,discussions, andcamaraderie have made this journeynot only productive

but also enjoyable.Iam truly fortunate to have hadthe opportunitytoworkalongside

such talented andsupportive individuals.

Lastly,Iwant to thank everyone whohas contributed,directly or indirectly,tothis

dissertation.Yoursupporthas been indispensable,and Iamdeeply appreciativeofall the

help Ihavereceived.

List of Publications

ListofPublications

Belowisthe listofpublications that resulted from theresearchworkinthis thesis:

H. Bešić, A. Demir, J. Steurer, N. Luhmann, andS.Schmid, "Schemes for Tracking

ResonanceFrequency forMicro-and Nanomechanical Resonators," Physical Review

Applied,vol.20, no.2,August2023, Art. no.024023.

DOI: https://doi.org/10.1103/PhysRevApplied.20.024023.

H. Bešić, A. Demir, V. Vukićević,J.Steurer,and S. Schmid, "Adaptable Frequency

Counter With PhaseFiltering for ResonanceFrequency Monitoring in Nanomechanical

Sensing," IEEE SensorsJournal,vol.24, no.6,March 2024, pp. 8094-8104.

DOI: https://doi.org/10.1109/JSEN.2024.3355026.

H. Bešić, A. Deutschmann-Olek,K.Mešić, K. Kanellopulos,and S. Schmid, "Optimized

Signal EstimationinNanomechanical Photothermal Sensing viaThermal Response

Modellingand Kalman Filtering," IEEE SensorsJournal,vol.24, no.19, October

2024.

DOI: https://doi.org/10.1109/JSEN.2024.3446369.

K. Schmerling,H.Bešić, A. Kugi,S.Schmid, A. Deutschmann-Olek, "Optimal

Sensing of MomentumKicks with aFeedback-Controlled Nanomechanical Resonator,"

arXiv:2411.0221.

DOI: https://doi.org/10.48550/arXiv.2411.02215.

VII

List of Symbols

ListofSymbols

Symbol Description

AArea

ADynamic matrixofastate-space

αThermalexpansion coeﬃcient

BMagnetic ﬂux density

BInput matrixofastate-space

BW Bandwidth

cDamping coeﬃcient

COutput matrixofastate-space

fFrequencyinhertz (Hz)

fxxProbabilitydensity function of x

FtTime-domaindrivingforce

HsTransferfunction of alinear system

kSpring constant

kBBoltzmannconstant

LLength

mMass in kilogram (kg)

meﬀ Eﬀective mass in kilogram (kg)

ntNoisesignal

NNoiselevel in whitenoise processes

PPower

QQualityfactor

RxτAuto-correlation function of x

SxfOne-sidedPSD of x,asafunction of linear frequency[V

/(Hz)]

SxωOne-sidedPSD of x,asafunction of angularfrequency[V2/(Hz)]

SII

xfTwo-sidedPSD of x,asafunction of linear frequency[V

/(Hz)]

SII

xωTwo-sidedPSD of x,asafunction of angularfrequency [V2/(Hz)]

xωTwo-sidedPSD of x,inthe unit [V2/(rad/s)]

TTime period

tTime

Temperature

τrResonatortime constant

uInput quantity of asystem

WEnergy

List of Symbols

Symbol Description

XfFrequency-domainrepresentationofasignal xtin linear frequency

XωFrequency-domainrepresentationofasignal xtin angularfrequency

Xrms Root mean square value of variable x

xtArbitrary signal

x0Amplitude of x

XVector

xIndicates that xis acomplexvariable

ˆxEstimateofx

∗Conjugatecomplexvalue of x

Electromotive force

Resonatorsusceptibility

yOutput quantity of asystem

ωAngularfrequency in radians persecond (rad/s)

ω0Eigenfrequency

ωrResonancefrequency

ΦMagneticﬂux

φDimensionless phase or angle

σStandard deviationofasignal

σ2Variance or normalized AC power

µMean value of asignal

µ2Mean or normalized DC power

ϕ, θ Phase, Angle

ϑPhaseintimeunits

ζDamping ratio

zDisplacement

Contributions to Original Knowledge

This thesispresentsthe following contributions to theoriginalknowledge in theﬁeldof

nanoelectromechanical systems (NEMS) andoptomechanics:

•

Arigorousand comprehensive study on frequencytracking schemesfor NEMS res-

onators, demonstrating throughbothexperimental andtheoretical resultsthatall

schemes(open-and closed-loop) canachieve comparable performance. Theﬁndings

suggest that theperformance limit is intrinsic to theNEMSresonator itself, allowing

thechoiceofschemetobebased on practicalconsiderationssuchascost, ease of

implementation,and application-speciﬁc requirements.

•

Thedevelopment of auniqueSelf-SustainingOscillatordesignwithpulsedfeedback,

notable forits ease of implementation, cost-eﬀectiveness, andhighconﬁgurability.

•

Thecreationofanadvanced frequencycounter architecture that is both adaptable

andcost-eﬀective, oﬀering enhanced usabilityfor frequencymonitoring taskswithin

theNEMScommunity.

•

Anovel thermalmodel forNEMSinfraredspectroscopy that accurately captures the

transient response of NEMS to infrared light absorption.Thismodel accounts forboth

keytimetransients: theintrinsic response of theresonatorand theframe-induced

response.

•

Theintroduction of an adaptableKalmanﬁlter that integrates theNEMSresponse

modeland thenoise proﬁlesinherent to NEMS devices, enablingfasterand more

precise measurements throughoptimal adaptive ﬁltering.

Introduction
Introduction
Background andMotivation
Nanoelectromechanical (NEMS) sensorshaveemerged as apowerfultoolfor awiderange of
applications due to theirhighsensitivity andthe abilitytodetect minute physical changes.
These sensorsoperate by monitoring resonancefrequency shifts,which canindicate various
environmental changesbythatinduceashiftinfrequency by changing parameters such
as change in mass[1
–
5], change in damping [6,7], or change in stiﬀness [8
–
10]. The
abilitytoaccurately track these frequencyshifts is crucialfor theeﬀectiveapplicationof
nanomechanical sensorsinbothresearchand industry.Itiscommonly perceivedwithin
thecommunitythatopen-loop frequency tracking schemesexhibit slower performance
comparedtotheir closed-loopcounterparts[11,12]. Moreover,literature[12]indicates that
thereisaninherent trade-oﬀbetween speed andaccuracy amongvarious conﬁgurations of
closed-loopfrequency tracking schemes. It is essential to thoroughly investigatebothopen-
andclosed-loopconﬁgurations andtoexperimentally validate whetherthe performance
limitations in speed andaccuracy of aNEMSsystemare due to thespeciﬁc trackingschemes
employed,orifthese limitations arefundamentally intrinsic to theNEMSitself.
Traditionalmethods formonitoring resonancefrequencies,suchasphase-locked loops
(PLL),havebeen widelyuseddue to theirprecision andstability. However, thesemethods
oftencomewithsigniﬁcantcostand complexity,makingthemless accessiblefor broader
applications.Frequency counters (FC) have been acost-eﬀectiveand simpleralternative,
oﬀeringcompetitive performancefor frequencyshift detectioninnanomechanicalsystems.
IncorporatingFCs as an alternativetoPLLs forfrequency detectioncould oﬀer signiﬁcant
beneﬁtstovarious research groups in this ﬁeld.
Whilevarious models existthatdescribethe behavior of NEMS in infrared sensing
(NEMS-IR)[10,13
–
20], thereremains aneed foracomprehensive modelthatincorporates
both transients—thoseinduced by theheating of themembraneand theheating of the
frame—intothe estimation of theabsorbedpower of theNEMSdevice.
3

Introduction

This dissertation exploresadvanced techniquesfor resonancefrequency monitoring in

nanomechanical sensing,novel tracking schemes, focusing on thedevelopment andapplica-

tion of adaptable frequencycounterswithphase ﬁltering,their integrationwithinfrared

spectroscopy,thermal modeling forinfraredspectroscopyand implementation of Kalman

ﬁlteringfor infraredspectroscopy.

Research Objectives

Theprimary objectives of this dissertationare:

To compare variousschemes fortracking resonancefrequency in micro-and nanome-

chanical resonators,evaluating theirtrade-oﬀs between speed andprecision.

To developanadaptable frequencycounter with phaseﬁlteringfor enhanced resonance

frequencymonitoring in nanomechanical sensing as an alternativetoPLL.

3. To modelthe thermalbehaviorofNEMSresonatorsininfraredspectroscopy.

To integrateKalmanﬁlteringtechniquesfor optimalsignalestimation in infrared

spectroscopy.

Structureofthe Dissertation

This dissertation is organized as follows:

•

Chapter 1:Theoretical Background andMethods -Oﬀers theoreticalfounda-

tionsthatare notextensivelycovered andmethods used in theexisting publications.

•

Chapter 2:Schemes forTrackingResonance Frequency -Exploresdiﬀerent

tracking methodsand comparestheir performanceboththeoretically andexperimen-

tally.

•

Chapter 3:Adaptable Frequency Counter withPhase Filtering -Presents the

design,implementation, andexperimentalresultsofthe proposed frequencycounter

forNEMSapplications.

•

Chapter 4:ThermalResponseModellingand KalmanFiltering -Proposes a

thermalmodel forNEMS-IR andexaminesthe useofKalmanﬁltering forimproving

frequencytracking accuracy,including experimental validation.

Introduction

•

Chapter 5:Conclusion andOutlook -Summarizes thekey ﬁndings, contributions,

andpotential future research directions in theﬁeldofnanomechanical sensing.

Throughthese chapters,thisdissertation aims to provideacomprehensive understanding

ofadvancedresonancefrequency monitoring techniquesand theirapplications,contributing

to theadvancementofnanomechanical sensing technology.

Chapter1

TheoreticalBackground andMethods

In this chapter, thetheoretical fundamentalsand methodsnecessary forunderstanding the

subsequent chapters arediscussed. Thechapter begins with an explorationofthe lumped

elementmodel of NEMS resonators,deﬁning theprimary quantities that will be used

throughoutthe dissertation. Following this,the main noisesources in NEMS resonators

aredescribed,providing afoundation forlater discussionsonnoise characterization.

Next, theAllanvariance, avaluable tool formeasuring frequencystabilityinresonators,

is explainedindetail. This section aims to equipthe reader with theknowledge required to

understand themethods used forassessing frequencystability.

Finally,the transduction andactuation schemes, used in this work,are brieﬂy described.

Most of thephysical quantities andanalyticalcalculationsinthis chapterare based on the

calculationsprovidedin[21].

1.1Nanomechanical Resonators

1.1.1DampedLinearResonator

Mechanical vibrations in physical systemsare associated with theperiodic conversion of

kineticenergytopotential energy andviceversa.Elastic elements such as beams, strings,

andplatesstore potentialenergyinthe form of deformationenergy, typically modeled

as aspring (Figure 1.1). Whenthe system is displaced from itsrest position,restoring

forces acceleratethe mass back toward equilibrium, converting potentialenergyintokinetic

energy.Asthe system traverses therestposition, kineticenergyistransformed back into

potentialenergy. This periodic energy exchange wouldcontinueindeﬁnitely if notfor the

presence of dissipative forces.

Chapter1 TheoreticalBackground andMethods

Figure1.1:Lumped elementmodel of aresonator

Forthe analysis of free vibration(eigenfrequency)ofcontinuum mechanical resonators,

dissipative forces areoftenneglected,assuming constant totalenergy. However, in real

systems, energy is dissipatedthrough mechanisms such as viscous damping, acoustic

transmission,surface losses, andinternaldissipation.Dissipative forces areapproximately

proportional to thevelocityofthe vibrationalmovementand areoften modeledasadashpot

(Figure 1.1).

Theforced vibrationofadamped system with asingledegreeoffreedom canberepresented

by thelumped elementresonatormodel.This system consists of alinearzero-mass spring,

alinear damping element, andamass. Whensubjected to aperiodic drivingforce

FtF

0cosωt

,the dynamics of thesystemare describedbyasecond-orderdiﬀerential

equation

m¨zc˙zkz Ft,(1.1)

where

is themass,

is thespring constant,and

is thedamping coeﬃcient. This

equation characterizes thesystem’sresponse to theappliedforce, taking into accountthe

eﬀectsofboththe spring anddamping elements.

1.1.2FreeUndamped Vibration

In thecaseofanondriven lumped-elementresonatorwithzerodamping (

c

0),the

totalenergyofthe system remains constant.The resonatorbehaves as an oscillator

withavibrationalamplitude expressed as

ztz

cosω0t

.During oscillation,energy

oscillatesbetween kineticand potentialforms.The eigenfrequency

ω0

of theundamped

free mechanical system is derivedfromthe equation

ω0k

m.(1.2)

1.1.3FreeDampedVibration

Foranondriven system,the equation of motion

(1.1)

reduces to thehomogeneousdiﬀerential

equation

¨z2ζω0˙zω2

0z0,(1.3)

wherethe dampingratio ζis deﬁnedas

ζc

km.(1.4)

Solving this equation with thetrial solution

ztz

γt

yields solutionsthatdescribethe

system’s behavior underdiﬀerentdamping conditions:

•Overdamped:ζ1.

•Critically damped:ζ1.

•Underdamped:ζ1.

1.1.4DrivenDampedVibration

Theresponse of alinear lumped-elementresonatortoaharmonic driving force

Ft

cosωt

combinestransient andsteady-stateresponses. Forthe steady-state response the

diﬀerential equation of motion is solved using complexvariablesfor theforce

FtF

iωt

anddisplacement zz0eiωt,resulting in expressionsfor thecomplexamplitude

z0F0m

ω2

0ω22iζω0ωz0eiϕ,(1.5)

Chapter1 TheoreticalBackground andMethods

where

is thedisplacementamplitude and

thephase diﬀerence between theforce

anddisplacement signals. The magnitude of thecomplexdisplacementamplitude canbe

calculatedas

0z

 F

m

ω

0ω

24ζ

2(1.6)

andthe phaseas

ϕargz0arctan 2ζω0ω

ω2ω2

.(1.7)

Theequations foramplitude andphase canbothberewrittenasfunctionsofrelative

frequency ωω0as

z0F0k

1ω

ω0224ζ2ω2

ω2

02(1.8)

and

ϕargz0arctan 2ζω

ω0

ω

ω021

.(1.9)

By deﬁningthe static deﬂectionas

sF

k

,whichisthe displacementamplitude that

wouldbecausedbyastatic or lowfrequency force, thedisplacementcan be writtenas

ztz

δz0cosωt ϕ,(1.10)

where δz0is therelativeamplitude response andisgiven by

δz01

1ω

ω0224ζ2ω2

ω2

02.(1.11)

Theamplitude andphase responses arecalculatedand plottedinFigure 1.2.

It canbeseenthatthe amplitudereaches amaximum near theeigenfrequency

ω0

,which

is calledthe resonancefrequency.Itcan be calculatedthrough therelation

∂z0∂ω 

whichyields

ωrω012ζ2.(1.12)

Forverysmall damping

ζ

1, theresonancefrequency is very closetothe eigenfrequency

ωrω0.The phase response of thesystembecomes

ϕrarctan 12ζ2

ζ.(1.13)

(a)

10−1100101

ω/ω0

10−2

10−1

100

101

δz0

ζ=1

ζ=0.1

ζ=0.01

(b)

10−1100101

ω/ω0

−1.0

−0.8

−0.6

−0.4

−0.2

0.0

ϕ/π

ζ=1

ζ=0.1

ζ=0.01

Figure1.2:

Frequencyresponseofthelineardampedresonator modelfor diﬀerentdamping

ratios

.(a) Shows theamplitude response of themodel accordingtoequation

(1.11)

.Atresonance frequency for

ζ

1the systemsexperiences maximum

ampliﬁcation, whichincreases with decreasing damping ratio. (b)Shows the

phase response obtainedbyequation

(1.9)

.Itshows that thephase hasa

π

phase shiftatresonance andanincreasingslope fordecreasing damping ration.

Chapter1 TheoreticalBackground andMethods

1.1.5QualityFactor

Thereare multiple deﬁnitions of thequalityfactor. It canbedirectly derivedfromthe

amplituderesponseequationbyexamining theheightofthe resonancefrequency peak.

Equation (1.8)atthe resonancefrequency yields

z0ω=ωrF0

2ζ1ζ2F0

kQ, (1.14)

with

being thequalityfactorthatalsorepresentsthe ampliﬁcationofthe displacement

amplitude at resonance. Forverysmall damping

ζ

1, thequalityfactorcan be simpliﬁed

Q1

2ζ.(1.15)

In termsofphase response,the phaseslope at resonanceincreases with thequalityfactor.

Theslope of thephase response is givenby

∂ϕ

∂ωω=ωr1

ζωr2Q

ωr

.(1.16)

Amorephysicalinterpretationofthe qualityfactoristhe ratiobetween thestoredand

dissipatedenergyofthe systeminone cycleatresonance

Q2πW

ΔW,(1.17)

where

is thestoredenergyand Δ

thelostenergyduring oneoscillation cycle. If the

displacementatresonance is givenby

zz

cosωrt

,the totalenergyofthe system canbe

calculatedas

Wmax









2m˙z2







1

2mz2

0ω2

r.(1.18)

Thelostenergyduring onecycle canbedescribed by theworkdonebythe dissipative force

Fdc˙zandisgiven by

ΔW2π/ω

d˙zdtπcz2

0ωr,(1.19)

whichyieldsthe qualityfactor

Qmωr

c.(1.20)

Incorporating thedeﬁnitionsprovidedinequations

(1.4)

and

(1.12)

,the qualityfactorcan

0.000 0.002 0.004 0.006 0.008 0.010

Time (s)

−1.0

−0.5

0.0

0.5

1.0

Amplitude

e−t/τr

e−t/τrcos(ωrt)

Figure 1.3: Qualityfactorcharacterizationbythe ring-downmethod

be rewritteninterms of thedamping ratio

Q12ζ2

2ζ,(1.21)

andcan be simpliﬁedtoequation(1.15)for slight damping (ζ1).

Amethodofmeasuring thequalityfactoristhe ring-downmethod. Oneway of doingit,

is to drivethe resonatoruntil an oscillation with steady-state amplitude is reached. After

turning oﬀ thedrive,the systemcan be modeledasafree damped vibrationdescribed by

theexponentially decaying oscillation

ztz

−t/τ

rcos ω01ζ2t,(1.22)

with thetime constant

τr

ζω0

.Equation

(1.15)

canbeexpressed in termsofthe time

constant as

Q1

2ωrτr.(1.23)

Thering-downmethodisparticularlyuseful forhigh

resonators andwill be employedin

this work fordetermining thequalityfactor.

Chapter1 TheoreticalBackground andMethods

1.2Noise in Nanomechanical Resonators

1.2.1Noise Fundamentals

Oneofthe main characteristicsofevery sensoristhe minimum quantity it candetect,also

knownassensitivity [21]. Thesensitivity is thedetectionlimit,which is usually constrained

by noisephenomena.Astationary ergodic noisesignal

xt

,observedinatime interval

is arandomprocesstypically describedbythe following quantities [22–26]:

1. Mean:The averagevalue of thenoise signal:

µlim

T→∞

TT/2

−T/2

xtdt. (1.24)

2. NoisePower:The mean square of thenoise signal:

Plim

T→∞

TT/2

−T/2

x2tdt. (1.25)

3. NoiseVariance:The spread or dispersionofthe noisesignal:

σ2lim

T→∞

TT/2

−T/2xtµt2dt. (1.26)

By taking thesquarerootofthe variance, thestandard deviation

of thenoise process

can be calculated.Inthis work, we will primarily deal with Gaussian noiseprocesses. A

Gaussian noiseprocessisarandomprocesswithaGaussian distributionasits probability

densityfunction (PDF). Figure 1.4 showsthe PDFofaPython-generated randomsequence

with µ1and σ1.

Noiseprocessescan also be describedinthe frequencydomain. Astandard tool to examine

noise signalsinthe frequencydomain is thepower spectraldensity (PSD)[22,24]. It describes

thepower distributionofastationary randomprocess across frequencies. According to

theWiener-Khinchin theorem, thetwo-sidedPSD

SII

xf

andthe auto-correlation function

Rxτof astationaryrandomprocess areaFourierpair, described by therelations:

SII

xf∞

−∞ Rxτe−i2πfτ dτ, (1.27)

Rxτ∞

−∞ SII

xfei2πfτ df. (1.28)

−2 0 2 4

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

fx(x)

Mean: 1.00

Std Dev: 1.00

Figure1.4:

Python simulation forvisualizationofthe probabilitydensity function of a

Gaussian noiseprocess, highlighting itsmeanand standard deviation.

Knowing oneofthese twopropertiesallows theother to be obtainedbythe Fourier

transform.

In theliterature, twoversionsofthe PowerSpectralDensity (PSD)are commonly used:

theone-sidedPSD,denoted as

Sf

,and thetwo-sided PSD, denotedas

IIf

.The

two-sidedPSD is symmetricaround

f

0and canbeconvertedintothe one-sidedPSD

using therelation:

Sf2S

IIffor f0.(1.29)

To clarifythis, consider thenormalized DC power, whichisthe square of themeanofthe

signal.For atwo-sided PSD, theDCpower is theareaunderthe impulseat

f

0and can

be expressedas[22]:

PDC µ20+

0−SIIfdf. (1.30)

In contrast,for aone-sidedPSD,the normalized DC powerisgiven by:

PDC µ20+

Sfdf. (1.31)

Thevariance, or thenormalized AC powerofasignal,can be obtainedbyintegrating

Chapter1 TheoreticalBackground andMethods

thetwo-sided PSDoverall frequencies, excluding theimpulseatf0[22]:

PAC σ20−

−∞ SIIfdf∞

0+SIIfdf. (1.32)

Similarly, forthe one-sidedPSD,the normalized AC poweris:

PAC σ2∞

0+Sfdf. (1.33)

Thepower spectraldensity (PSD)isdirectly relatedtothe magnitude spectrumofa

signal.The spectrumofasignal

xt

canbeobtainedbyperformingthe Fouriertransform,

yielding:

Xf∞

−∞ xte−i2πft dt, (1.34)

Xω∞

−∞ xte−iωt dt, (1.35)

where

Xf

and

Xω

representthe signal spectruminthe linear frequency

(Hz) and

angularfrequency ω(rad/s) domains,respectively.

Forexample,if

xt

hasunitsofVolt(V),both

Xf

and

Xω

will have unitsof

Volt-seconds (Vs) due to thetime integration. Thetwo spectra arerelated by substituting

ω2πf in Xω:

XfXω,where ω2πf. (1.36)

Althoughthe spectra areequivalent, they areexpressed on diﬀerentfrequency scales.

This distinction is crucial, particularlywhenintegrating over thefrequency axis,suchasin

theinverse Fouriertransform:

xt∞

−∞ Xfei2πft df, (1.37)

xt1

2π∞

−∞ Xωeiωt dω. (1.38)

In Eq.

(1.37)

,the integrationoverthe linear frequency

restores theoriginalsignal

with unitsof(V).However,inEq.

(1.38)

,the factor 1



π

is required to accountfor

therelationship d

ω

,ensuring theunitsremainconsistentand thetime signal is

recoveredin(V).

Thetwo-sidedPSD canbecalculatedfromthe corresponding magnitude spectra as

follows:

SII

xflim

T→∞

TXf2,(1.39)

SII

xωlim

T→∞

TXω2,(1.40)

wherebothversionsofthe PSDhaveunitsof(

Hz

)and diﬀeronlybythe frequency

scaling. Thetotal signal powerisobtainedbyintegrating thePSD over allfrequencies:

P∞

−∞ SII

xfdf1

2π∞

−∞ SII

xωdω. (1.41)

Again, when integrating over angularfrequency

,the correctionfactor1



π

is necessary

to maintain unit consistency, similartothe inverseFourier transform.

It is worth noting that in some literature[27], thePSD with respect to angularfrequency

is expressed in unitsof(

(rad/s)

). In such cases, we denote this PSDas

ω

,which

includesthe 12πscaling inherently in theunits. Thetotal poweristhencalculatedas:

P∞

−∞ Sc

xωdω. (1.42)

Onespecial type of noiseprocess that is very commonly used in many models is white

Gaussian noise(WGN).Ithas aGaussian-distributedPDF andaconstant PSD(

Sf

const.

). Theconstantvalue of thePSD forsuchaprocess, referredtoasthe noiselevel,is

typically denotedas

SfN

forasingle-sided spectrum, or

SIIfN

2for adouble-sided

spectrum.

When anoise process passes throughalinear system,spectralanalysiscan be useful.

Figure 1.5 showsalinear system with atransfer function

Hs

that hasanoisesignalwith

apower spectraldensity

Sxs

at itsinput (where

sαiω

is thecomplexfrequency).

Theresulting PSDatthe output of thesystemcan be calculated by multiplyingthe PSD

of theinput signal with thesquared magnitude response of thesystem[25]

SysHs2Sxs,(1.43)

where

Hs

is thetransfer function of thesystem. If thetransferfunction

Hs

is known

asimple wayofdeterminingthe square magnitude response is by using therelation

Chapter1 TheoreticalBackground andMethods

Property Description

SII

xfS

xfSII

xfhasevensymmetryaround f0.

SII

xfRS

xfis real.

∞

−∞ SII

xfdfP

Theareaunder

SII

is themeanvalue of the

powerofxt.

xtVS

xfV

Hz If xthasunitsV,then SII

xhasunits V2

Hz .

0+

0−SII

xfdfPDC µ2

Theareaunderanimpulsein

f

0is

thesquareofthe mean or thenormalized DC

powerofxt.

0−

−∞ SII

xfdf∞

0+SII

xfdfPAC σ2

Theareaunder

SII

excluding theimpulseat

f

0isthe variance or normalized AC power

of xt.

SII

xf0S

xis positive forall frequencies.

Table 1.1: Properties of thepower spectraldensity SII

xfof thesignal xt.

Hs2HsH∗s,where H∗sis theconjugatecomplexofHs.

Hs

sS

sHs2Sxs

Figure1.5:

Blockdiagram representation of theresponseofalinearsystemtonoise input.

In this work thenoise processeswill forthe most part been treated in thespectraldomain

thereforesomepropertiesofthe PSDwillbesummarized in Table 1.1 (takenfrom[22]).

1.2.2Relationbetween Amplitude, Phaseand FrequencyNoise

Noiseprocesses areacritical limiting factor in theperformance of NEMS sensors,making it

essential to understand howtheyimpact thesignalofinterest. High-

resonators,due to

theirwell-deﬁned resonancefrequency andsteep phaseresponse, areparticularlywell-suited

as frequencytransducers, whereaphysical quantity is mapped to aresonancefrequency

shift. To accurately model systemperformance,itiscrucial to comprehend theinterplay

between amplitude noise, phase noise, andfrequency noise.

Consideranarbitrary noiseless signal

xtAcosω0t

,withamplitude

andangular

frequency ω0.Various typesofﬂuctuations canbeintroduced as follows [26]:

•Amplitude ﬂuctuations:

xtAA

cosω0t(1.44)

where Anrepresentsthe amplitude noise.

•Phaseﬂuctuations:

xtAcosω0tϕn(1.45)

where ϕndenotesthe phase noise.

•Frequency ﬂuctuations:

xtAcosω0ωnt(1.46)

where ωnsigniﬁesthe frequencynoise.

Amplitude,phase, andfrequency noiseare generally notcompletelyindependent.Amore

comprehensive expression forthe signal canbewrittenas:

xtAcosω0tϕnt,(1.47)

where

xt

represents the signal,consistingofacosine wave with amplitude

,angular

frequency ω0andarbitrary phase ϕ,and ntdenotesthe additivenoise.

Quadrature (I/Q)Demodulator

Theamplitude andphase noiserelationcan be shownonastandard quadrature (I/Q)

demodulator[22,24,28]. This type of demodulator,shown in Figure 1.6,isalsoapartofmany

lock-inampliﬁers, such as theZurich InstrumentsMFLI. It takesasignal

xtAcosω0t

andmultipliesitwithits quadrature components

cosω0t

and

sinω0t

.Multiplying

thosevaluesby2will yieldthe quadrature components

sIt

and

sQt

in theamplitude

domain. Multiplying them by

2

will yieldtheir valuesinthe RMSdomain. We chosethe

ﬁrst becauseitismoreconvenient to work in theamplitude domain,eventhoughsome

commercialdevices likethe Zurich InstrumentsMFLIuse the 2normalization.

Chapter1 TheoreticalBackground andMethods



xref,It2cosω0t



xref,Qt2sinω0t

Low-pass

ﬁlter

Low-pass

ﬁlter

xt

x′

Figure1.6:

Quadrature demodulatorthatcreates an analytical signal from asinusoidat

frequency ω0

I/QDemodulatoion of aSinusoidalSignal

Firstly, anoiseless sinusoidal signal is appliedatthe input of theI/Q demodulator

xtx

0cosω0tϕ0.(1.48)

Theﬁrststepincalculating thein-phase componentistomultiply it by xref,I2cosω0t:

x′

Itx

0cosω0tϕ02cosωt(1.49)

x0cosϕ0cos2ω0tϕ0.(1.50)

Applyingalow-pass ﬁlterto

′

t

eliminatesthe doubled frequencycomponent (ifthe

low-pass cut-oﬀ frequencyislower than 2ω0), yielding thein-phase component:

xItx

0cosϕ0.(1.51)

Analogously, thequadraturecomponent is derivedas:

xQtx

0sinϕ0.(1.52)

Thein-phase andquadraturecomponentscreateacomplexanalyticalsignal:

xxIixQ.(1.53)

Figure1.7:

Eﬀect of quadrature demodulation on noiseinthe in-phase (I)and quadrature

(Q)components. Theinput noisespectrum

Sxf

is convolvedwiththe reference

spectrum, doubling thenoise PSDinbothcomponents:

Sx,IfS

x,Q

f

f

.The bandwidth

represents thelow-passﬁlter bandwidthinthe

demodulator.

If thephase of theinput signal

xt

nπ

,the wholesignalwill be containedinthe in-phase

component

xxI

.Onthe otherhand, if thephase of theinput signal is

nπ

2, thewhole

signal will be in thequadraturecomponent.For otherphases, thesignalwill be split

between thetwo.The complexsignal xcanberepresented in polarformas:

xtrte

iϕ0(t)(1.54)

with

rtx

tx

t,(1.55)

ϕ0tarctan xQt

xIt(1.56)

representing theamplitude andphase of theoriginalinput signal xt.

Chapter1 TheoreticalBackground andMethods

I/QDemodulationofaNoiseSignal

Secondly,awhiteGaussiannoise signal is appliedtothe input of thequadraturedemodulator,

representedby:

xtnt,(1.57)

where

nt

is thenoise signal.Whenanoisesignalwithapowerspectraldensity (PSD)

SII

xN

2isappliedtothe input,spectralanalysisbecomesauseful tool forpredicting the

noiseinthe quadrature components.Asdescribed in Section 1.2.1,where thePSD of the

noise at theoutput of alineartime-invariant(LTI) system is calculated by multiplyingit

by thesquared magnitude of thesystem’s frequencyresponse, themaindiﬀerence here is

that thenoise is instead multipliedbythe referencesignal. Thein-phase referencesignalin

thefrequency domaincan be describedby[22,28]

Xref,Ifδff

δff

,(1.58)

whichyields thesquaremagnitude

Xref,If2δff

δff

.(1.59)

Analogously, thesameresult is obtainedfor thequadraturereference

Xref,Qf2X

ref,If2

Multiplicationofthe noise at theinputwiththe reference in thetime domain resultsinthe

convolutionofthe twosignals in thefrequency domain. This is described by therelation

SII

x,I/QfS

xfX

ref,I/Qf2SII

xfδff

δff



SII

xfS

xf2S

xf.(1.60)

Thenoise at theinput,withaconstant PSD, is convolvedwithtwo Diracdelta impulses,

each with amagnitude of 1. Consequently,the noisePSD at theoutput of themultiplier

is twicethatofthe input noisePSD,asillustratedinFigure 1.7.The low-pass ﬁlter

following themultiplierisequivalent, forthe noisesignal, to aband-pass ﬁlter with the

same bandwidth

placed before themultiplier. Althoughthe low-pass ﬁltered signal

exhibitstwice thePSD of theinput,ithas thesametotal noisepower (variance) as the

band-pass ﬁltered signal centeredaround

with bandwidth

.This equivalence arises

becausethe band-passsignalcontains twospectralcomponentsatthe positive andnegative

frequencies, each with bandwidth

,matching thebandwidthofthe low-pass ﬁltered

signal.

Sincethe noiseisconvolved with diﬀerent signals, thenoise processes in thetwo arms of

thedemodulatorcan be treatedastwo uncorrelatedprocesses:

xItn

t,

tn

t,

tn

t,(1.61)

whereboth

nIt

and

nQt

have thesamestatistical properties likePDF (inthis case

Gaussian), PSDand variance described by therelations

fnInIf

n

 1

2πσ2exp n2

2σ2,(1.62)

SII

x,IfS

x,Qf2S

xf,(1.63)

σ2

Iσ2

Qσ2.(1.64)

Thequadraturepair nIand nQtogether form theanalyticalnoise signal

ntn

tinQt,(1.65)

whichisacomplexrandomvariable.The PDFofacomplexrandomvariablecan be deﬁned

asjoint PDFofits real andcomplex part [28]. Thejoint PDFof

and

canbewritten

as theproductofthe PDFs from equation (1.62)

fnI,nQnI,n

 1

2πσ2exp n2

In2

2σ2.(1.66)

Figure 1.8 showsathree-dimensionalplotofthe jointprobabilitydensity function (PDF),

describedbyatwo-dimensionalGaussian distribution. Sincethe quadrature components

have equalvariances, thecontoursofthe jointdistributionare concentriccircles.The joint

distributioncan be projectedontothe twocoordinate planes, yielding themarginalPDFs

(

fnInI,f

n



of thequadraturecomponents, as also showninFigure 1.8.Itisoften

usefultorepresent thethree-dimensionalplotintwo dimensions by taking thecontour at

radius

,whichcorresponds to thestandard deviationofthe quadrature noiseprocesses.

This is depictedinFigure 1.9a.

Chapter1 TheoreticalBackground andMethods

PDF

Figure1.8:

Thejoint probabilitydensity function (PDF)ofthe quadrature components

is representedbyatwo-dimensionalGaussian distribution. By projecting the

jointPDF onto the twocoordinateplanes, themarginaldistributions of the

quadrature components areobtained, each following aone-dimensionalGaussian

distribution.

It is oftenmorepractical to representsignals in termsofamplitude andphase (polar coor-

dinates) rather than in termsofthe real andimaginary components (Cartesian coordinates).

To achieve this,acoordinate transformationisrequired.For thepolar representation of

theamplitude andphase arecalculatedasshown in equations(1.67)and (1.68):

rtn

tn

t,(1.67)

ϕtarctan nQt

nIt.(1.68)

Thejoint PDFfromequation

(1.66)

canbetransformed from Cartesian to polarcoordi-

natesusing theJacobian[29]

fr,ϕ r, ϕf

,nQnI,n

J,(1.69)

wherethe Jacobiandeterminant Jris givenbythe transformation

nIrcosϕ,(1.70a)

nQrsinϕ.(1.70b)

This resultsinthe jointPDF in polarcoordinates

fr,ϕ r, ϕ r

2πσ2exp r2

2σ2.(1.71)

It is evident that thejoint PDFinpolar coordinateshas no dependenceon

,indicating

that

and

arestatistically independent.Therefore,the jointPDF canbefactorized as a

productofthe amplitude andphase PDFs

fr,ϕ r, ϕf

rf

ϕ.(1.72)

Foracircularly symmetric Gaussian distribution, thephase

is uniformly distributed

between πand π,asexpressed by

fϕϕ









2π

,πϕπ,

0,otherwise.

(1.73)

This leadstothe amplitude PDFfollowing a Rayleighdistribution [24,28]:

frrr

2exp r2

2σ2,r0.(1.74)

with themean

µrσπ2

andvariance

σ2

r(4−π)

2σ2

.Fromthe generalconservation of

powerlaw fornoise signals,if

and

arezeromeanprocesses, thefollowing relation can

be written

µ2

rσ2

rσ2

Iσ2

Q,(1.75)

wherethe variancesofthe I/Qarms contributetothe sumofthe mean squaredand variance

of theresulting amplitude noise. Theamplitude variance

σ2

is smallerthanthe sumofthe

initialinput variances,because some of thenoise powerisconvertedtothe mean value

µr

I/QDemodulationofaSinusoidalSignalWithAddedNoise

Nowthatthe demodulatoroutput is treated forsignals

xt

andnoise processes

nt

separately,their sumcan be considered as asuperposition.Assuming theinput signal is

givenas

xtx

0cosω0tϕ0nt,(1.76)

Chapter1 TheoreticalBackground andMethods

(a) (b)

Figure1.9:

Cartesian representation of thesignals in theI/Q arms.(a) When only noise

is presentatthe input of theI/Q demodulator(

xtnt

), thedistribution

is circularlysymmetric around theorigin, resulting in an evenly distributed

phasenoise as described in relation

(1.73)

.(b) Whenacombination of a

sinusoidalsignaland noiseispresentatthe input of theI/Q demodulator

(

xtx

cosωt ϕ0nt

), thesinusoidalsignalintroduces ameanoﬀset

µx0

in thejoint Gaussian distributiondue to thenoise. As theamplitude

of thesinusoidalsignalincreases, thedistributionshifts further away from the

origin, therebyreducing thephase noise ϕwhen µσ.

thequadraturecomponentsatthe output of thedemodulatorare givenby
x
Ix
0
cosϕ0n
I
,
x
Qx
0sinϕ0n
Q
.(1.77)
Theamplitude andphase can be calculatedasinequations (1.67)and (1.68), resulting in
themeanamplitude
µ2
rµ2
Iµ2
Qx2
0.(1.78)
Thequadraturepairforms an analytical signal
xtx
I
tixQt,(1.79)
with ajoint Gaussian distribution
fxI,xQxI,x
Q
 1
2πσ2exp xIx0cosϕ02x
Qx
0sinϕ02
2σ2,(1.80)
whichsharesthe same properties as thejoint PDFinequation
(1.66)
,but is shiftedby
x
0
cosϕ0
in thein-phase directionand
x0sinϕ0
in thequadraturedirection, as shownin
Figure 1.9b.
By applying aproceduresimilartothe onedescribed in theprevioussection forderiving
theRayleighdistribution, it canbeshown that theamplitude probabilitydensity function
forasinusoidalsignalwithaddednoise follows a Rice or Rician distribution[24,28,30]:
frrr
σ
2exp r2µ2
r
2σ2I0µrr
σ2,(1.81)
where
σ2σ2
Iσ2
Q
is thevarianceofthe quadrature components,and
I0
is themodiﬁed
Bessel function of theﬁrstkind of zeroorder.Equation(1.81)can be normalized as
fvvvexp v2a2
2I0av(1.82)
with
fvvσfrr,a
µ
r
σ
,v
r
σ
.(1.83)
Figure 1.10a showsthe Rice distributionfor diﬀerentvaluesof
a
.Fromthe curves, two
observations canbemade:
27

Chapter1 TheoreticalBackground andMethods

(a)

0.0 2.5 5.0 7.510.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

fv(v)

a=0

a=1

a=2

a=3

a=4

a=5

(b)

−2 0 2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

fϕ(ϕ)

µ=0

µ=1

µ=2

µ=3

Figure1.10:

TheRicedistribution.(a) Rice amplitude distributionfor diﬀerentvaluesof

(b)Ricephase distributionfor diﬀerentamplitudesofsignal µ

•

If only noiseispresent at theinput (

a

0),the Rice distributioncollapses to the

Rayleigh distribution.

•

a

1, thedistribution becomesGaussian in thevicinityof

va

.Thisoccurs when

thesinusoidamplitude is much larger than thenoise deviation

.Inthis case the

amplitude noiseisapproximately Gaussian with adeviation of σAσ.

Thephase probabilitydensity function (PDF)isdescribed by theRicephase distribution[30]

fϕϕ1

2πexp µ2

2σ21µ

σπ

2erfcx  µ

σ2cosϕϕ0,πϕπ, (1.84)

where

ϕ0arctanµQµI

is themeanphase determined by thephase of thesinusoid, and

erfcxisthe scaled complementary errorfunction.

Figure 1.10b showsthe phasedistributionfor diﬀerentsignalamplitudes. Thefollowing

observations canbemade

•

Whenthe mean amplitudeiszero(

µ

0),the phase distributionreduces to aconstant

12πbetween πand π.

•

As theamplitude of thesignalincreases, the

erfcx

function part of theequation

becomeslarger, leading to thephase anglebeing more concentratedaround themean

phase angle

ϕ0

.This concentration eﬀect causes thedistributiontobecome more

peakedand symmetric, resembling aGaussian distributioncenteredaround

ϕ0

for

µσ.

•

Withincreasing amplitude expression

(1.84)

asymptotes theGaussian distribution

yielding thelimiting case [29]

lim

µ→∞ fϕϕ 1

2πσ

exp 



ϕϕ02

2σ

µ2



.(1.85)

whichresultsinaphase variance σϕσµ.

In summary,fromequation

(1.85)

theamplitude-to-phase noiseconversionofasinusoid

signal with additive whiteGaussian noisecan be simpliﬁedfor alarge amplitude compared

to noise(x

0σ

)as

ϕσ

,(1.86)

where

σx

is thenoise deviationofthe noisecomponent of thesignal

xt

,and

is the

amplitude of thesinusoidalpartofthe signal.Ifoperating within thebandwidth

thedemodulator, an equivalentexpressioncan be derivedinthe frequencydomainfor the

phase noisePSD

Sϕω2S

ωω



,for πBW ωπBW. (1.87)

Thereisadirect relationship between phase noiseand frequencynoise,asdescribed by

equations

(1.45)

and

(1.46)

.Speciﬁcally,frequency noisecan be derivedfromphase noiseby

taking thederivativeofthe phase ﬂuctuations with respect to time [31]. This relationship

is expressedas

t∂ϕnt

∂t ,(1.88)

where

ωnt

represents thefrequency ﬂuctuations and

ϕnt

denotesthe phase ﬂuctuations.

Consequently,the powerspectraldensity (PSD)ofthe frequencynoise increases with

frequency, andisgiven by

Sωωω

ω,(1.89)

where Sωωis thefrequency noisePSD and Sϕωis thephase noisePSD.

Chapter1 TheoreticalBackground andMethods

Another wayofacquiring thefrequency in mechanical oscillatorismapping it to the

phase, resulting in phase noisebeing directly proportional to thephase noise[21,31,32]

Sωω1

ω.(1.90)

Both phase to frequencytechniqueswill be furtherdiscussed in Chapter 2.

NoiseRelations in aFrequencyCounter

Afrequency counterisaninstrument that measures thetime period of afullsignalcycle.

Letusassume that anoiseless sinusoidalsignalisapplied to theinput of thefrequency

counter:

xtx

0sinω0t,(1.91)

where

is theamplitude and

ω0

theangularfrequency.The frequencycounter typically

uses zerocrossingsonthe positive slopeofthe signal to measurethe time period,which

canbedeﬁnedbythe following conditions:

xt0,(1.92a)

˙xt0,(1.92b)

yielding samplesattimes:

tkk2π

ω0kT0,(1.93)

where

is thesignalperiodand

kN

is an integer. Thefrequency counterestimates the

period of onecycle as:

T0tktk−1,(1.94)

which, foranoiselesssine wave andaperfect time resolutionofthe counter, will result in

T0T0.The frequencyisthencalculatedasthe inverseofthe measured period.

Now, considerthe case wherethe frequencycounter measures asinusoidalsignalwith

addedwhiteGaussian noise:

xtx

0sinω0tnt,(1.95)

where

nt

is theadditivenoise.The detected zerocrossing will occurwhenthe measured

signal satisﬁes

xt

0. Duetothe presence of thenoise

nt

,the detected phase

ˆϕ

Figure1.11:

Schematicrepresentationofthe signal

xt

linearized around thevicinityofthe

zerocrossing.The sine signal is shownalong with theadditivenoise standard

deviationinthe verticaldirection. Increasing theamplitude

enhances the

slopeofthe signal, therebyreducingthe phase error, described by therelation

σϕσ

x0.

will deviatefromthe actual signal phase ϕω0t.This phase errorcan be expressedas:

ϕeˆϕϕ. (1.96)

Combining equations(1.95)and (1.96)for xt0and ˆϕ0yields:

x0sinϕetnt.(1.97)

Foramplitudesmuchlargerthanthe noise, we canapproximatesinϕeϕ

,resulting in:

ϕetnt

.(1.98)

Thevarianceofthe phase errorcan then be calculatedas:

σ2

eVarϕeVarn

0σ2

,(1.99)

Chapter1 TheoreticalBackground andMethods

where

σ2

is thevarianceofthe noise process

nt

.The phase deviationcan be expressed as:

σϕσ

,(1.100)

whichisthe same result obtainedin(1.86).

1.2.3Thermomechanical Noise

Oneofthefundamental noisesources in NEMS resonators is thermomechanical noise

[12,21,31]. This noisearises due to thecoupling (interaction)between theresonator anda

thermalreservoir ﬁlledwithrandomly distributed phonons.Itmanifests as a noisyforce

appliedtothe resonator, described by theﬂuctuation-dissipation theorem(FDT).The

NEMS resonatorcan be modeledasalumped-element system,asdiscussed in Section 1.1.1.

ThewhiteGaussian forcenoise with thepower spectraldensity (PSD)

SFωN

,acting

on theresonator, leadstoanamplitude noisePSD described by

SzωH

ω2SFω,(1.101)

where

HRω2

is thesquared magnitude of theresonator’stransfer function.For small

damping,this canbeapproximatedbyaLorentzian function

HRω2ω21

4m2

eﬀ ω2

0

ω0ω2ω

2



,(1.102)

where

ω

is theresonator’s susceptibility. Accordingtothe equipartition theorem[21],

thetotal energy of amechanical system with onedegreeoffreedom in thermalequilibrium is

kB

,where



is thetemperature and

is theBoltzmann constant.For alumped-element

modelwithelastic restorativeenergy, theconservationofenergyequationis:

2kB1

eﬀ z2

th1

eﬀ ω2

0z2

th,(1.103)

where

meﬀ

and

keﬀ

arethe eﬀectivemassand spring constant of theresonator,and

z2

th

themeansquaredisplacement due to thermalnoise.Solving this for z2

thgives

z2

th k



eﬀ ω2

.(1.104)

Sincethermomechanical noiseisazero-mean process,the totalnoise poweriscontained

within thevariance, implying that thevarianceofthe displacement ﬂuctuationis
σ
2
zz
2
th
.
Thevariancecan be determinedbyintegrating thePSD over allfrequencies. Usingthe
variance-PSDrelationfromTable 1.1 andtreating it in angularfrequency,weobtain
σ2
z1
2π∞
0
Szωdω1
2π∞
0
SFωω2dω0NFQ
4ω3m2
eﬀ
.(1.105)
Solving equations(1.104)and (1.105)yields thethermomechanicalforce noiselevel
NF4meﬀ ω0kB
Q.(1.106)
Using equation
(1.101)
,the peak of thethermomechanical displacementnoise at the
Lorentzian resonancecan be calculatedas:
Szω04k
B
Q
m
eﬀ ω3
0
.(1.107)
It is important to note that,althoughthe thermalnoise force
NF
is inverselyproportional
to Q,the peak displacementnoise scalesproportionally with Q[21].
1.2.4Detection (Transduction)Noise
Detection or transduction noise,inthiscontext,referstoany noiseintroduced duringthe
conversion of mechanical displacement(or velocity)intoanelectricalsignal[12,21,31]. This
noisetypically manifests as additiveamplitude noisethatissuperimposed on thesignal
afterthe resonator’smotionhas been convertedintoanelectricalform, as illustrated in
Figure 1.12a.Assuming thetransduction process does notlimit thesignalbandwidth,it
canbemodeled as asimple gainelement with aconversionfactordenoted by
G
.Sincethe
transduced signal is electrical, it canbeanalyzed within theframework of circuit theory.
Thenoise canbemodeled as aparasitic voltage source connected in series(Figure 1.12b)or
aparasitic currentsourceconnected in parallel(Figure 1.12c)tothe networkits coupling
to,e.g.equivalentresistance. In circuit theory,noise sources aretypically expressed in
termsoftheir root mean square (RMS)values, whichcorrespondtothe standard deviation
ofthe noisesignals (
VnσV
or
InσI
). Theprimary transduction noisesources relevant to
this work are
•
Johnson-Nyquistthermalnoise –Commonly encounteredwhere metaltraces
with ohmic resistance areused fortransduction.
33

Chapter1 TheoreticalBackground andMethods

(a)

G

ndt

xtututn

t

(b)

Rout

(c)

InR

out

Figure1.12:

Transduction mechanism. (a)Equivalentblock diagram.(b) Noisevoltage

source equivalentcircuit.(c) Noisecurrent source equivalentcircuit.

•

Hooge 1

f

noise –Observedinresistors, semiconductordevices, andampliﬁer

circuits.

•Shotnoise –Asigniﬁcantlimitingfactorinlaser-based transduction systems.

Johnson–Nyquist ThermalNoise

Themostbasic andintrinsic type of noiseinresistors is knownasJohnson-Nyquistnoise,

or thermalnoise [21,33,34]. This type of noiseischaracterized as whiteGaussian noise,

meaning it hasaconstant powerspectraldensity (PSD)acrossall frequencies(white) and

follows aGaussian probabilitydensity function (PDF). ThePSDsfor theequivalentnoise

voltageand noise current were ﬁrst describedbyJohnson [35]and aregiven by:

SV,thm 4kBRV2

Hz,(1.108)

SI,thm 4kB

RA2

Hz,(1.109)

wherethe root mean square (RMS)valuesofthe thermalnoise canbederived as shown

in [21]

Vthm f1

4kBRdf 4k

RBW, (1.110)

Ithm f1

4kB

Rdf 4kB

RBW, (1.111)

where

represents Boltzmann’s constant,



is thetemperature of theresistor,

is the

resistance,and BW denotesthe bandwidthofthe measurement.

Hooge 1fNoise

Hooge 1

f

noise, also knowninliteratureasexcess noise, ﬂicker noise, or pink noise,

exhibitsaPSDthatincreases as thefrequency decreases. Theoriginofthisnoise is notfully

understood,but it is knowntooccuronlywhencurrent ﬂows throughthe device [33,36].

Althoughcommonlyreferred to as 1

f

noise, itsPSD follows a1

f

dependence, where

α

3, with

α

1being themosttypical value in electronic devices [21]. Unlike

thermal noise, excessnoise does nothaveaGaussian PDF[33]. Hooge developeda

semi-empirical modelfor this noise, expressedas

V,1/fγ

bias

fV2

Hz,(1.112)

SI,1/fγ

bias

R2fA2

Hz,(1.113)

where

is an empirically determinedproportionalityconstantspeciﬁc to thedevice,

thenumberofchargecarriersinthe device, and

Vbias

is theappliedbiasvoltage across the

resistance

.According to [37], 1

f

noiseisinverselyrelated to thevolume of theresistor,

while [21]notes that it increases with temperature. Both factorsinﬂuencethe number of

charge carriers

.The RMSvalue of this noise, as discussedin[21], canbecalculatedas

Chapter1 TheoreticalBackground andMethods

V1/fVbiasγ

ln f1

f0,(1.114)

I1/fVbias

Rγ

ln f1

f0.(1.115)

ShotNoise

Shotnoise is afundamental limit of anyphysical quantity that hasquantized carrier [21].

Suchquantitiesfor example canbeelectrons in electronic,photons in opticalsystems or

phonons in acoustic systems. This eﬀect hasbeen studiedinmanypreviousworks [38

–

41].

Thespectrumofthe shot noise is whitewiththe equivalent noisevoltage andcurrent source

PSDs [21]

SV,shot 2eξIbiasR2V2

Hz,(1.116)

SI,shot 2eξIbias A2

Hz,(1.117)

where

is theelectroncharge,

bias

thecurrent that passes throughthe resistance

and

the Fano or correlation factor.Ifthe material is aperfect conductor

ξ

0, howeverifthe

material is afull partition (tunneljunction,fully depletedregion)

ξ

1. Usually 0

ξ

formostcases.

1.2.5Temperature FluctuationNoise

ANEMSresonator is amechanical system that is thermally coupledtoits environment.

Themicroscopical thermalequilibrium is in themicroscopic worldquantum andstatistical

in nature.The coupling betweenthe resonatorand theenvironment in vacuum is realized

viaradiation andconduction.The radiation includesemission andabsorptionofphotons

on theresonator surface, while theconduction includesthe exchange of phonons between

resonatorand frameatthe clamping.The noiseofthe heat transfer is manifested as shot

noise[42]. Theﬂuctuations in temperatureofthe resonatorcan be modeledasaresonator

response to thetemperature ﬂuctuationnoise

SΔTrsS

thHths24kB2RthHths2,(1.118)

where

Rth

is theequivalentthermal resistance of theresonator (combinedofradiation and

conduction)and

Hths 1

1sR

thCr

(1.119)

is aﬁrst-orderlow-passﬁlter thermalresponsecaused by theresonator’s thermalproperties.

1.3AllanVariance

Theanalysisofresonant frequencynoise in aresonatorcan be conductedbycalculating the

power spectraldensity (PSD)fromthe acquired time seriesdata. Althoughthe PSDprovides

comprehensive informationabout thespectraldistributionofnoise power, determiningthe

variance of thetime signalrequiresintegrating over theentirespectrum. Additionally,PSD

spectra canbechallenging to interpretdue to theinherent roughness that arises during its

calculation.Toaddress these issues,the useofthe Al lan variance (AV) is beneﬁcial.

Allanvarianceisatime-domaintoolused foranalyzing frequencyﬂuctuations.Similar

to PSD, andunlikeclassicalvariance, AV candistinguishbetween diﬀerenttypes of noise

basedontheir spectralbehavior. While classicalvariancemeasuresthe extent of frequency

ﬂuctuations within themeasurement time frame, Allanvariancecomparesdiﬀerentintervals

within themeasurement to each other. By varying thelengthofthese intervalsand

plotting thecorresponding variances,one canextract informationabout thenoise’sspectral

characteristics. TheAllanvarianceiscomputed as follows [12,21,32]

σ2

yτ 1

2N1

N−1



i=1y

i+1,τ yi,τ 2,(1.120)

where

is the

ith

sample of theaveragedfrequency over theaveraging interval time

deﬁned as

yi,τ 1

τiτ

(i−1)τ

ytdt. (1.121)

Whencomputing Allan variance,frequency values aretypically normalized,resultinginthe

fractional frequency

ytωtω

.(1.122)

TheclassicalAllanvariancecan be extendedtothe overlapping Allanvariance. In this

approach, data segmentsoverlap insteadofbeing treatedcompletely separately,resulting

Chapter1 TheoreticalBackground andMethods

in alargernumberofpossible combinationsfor computing thevariance. This improvesthe

resolutionand precisionofthe resultingplot, albeit at thecostofincreased computational

resources [43,44]. Theoverlapping Allanvarianceisgiven by

σ2

y1

2m2

sN2ms1

N−2ms+1



i=1i+ms−1



j=iyj+msyj2

,(1.123)

where

is thenumberofoverlappedsamplesout of thetotal original non-overlapped

sample.Inthe literature,the Allan deviation(AD)iscommonlyusedfor data presentation

sinceitismoreintuitive to interpretthe data in termsofrootmeansquare(RMS) values

rather than power. TheAllandeviation is simply thesquarerootofthe Allanvariance,

σyτσ

τ.

Allanvarianceand powerspectraldensity oﬀer twodiﬀerentapproachestointerpreting a

statisticaldataset in thespectraldomain: AV by creating aspectrumbased on diﬀerent

integrationtimes

,and PSDbycreatingaspectruminthe frequencydomain. If thePSD

is known, Allanvariancecan be directly calculated from it.The relationship between Allan

variance andPSD is givenby[12]

σ2

yτ1

2π

2∞

0sin ωτ

24

ω2Syωdω. (1.124)

Thesimilarity between PSDand AD is displayedinFigure 1.13.Itcan be observedthat

both toolsare capableofcapturing diﬀerentnoise typeslikewhitefrequency noise, white

phase noiseand drift.

(a)

10−1101103

f(Hz)

10−10

10−8

10−6

10−4

10−2

100

102

Drift

(∝1/f2)

White

frequency

(∝const.)

White phase

(∝f2)

(b)

10−410−310−210−1100101

τ(s)

10−3

10−2

10−1

100

101

σy

Whitep

hase

(∝1/τ)

White

frequency

(∝1/√τ)

Drift

(∝τ)

Figure1.13:

Comparisonbetween PSDand AD foranoisesignalwithdiﬀerentnoise

components (python simulation)(a) PSDofthe mixed noisesignal, with Drift

having a1

f

slope, whitefrequency noisebeing constant andwhitephase

noise having a

slope. (b)Allandeviation of themixed noisesignalwith

whitephase noisehaving a1

τ

slope, whitefrequency noisehaving a1

τ

slopeand drifthaving a τslope.

Chapter1 TheoreticalBackground andMethods

1.4Noise in Closed-Loop Systems

Resonators areoften operated in closed-loopsystems [12,21,31], whichintroduces the

potentialfor noisetobecomepartofthe closed-loopdynamics. Thenoise sources in such

systemstypically fall into twocategories:

•

Noiseatthe input of theresonator—typically thermomechanical noise, as shownin

Figure 1.14a.

•

Noiseatthe output of theresonator—commonly detectionnoise, as showninFig-

ure 1.14b.

Whenthe resonatorisdrivenatits resonant frequencyinanopen-loop conﬁguration, it

behaveslikeaﬁrst-order low-pass ﬁlter with atimeconstant

τr

,whichﬁltersphase and

frequencyﬂuctuations around resonance[12,26]. Thetransferfunction forthis behavior is

givenby:

Hrs 1

sτr1.(1.125)

In thecaseofthermomechanical phase noise(S

thm s), theopen-loop response is

SϕsH

s2Sϕthm s.(1.126)

Withthe introduction of feedback,withthe transfer function

Hfs

,the system’s response

(a)

Sϕthm s

Hrs

Hfs

Sϕs

(b)

Sϕds



Hrs

Hfs

Sϕs

Figure1.14:

Diﬀerentclosed-loop casesofnoise coupling.(a) Noisecoupling at theinput

of theresonator, typically thermomechanical noise. (b)Noise couplingatthe

output of theresonator, usuallydetection noise.

(a)

100102

10−3

10−1

101

103

Sθ

1/τr

Open-loop

Closed-loop

(b)

100102

10−1

100

101

102

103

104

Sθ

1/τr

Open-loop

Closed-loop

Figure1.15:

Powerspectraldensities (PSDs) of diﬀerentnoise sources foropenand closed-

loop conﬁgurations with

Hfs

1. (a)Thermomechanicalnoise. (b)Detection

noise.

becomesaclosed-loopresponse

Sϕs H

s

1H

sH

s2

thm s.(1.127)

Similarly, fordetectionnoise (Sϕds), theopen-loop response is simply

SϕsS

s.(1.128)

In aclosed-loop system, this response becomes

Sϕs 1

1H

sH

s2

s.(1.129)

Figure 1.15 illustrates thePSDsofvarious noiseprocesses in both open andclosed-loop

conﬁgurations,assuming

Hfs

1. Theresonatoractsasalow-pass ﬁlter,attenuating

high-frequencynoise introduced by thefeedback.However,low-frequency noise(frequencies

slower than

τr

)originating from thefeedback loop is notﬁltered by theresonator,leading

to noiseintegration over theloop—aphenomenonknown as theLeesoneﬀect [26].

Chapter1 TheoreticalBackground andMethods

As showninFigure 1.15a,awhiteinput phase noisecausesthe thermomechanical noise

in theclosed-looptoexhibit a1

ω

characteristic,which,whendiﬀerentiated into frequency,

resultsinawhitenoise process with aPSD givenby:

Sfω1

thm ω.(1.130)

In contrast,phase detectionnoise in closed-loopcomprises acombinationofwhite noise

anda1

ω

component. When convertedintofrequency,the detectionnoise PSDisgiven by

Sfωω

21

rS

ω.(1.131)

It canbeobservedthatfrequency ﬂuctuations due to detectionnoise increase with frequency.

1.5Transduction andActuation

Transduction refers to theprocess of converting mechanical displacementintoanelectrical

signal,where actuationrefers to theprocess of converting an electricalsignalintomotion.

Varioustransduction andactuation methodsexist,whichcan be broadly categorized as

follows[21]:

•Electrodynamic transduction andactuation

•Electrostatic transduction andactuation

•Thermoelastic actuation

•Piezoresistivetransduction

•Piezoelectric transduction andactuation

•Optical transduction andactuation

Among these, opticaltransductionprovidesthe highest sensitivity;however,itoften

comeswithhighercosts andbulkier designs. Conversely,electrodynamic transduction,such

as magneto-motive transduction,oﬀersamore practicaldesignthat, while less sensitive, is

easier to implement across variousapplications.

Figure1.16:

Schematicrepresentationofaconducting string oscillating in amagnetic ﬁeld,

illustrating theconversionofmechanical motion into an electricalvoltage.

1.5.1Magneto-motive Transduction andActuation

Themagneto-motive transduction scheme utilizes amagnetic ﬁeld to convertmechanical

motion into an electricalsignal. This process is based on Faraday’slaw of induction [45,46],

whichstatesthatthe electromotive force(EMF) or induced voltage in aclosed loop is

proportional to thenegativerateofchangeofmagnetic ﬂux throughthe loop,mathematically

expressed as: tdΦt

dt,(1.132)

where trepresentsthe EMF, andΦtdenotesthe magnetic ﬂux,given by:

ΦBA, (1.133)

with

being themagnetic ﬂux densityand

theareaperpendiculartothe magnetic ﬁeld.

Figure 1.16 depicts aconducting string resonatorsituatedwithin astaticmagnetic ﬁeld.

Assuming theresonator’s mode shapeisdescribed by

zx, tz

0sin nπ

Lxcos 2πft,(1.134)

where

is theoscillation amplitude,

themodenumber, and

thestring length,the

Chapter1 TheoreticalBackground andMethods

inducedEMF canbederived as

d

dtL

BdA

d

dtL

Bz

0sin nπ

Lxcos 2πftdx

2Bz0Lf

n1cosnπsin2πft.(1.135)

This result indicates that themagneto-motive transduction is eﬀectiveonly forodd modes,

as theEMF cancelsout forevenmodes. Forodd modes, theresulting EMFis

4Bz0Lf

nsin2πft.(1.136)

It is important to note that magneto-motive transduction introduces aphase shiftof

π

2in

thedetected signal.

Similarly, magneto-motive actuationfollows thesameprinciple,where thedisplacement

is driven by theLorentz forcegenerated by thevoltage on thewire, expressed as [46]

z0n

4BLf .(1.137)

1.5.2Optical Transduction With Laser-Doppler Vibrometer

TheLaser-Doppler Vibrometer (LDV) is an opticaltransduction technique used to convert

mechanical motion into an electricalsignal[21,44,46]. This method leveragesthe Doppler

frequencyshift observed in light reﬂected from amovingresonator. In this work,weutilize

theMSA-500 from PolytecGmbH, whichoperateswithalaser wavelength of

λ

633

Thefrequency shiftofthe reﬂected light due to theDopplereﬀect canbeexpressed as

ΔfDoppler 2∂U

λ∂t ,(1.138)

where

∂U∂t

represents thevelocityofthe resonator. Unlikeinterferometricdetection

methods, theLDV measures velocity rather than displacement. Theresolution of the

vibrometer depends on variousfactors,including thedecoderused,the objectivelens, the

surfacecharacteristics of theresonator,and itsreﬂectiveproperties. Forsilicon nitride

structures,sub-picometer resolutioncan be achievedwiththissetup.

Chapter2
Schemes forTrackingResonance
Frequency
This sectionprovidesaconcisesummary of thework Schemes for Tracking Resonance
Frequency forMicro- and Nanomechanical Resonators [31].
Micro- andnanomechanical resonators arehighly sensitive to changesinenvironmental
conditions such as mass [1
–
5], damping [6,7], andstiﬀness [8
–
10], typically resultinginshifts
in theirresonance frequency. To accurately track these shifts,various frequency-tracking
schemesare employed, including feedback-free(FF)oropen-loop,self-sustaining oscillator
(SSO), andphase-lockedlooposcillator(PLLO)schemes.
In theFFscheme, theresonator is driven by asinesignalmatchingits resonant frequency.
In theclassicalconﬁguration, thephase diﬀerence between thedriving signal andthe
resonator’sresponse is monitored to mapthe phase diﬀerence to thefrequency shiftof
interest.This approachislimited by theresonator time constant [11,12,21,31], whichcan
be very slow forhigh Qresonators.Wepropose anew conﬁguration, showninFigure 2.1,
where, in additiontothe traditionalphase-to-frequency mapping (theslowresponse),the
derivative of thephase (thefastresponse) [47,48]isalsoadded. This resultsinacombined
response whosespeed depends on thephase detector rather than theresonator. The
thermomechanical anddetectionnoise transferfunctionsfor this conﬁgurationare givenby
H
FF
ϕth s1
τ
r
H
L
s,
H
FF
ϕds1
τ
r
1
H
r
sH
L
s,
(2.1)
where
HLs
is thelow-passﬁlter transfer function of thephase detector,and
Hrs
is a
45

Chapter2 Schemesfor Tracking Resonance Frequency

Resonator

noise

Detection

noise

Resonator

Phasetofrequencyconverter

Figure 2.1:

Theblock diagram illustratesthe FF frequency-trackingscheme, whichincludes

asignal generator(SG)thatexcites amicro-ornanomechanicalresonator.The

resonator’soutput is routed to aphase-diﬀerencedetector (PD),where the

phase diﬀerenceisconvertedintoafrequencysignalvia aphase-to-frequency

conversion process. Thermomechanical noiseisintroduced at theresonator’s

input,while detectionnoise is addedatthe phase detector’s input.. (Figure

takenfrom[31])

single-polelow-passﬁlter withthe time constant of theresonator, deﬁnedas:

Hrs 1

1sτr

.(2.2)

TheSSO scheme,shown in Figure 2.2,introduces afeedback loop to sustainoscillations

within theresonator. Accordingtothe Barkhausen criterion, theloopgainand phase shift

areadjustedtoensurestable oscillations. Thespeed of theSSO scheme is limitedbythe

frequencydetector used in conjunction with theSSO.The noisetransfer functionsfor the

SSO aregiven by:

HSSO

ϕth s1

s,

SSO

ϕds1

sH

s,

(2.3)

Saturating

amplifier Phase shifter

SSO

Freq. detector

Resonator

noise

Detection

noise

Figure2.2:

Blockdiagram of thestandard SSO scheme,incorporating aphase shifterand a

saturating ampliﬁer to satisfy theBarkhausen criterion. (Figuretaken from [31])

where

Hrs

is asingle-polelow-passﬁlter with thetime constant of theresonator.

HLs

has

low-pass characteristicsand represents thebandwidth-limiting (noise ﬁltering)mechanisms

in thefrequency detectiondevice.

PLL

LO PD

Resonator

noise

Detection

noise

Resonator

Figure2.3:

Blockdiagram of thePLLO scheme,featuring aproportional-integral(PI)

controllerthatregulatesthe localoscillator(LO). (Figuretaken from [31])

ThePLLO scheme,illustratedinFigure 2.3,isessentially aclosed-loopversion of theFF

Chapter2 Schemesfor Tracking Resonance Frequency

scheme,where thedriving signaladaptstothe resonant frequencychanges of theresonator.

This is achievedbyforming acontrolloop, connecting theoutput of thephase detector

throughaproportional-integral(PI) controllertothe localoscillatorthatgenerates the

driving signal.The noise transferfunctions forthe PLLO canbederived based on theloop

dynamicsasdescribed in [12]:

HPLL

ϕth s1

rskpkiHLs

s2s

τrskpkiHLs,

HPLL

ϕds1

sskpkiHLs

s2s

τrskpkiHLs.

(2.4)

ThePLLO hastwo bandwidth-limiting components:the phase detector andthe PI controller.

Theresulting PSDatthe output of each scheme canbecalculatedfromthe thermome-

chanical anddetectionnoise as [12]

SyωS

Δω

ω

S

th ωHϕth jω2SϕdωHϕdjω2

ω2

Sϕth ω

ω2

0Hϕth jω22

H

jω2.

(2.5)

Experimental comparisons of these threeschemes were conductedusing aNEMSresonator.

Theresults(Figure 2.4)showedthatthe FF scheme,despiteits simplicity, canachieve

aperformancelevel comparable to theclosed-loopSSO andPLLO schemeswhenusing

anoptimized design.The data indicatethatthe trade-oﬀs between speed andaccuracy

aresimilaracrossall threeschemes. Theﬁndingssuggest that thelimitations areintrinsic

to theresonator andnot theschemeusedfor itsfrequency tracking.The choice between

thesefrequency-trackingschemes should be basedonpractical considerationssuchasease

of implementation, cost,and thespeciﬁc applicationneeds.

This study highlightsthatwhile each scheme hasuniqueadvantages, thefundamental

diﬀerences in performanceare notsigniﬁcant. Instead, thepractical aspectsofeachscheme

should guide theselectionprocess forfrequency tracking in micro-and nanomechanical

resonators.

Integrationtime,τ(s)

Measurements

Slow

Fast

Sum

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−6 10−5 10−4 10−3 10−2 10−1 100101

Theory

Slow

Fast

Sum

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−6 10−5 10−4 10−3 10−2 10−1 100101

PLLO

SSO

Allandeviation, σy

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−6 10−5 10−4 10−3 10−2 10−1 100101

PLLO

SSO

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−6 10−5 10−4 10−3 10−2 10−1 100101

PLLO

SSO

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−6 10−5 10−4 10−3 10−2 10−1 100101

PLLO

SSO

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−6 10−5 10−4 10−3 10−2 10−1 100101

Figure2.4:

Comparisonofexperimental Allandeviationswiththeory. (a)Allandeviations

forthree FF tracking modeswithsecond orderdemodulationﬁltersofbandwidth

fL

100

Hz,

kHz

(b)ComparisonofPLLO andSSO tracking schemesfor

fPLL 

Hz,

500

Hz,

kHz

,where aPLLFDisused as thefrequency

detectorfor theSSO scheme.c)ComparisonofPLLO andSSO with

fPLL 

500

andFFscheme(combined/sum mode)with

fL

kHz

d)-f)Theoretical

modelbased computations using equations(1.124)and (2.5)for thesame

settingsasinthe measurements a)-c), andwithcalibration fordetectionnoise

level. (Figuretaken from[31])

Chapter3

AdaptableFrequencyCounter with

PhaseFiltering

This sectionprovidesaconcisesummary of thework Adaptable Frequency Counter With

Phase Filtering for Resonance Frequency MonitoringinNanomechanical Sensing [49].

Frequencycountersare essentialtools forassessing thefrequency stabilityofoscillators

andclocks, particularlywhencalculating theAllanDeviation (AD) [50,51]. Traditional

countersaverage instantaneous frequency over agatetime to computeAD. This work

proposes an enhanced frequencycounter with highresolutionand accuracy [52], used as a

frequencyshift monitorfor signals generatedbySSO-NEMSdevices.The goal is to quickly

andpreciselydetect smallfrequency shifts due to interactions with stimuli likemassor

temperature with acomparableorbetterspeed versus accuracy performancethanother

frequencydetectorslikethe PLL.

We consideranadvanced counter, the interpolating reciprocalcounter with continuous

time stamping [52,53]. To analyzeits speedand precision as afrequency shiftmonitor,

we developamodeltocapture itscharacteristics. Let

fst

representthe instantaneous

frequencyofthe signal source, including noiseand shifts of interest.The normalized phase

is deﬁnedasφtf

tdt.

Time stamps

,where

φtnn

(aninteger

), aregenerated at everypositivesignal

edge using areference high-frequency clockand an interpolator [53]. Thecounter estimates

frequencyfromthese time stamps forasingle signal period using

fctnφt

φt

n−1



nt

n−11

nt

n−1

.(3.1)

This equation givesthe averagefrequencyoverone signal cycle, with thecounter’s output

Chapter3 AdaptableFrequency CounterwithPhase Filtering

rate limited by thesignalfrequency.

Foragatetime of kcycles, thefrequency estimate is

f(k)

ctn k

nt

n−k

.(3.2)

This allows atrade-oﬀbetween speed andprecision,where longer gate timesoﬀerhigher

precisionbyaveraging outrapid ﬂuctuations.

We further examine theeﬀect of frequencyﬂuctuations on thecounter’s estimates. Given

aconstantnominalsignalfrequency f0,the phase with noisecan be expressedas

φtf

tϑt,(3.3)

where ϑtis thephase (time)noise. Thecorresponding fractional frequencyisgiven by

yst˙

φt

01˙

ϑt(3.4)

Thereciprocal counter’sfractionalfrequency estimate forasingle signal cycle, with the

nominalperiod T0,is

t

 1

1ϑ(t

)−ϑ(tn−1)

T01

1z.(3.5)

Here,

yctn

involves anonlinear transformation,asopposed to theideal linear transforma-

tion 1 zas in (3.4). Thenonlinear expressionspower seriesexpansion

1z1zz2... (3.6)

indicates that forsmall

theexpression behaveslinearly. High-frequencynoise components

z˙

ϑt

canintroduceintermodulationnoise throughthe nonlinear

term.Tominimize

this eﬀect,alow-pass digital ﬁlter canbeappliedtothe time stamps tnbefore converting

them to frequencyestimates, allowing controloverthe speed-precision trade-oﬀwhenused

as afrequency shiftmonitor.

In areciprocal counter, theoutput sampling rate is directly tied to theinput signal

frequency,given by

frate fs

,where

is thenumberofcycleswithin onegatetime.

This dependencycan complicatesubsequent digitalsignalprocessing,asvariationsinthe

inputfrequency aﬀect thesampling rate,altering ﬁlterdynamics. To address this,the

input-dependentsampling rate should be convertedtoaﬁxedrate.

(a)

(b)

N=2, D=2

Magnitude(dB)

−120

−100

−80

−60

−40

−20

Normalized frequency (f/frate)

00.1 0.20.3 0.40.5

Figure3.1:

(a)Block diagram of thesecond-orderCIC decimatorwithdecimation factor

andcombdelay

.(b) Transfer function of thedecimatedoutput.(Figure

takenfrom[49])

Onemethodinvolvescontinuousevent-triggeredtimestampcounting[53], wheretrigger

eventsare generatedatregularintervals

Tint

,setting thesampling rate.However,samples

aretaken at thenextrising edge of theinput signal,introducing uncertainty up to one

signalperiod. Theimpact of this uncertainty diminisheswithahighernumberofsignal

periodsper interval but canaﬀectaccuracy with fewerperiods.

While thesampling rate

dependson

,the sampling instantsalignwiththe frequency

counter’sinternalclock

fCLK

.The rate canbeupsampledto

CLK

using azero-order

hold, but this introduces high-frequencyharmonics, whichcan be mitigated by applying a

low-pass ﬁlter.The combinedlow-passﬁlteringand decimation,downtoafraction of

fCLK

canbeeﬃciently performedusing acascadedintegrator-comb (CIC) ﬁlter [54].

Figure 3.1a showsthe second-orderCIC ﬁlter used,withtwo integrator sections and

twocombsections (delay

N

2).The downsampler, with

R

,isplaced between the

integrator andcombsections,giving aﬁnaloutput sampling rate

fnew fCLK

,independent

.Figure 3.1b showsthe CIC ﬁlter’s transferfunction afterdecimation, whichlacks a

sharpseparationbetween pass-band andstop-band, requiring further ﬁltering with an FIR

or IIRﬁlter.

Theresponse speed to frequency jumps is limited by theinput signal frequency, as

described in equation

(3.1)

.Resampling with theCIC ﬁlter adds additionalﬁltering,

potentially slowing theresponse if

fnew

is lowerthanthe originalsampling rate

frate

However, this methodcan handle frequencysteps of anymagnitude,making it ideal for

monitoringdevices across awide frequencyrange.

Experimental results, illustratedinFigure 3.2a,wecompare twomethods forprocessing

frequencycounter data:converting time stamps to frequencydataeither before or after

Chapter3 AdaptableFrequency CounterwithPhase Filtering

(a)

Raw

LPFtime

LPFfreq.

Theory

Allandeviation, σ

10−8

10−7

10−6

10−5

10−4

10−3

10−2

Integrationtime, τ(s)

10−5 10−4 10−3 10−2 10−1 100101

(b)

FC raw

FC LPF

PLLFD

Theory

Allandeviation, σ

10−8

10−7

10−6

10−5

10−4

10−3

Integrationtime,τ(s)

10−5 10−4 10−3 10−2 10−1 100101

Figure3.2:

Experimental results. (a)Impact of low-pass ﬁlter placement(before or after

time-to-frequencyconversion) with acutoﬀ frequencyof200

on theAD. The

"Raw"curve represents unprocessed output from thefrequency counter, while

"LPFtime" and"LPFfrequency"showthe resultsofﬁlteringbeforeand after

time-to-frequencyconversion, respectively. (b)ADs of raw(unﬁltered)and low-

pass ﬁltered frequencycounter output comparedwithaPLLFDwiththe same

bandwidth. Resultsshownodiﬀerence in performancebetween thecommercial

PLLFDand theproposedfrequency counter. (Figuretaken from [49])

low-pass ﬁltering.Aspreviouslydiscussed,the non-linearityinthe time-to-frequency

conversion introduces intermodulation noise, altering theAllanDeviation (AD) behavior

from theexpected 1

τ

to 1

τ

forlarge

.Thiseﬀect is mitigated by applying low-pass

ﬁlteringtothe time stampseriesbeforefrequency conversion.

Figure 3.2a showsthatﬁlteringtime stampdataﬁrst, then converting to frequency,

resultsinabetterADcomparedtothe reverseorder,conﬁrming thetheoretical prediction.

In thecomparisonexperiment, we compare theproposed frequencycounter with a

commercialPLL-basedfrequency detector (PLLFD).The PLLFDisconﬁgured with a

200

loop bandwidthusing PIDcoeﬃcients

kp

Hzdeg

and

ki

Hzdegsec

ThePLL operates with a1

kHz

low-pass ﬁlterand a27

kHz

sampling rate.The frequency

counterprocesses theNEMSoutput throughaband-pass ﬁlter(5

kHz

bandwidth)and uses

aCIC decimator(decimation factor

R

8192)toachieve aﬁnalsampling rate of 9

kHz

Thedecimatedoutputisthenlow-passﬁltered with acutoﬀ of 200 Hz.

Figure 3.2b demonstratesthatthe proposed frequencycounter matchesthe performance

of thePLLFD, both in measurement andtheory.

Chapter4

ThermalResponseModelling and

Kalman Filtering

This chapterprovidesacomprehensive summary of theworktitled OptimizedSignal

Estimation in Nanomechanical Photothermal Sensing via Thermal Response Modelling and

Kalman Filtering [55].

Nanomechanical systems, particularlyinthe contextofphotothermalsensing,demand

precise signal estimation techniques to enhancetheir performanceinapplications such as

infrared(IR)spectroscopy, commonly referredtoasNEMS-IR.Asdiscussed in Chapter 2,

afundamental trade-oﬀexistsbetween speed andaccuracyinNEMSresonators,where

precision canbeincreased at theexpense of speed by applying signal ﬁltering.The primary

challenges faced by NEMS-IRsystemsinclude:

•

Slow response speed: Typically ranging from 5msto100 ms.Beyond100 ms,

driftand randomwalkbecome thepredominant noisesources, signiﬁcantly limiting

precision at lowfrequencies.

•

Complexstepresponse: Thestepresponseischaracterized by twodistincttransient

responses, with theslowertimeconstantoften exceeding 100 seconds [56].

This work introduces an advanced thermalresponse modelspeciﬁcally designed formicro-

andnanomechanical resonators.The modelcapturesthe dualtime constant nature of the

device andintegratesitwithanadaptiveKalmanﬁlter to achieverapid andaccurate signal

estimation,thereby improving theintrinsic speed versus accuracy limitations of NEMS-IR

devices.

Theproposedmodel,asdepictedinFigure 4.1,accounts fordualthermal time constants:

oneassociated with theresonator andthe otherwiththe supporting chip. These time

Chapter4 ThermalResponseModellingand Kalman Filtering

(a) (b)

PaCr

Δr

Rrad

Δf

Figure4.1:

(a)Schematic illustrating theresonator andframe components involved in the

thermalcircuit model. (b)Linearized thermalequivalentcircuit model, showing

thethermal interactionbetween theresonator, frame, andthe environment.

Theground symbol represents theambient temperature

0

.(Figureadapted

from [55])

constantsare critical foraccurately capturing thethermal dynamicsofthe system.The

model is then utilized in conjunction with aKalmanﬁlter to enhancesignalestimation

performance, addressing thespeed-accuracy trade-oﬀinherent in NEMS-IRdevices.

Thecorresponding state-spacerepresentationofthe modelisdescribed by



Δ˙

r

Δ˙

f



xsRr+Rrad

RrRradCr

RrCr

RrCfRr+Rf

RrRfCf





Δr

Δf



xs

0



Paw,(4.1)

where

is thedynamic matrix,

theinput matrix,

thestate vector,

˙x s

itsderivative,

wtheprocess noiseinthe system andΔ

r



f

thedynamic states of themodel,which

are thetemperatures of theresonatorand theframe.Additionally,let

denote the

absorbed power,

Rrad

theradiatedheat,

thethermal resistance of themembrane,

its

thermalcapacitance, and

and

representthe thermalresistance andcapacitance of

theresonator frame, respectively.

Therelationbetween themeasuredfrequency andthe dynamic states aregiven by the

measurementequationofthe state-spacemodel as [55]

Δω0



yω

gαrω0gαf





Δr

Δf



xsv(4.2)

where

is theoutput matrix of thesystem,

themeasuredquantity,

αrαf

thethermal

expansioncoeﬃcient of theresonator/frame,

ascaling factor described in [55]and

the

measurementnoise.

Forthe Kalmanﬁlter to performoptimalﬁlteringthe noisesources need to be described.

Theprocess noisecan be modeledinstate-space as [55]

˙x wt1

th 



xwt1





t,(4.3a)

wt





0



xwt.(4.3b)

where

is thewhite measurementnoise sequencewithits auto-correlation

Euwtuwτ

Sthδtτδtτ

,with

Sthδtτ

being thethermal ﬂuctuations noiselevel described in

(1.118)

and

δ

is theDirac delta function.Inasimilarway themeasurement noisecan be

modeledinstate-space as [55]

˙x vt







BPτFC 1

τBP 1

τFC 



xvt













t,(4.4a)

vt√

BPτFC K

τBPτFC 



xvt,(4.4b)

where

is thewhite measurementnoise sequencewithits auto-correlation

Euvtuvτ

Sθthm δtτ,where Sθthm is thethermomechanicalphase noiselevel (obtainedfrom(1.87)

and

(1.107)

)and



is thesquarerootofthe ratiobetween thethermomechanical and

detectionphase noiselevels.

In ordertoimprovethe speed performancewiththe Kalmanﬁlter withoutlosingaccuracy

we proposeanadaptable Kalmanﬁlter (AKF), wherethe tuning laserthatgenerates the

IR spectrum is operated in thestepregime.For everywavelengththe laser is tunedon

andrapidly oﬀ,yieldinganinput (absorbedpower)signalofknown shapeand unknown

intensity. By knowing thetiming of theinput signal theKalmanﬁlter bandwidthcan be

increased making it able to followthe signal at ahighspeed.The Kalmanﬁlter converges

very fast to therealvalue of thesignal, yielding precise input powerestimates. In orderto

Chapter4 ThermalResponseModellingand Kalman Filtering

(a)

0 2 4 6 8 10

(s)

0.25

0.00

0.25

0.50

0.75

1.00

1.25

(kHz)

Measured data

Fitted data

(b)

0 1 2 3

(s)

0.5

0.0

0.5

1.0

1.5

(kHz)

0.5

0.0

0.5

1.0

1.5

Normalized power

(c)

0.8 0.9 1.0 1.1 1.2 1.3

(s)

200

100

200

(Hz)

0.4

0.2

0.0

0.2

0.4

Normalized power

(d)

1100 1200 1300 1400 1500 1600 1700 1800

(cm 1)

Normalized spectrum

LPF

AKF

5.56.06.57.07.58.08.59.0 (

)

Figure 4.2:

Resultsfromthe NEMS-IRexperiments: (a)Systemidentiﬁcation forthe NEMS-

IRsetup. (b)Adaptive Kalmanﬁlteringappliedtoalaser powerstep, illustrating

arapid powerestimate, acomparatively slower measurementestimate,and

an even slower response from alow-passﬁlter possessing equivalentﬁltering

strength to theAKF.(c) Adetailedviewofasegment from theQCL step scan,

comparing AKFand low-pass ﬁlterdata. (d)Comparisonofthe polystyrene

spectrumobtainedusing theadaptiveKalmanﬁlter estimate andthe low-pass

ﬁlter,withred lineshighlightingthe characteristic peaks. (someaxesare

invertedtoprovide an easier visualization, andﬁguresare takenfrom[55])

performprecise estimateswiththe Kalman ﬁlterthe system parameters need to be identiﬁed.

Forsystemparameter identiﬁcationasteep response ﬁt with the Levenberg–Marquardt

algorithm is used.Figure 4.2a showsacomparisonbetween themeasuredand ﬁttedstep

response. Thestepisgenerated with thetunableIRlaser.Figure 4.2b showsacomparison

between aabsorbed powerstepestimate of theAKF andasimple low-pass ﬁlter (LPF)

with comparableﬁlteringstrength (approximately 0.4Hzstandard deviationovera300 ms

recording period). To generate aspectrumthe resonatoriscoated with polystyreneand

illuminatedbythe laser, whichistunedtowavelengths from 1789

cm−1

to 1122

cm−1

incrementsof1

cm−1

,witha100 ms "on" and150 ms "oﬀ" period.Figure 4.2c showsapart

of thetime data obtained from thescanand thecomparisonofthe estimatedAKF signals

to asimple low-pass ﬁltersignal, showingthe superiorityofthe AKFtothe LPFinterms of

signal preservation.LastlyFigure 4.2d showsthe resultingspectrumcreated by subtracting

thedataathe pointright before thelaser is turned on from thepoint rightbeforethe

laseristurnedoﬀfor each wavelength. TheAKF generatedspectrumismuchcleaner and

precise in comparisontothe LPFgenerated spectrum. Duetothe fast anddiﬀerential

sampling theslowrandomwalkand drifteﬀectsare minimized yielding fast andprecise

measurements.

Theintegration of thermalmodeling andKalmanﬁlteringled to asigniﬁcantimprovement

in thespeed andprecision ofphotothermalsensing,asdemonstratedbythe experimental

results.

In conclusion,the proposed approachoﬀersasubstantialenhancementinnanomechanical

photothermalsensing by eﬀectivelyreducing theeﬀectsofthe second slow NEMS time

constant,drift andrandomwalkmaking it highly suitable forapplications requiring rapid

andprecise measurements.

Chapter5

Conclusion andOutlook

5.1Conclusion

This dissertation hasadvanced theﬁeldofnanomechanicalresonators by examiningand

improving upon thefundamental aspectsofresonancefrequency tracking,frequency counter

(FC)design,and theirapplicationsinphotothermalinfrared(IR)spectroscopy.The research,

spanning threekey studies, hasnot only introduced newmethodologies but also debunked

commonmisconceptionswithin theﬁeld.

Theﬁrstpaper rigorouslyanalyzedvarious frequencytrackingschemes, including both

open-loopand closed-loopconﬁgurations,challengingthe prevailingbeliefthatclosed-loop

schemes inherently outperform open-loopones. Thestudy conclusively demonstratedthat,

contrary to popular opinion,all frequencytrackingschemes canachieve similarperformance

levels.Itrevealedthatthe ultimatelimiting factor in performanceisintrinsic to the

nanomechanicalresonator itself, rather than thechoiceoftracking scheme.This ﬁnding

is crucialasitbroadensthe scope of viable tracking techniquesfor diﬀerent applications,

oﬀering more ﬂexibility in systemdesign withoutcompromising on performance.

In thesecond paper, anovel adaptableFCarchitecturewas introduced,speciﬁcally

designed forresonance frequencymonitoringinnanomechanicalresonators. This architec-

ture,featuring phase ﬁltering,oﬀered arobustalternative to traditionalphase-lockedloop

frequencydetectors(PLLFDs). By integratingphase ﬁltering andresampling, theproposed

system matchedthe precision andresponse speed of existing PLLFDs.The ﬂexibilityand

cost-eﬀectiveness of this FC design make it apromisingcandidate forwidespreadadoption

in variousnanomechanical sensingapplications.

Thethirdpaper applied thedeveloped athermal modelcapturing thedualtime constant

of NEMS sensors in photothermalIRspectroscopy.Additionally,byutilizing thethermal

Chapter5 Conclusion andOutlook

modeland noiseproﬁles, we introduced Kalmanﬁltering forfasterand optimalsignal

estimation.The resultsdemonstratedsigniﬁcantimprovements in spectralresolutionand

detectionspeed,madepossible by theprecise andreal-time tracking of frequencyshifts.

5.2Outlook

This dissertation hasintroduced signiﬁcantadvancementsinresonance frequencymonitoring

andinfraredspectroscopywithin nanomechanical sensing.However,there areseveral

promising avenuesfor futureresearchthatcould extend andenhancethe contributions

made in this work.

Oneofthe most exciting possibilities forfutureresearchliesinthe integrationofmachine

learning techniqueswiththe models developedinthis dissertation.Machine learning,

particularlydeep learning algorithms,has shownconsiderable promiseinsignalprocessing

andpattern recognition. Applying these techniquestonanomechanicalsensing could lead

to more adaptive androbustfrequencytracking algorithms.For instance, neural networks

could be trainedtopredict resonancefrequency shifts or to optimizethe parameters of

Kalmanﬁltersdynamically based on real-time data,potentially improvingaccuracyand

reducing computationalcomplexity.

Expanding thecurrent models to address nonlinear systemsisanother critical area

for future work.While this dissertation hasfocused primarily on linear systems, many

real-world applicationsinvolve nonlinearitiesthatcannotbeignored.Developingmodels

that accountfor thesenonlinear behaviorscould signiﬁcantlybroaden theapplicabilityof

nanomechanical sensors, allowing them to performreliably in more complexenvironments.

This could involvethe useofadvanced mathematical techniquesorthe integrationof

machine learning models trainedtorecognizeand compensate fornonlinear eﬀects.

Furthermore, thetechniquesdiscussed in this dissertation could be extendedtoother

typesofsensors,particularlythose that measureamplitude rather than frequency. In these

systems,amplitude modulation anddemodulationcan be crucialfor detectingsmall changes

inthe measured quantity.Extending theKalmanﬁlteringand thermalmodelingtechniques

toamplitude-based sensingcould lead to newapplications in ﬁeldssuchasacousticsensing,

opticalmetrology,and even biomedical diagnostics.

Additionally,futureresearchcould explorethe development of hybrid sensors that combine

frequencyand amplitude measurements,leveraging thestrengthsofbothapproaches. Such

sensors could oﬀer enhanced sensitivityand abroader rangeofapplications,particularlyin

environmentswhere both frequencyand amplitude changesare relevant to themeasured

phenomena.

In summary,whilethis dissertation hasmadesigniﬁcantstridesinthe ﬁeld of nanomechan-

ical sensing,there remain numerous opportunities forfurther research.Integrating machine

learning,expanding to nonlinear systems, andapplying thesetechniquestodiﬀerenttypes

of sensorsare allpromisingdirections that could signiﬁcantlyadvance theﬁeld. Continued

exploration in these areaswill be essential fordeveloping thenextgenerationofsensing

technologies.

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PHYSICAL REVIEW APPLIED 20, 024023 (2023)

Schemesfor Tracking Resonance Frequencyfor Micro- andNanomechanical

Resonators

Hajrudin Beši´

c ,1,*AlperDemir ,2Johannes Steurer,1Niklas Luhmann,1and Silvan Schmid1

1Institute of Sensor and Actuator Systems, TU Wien, Gusshausstrasse 27–29, Vienna 1040, Austria

2Department of Electrical Engineering, Koç University,Istanbul 34450, Turkey

(Received 13 May2023; accepted 13 July 2023; published 9August2023)

Nanomechanical resonators can serve as high-performance detectors andhave the potential to be widely

used in industryfor avariety of applications. Most nanomechanical-sensing applications rely on detecting

changesofresonance frequency.Incommonly used frequency-tracking schemes,the resonator is driven

at or close to its resonance frequency.Closed-loop systems can continually check whether theresonator

is at resonance andadjust the frequency of thedriving signalaccordingly.Inthis work,westudy three

resonance-frequency-tracking schemes, afeedback-free (FF) scheme,aself-sustaining oscillator (SSO)

scheme, andaphase-locked loop oscillator (PLLO)scheme. We improve and extend thetheoretical models

for theFFand theSSO tracking schemes and test themodels experimentally with ananoelectromechanical

system (NEMS) resonator.WeemployanSSO architecturewith apulsed positive-feedback topology and

compareittothe commonlyused PLLO and FF schemes.Weshow that all tracking schemes aretheoret-

ically equivalent and that they allare subject to the same speed versusaccuracy trade-oﬀcharacteristics.

In ordertoverify thetheoretical models,wepresent experimentalsteady-state measurements for allofthe

trackingschemes.The frequency stability is characterizedbycomputing theAllandeviation. We obtain

almost perfect correspondence between the theoretical models and theexperimental measurements.These

results show that the choice of the tracking scheme is dictatedbycost, robustness, and usability in practice

as opposed to fundamental theoretical diﬀerences in performance.

DOI: 10.1103/PhysRevApplied.20.024023

I. INTRODUCTION

Micro- andnanomechanical resonators areexceptional

sensorsthe resonance frequencyofwhich detunes with

parameter changes,which can be achange in mass [1–5],

achange in damping [6,7], or achangeinstiﬀness [8–

10]. Resonance-frequency trackingcan beaccomplished

via open- and closed-loop schemes:

(a)The feedback-free (FF) or open-loop approach.

In thestandard conﬁguration, this scheme has an infe-

rior speedperformance when compared to closed-loop

and self-adjusting approaches,since it is limited by the

mechanical response time of the resonator.

(b) Thephase-locked loop oscillator (PLLO) approach.

This is commonly used since it can be easily realized

digitally with setups basedondigital signalprocessing

(DSP)orﬁeld-programmablegatearrays (FPGAs). Having

awidefrequency range, one setup can be suitable for most

micro- and nanomechanical resonators. Thereare many

commercial devices available that can be usedfor PLLO

implementations.

*hajrudin.besic@tuwien.ac.at

feedback approach. Theclassical realization of the SSO

has found limited use, mostly because they aretypically

implemented as analogcircuits, resulting in anarrow

frequency range.

The above frequency-tracking schemes have been com-

pared both theoretically [11]and experimentally [12]and

it has been concluded that with theclosed-loop PLLO and

the self-adjusting SSO schemes, better speed performance

can be obtained at theexpense of degraded precision in the

presence of signiﬁcant transduction noise, as compared to

the FF approach. We show that one can, in fact, obtain the

samespeed versus precisiontrade-oﬀusingthe open-loop

FF scheme with asimple modiﬁcation to the standardcon-

ﬁguration. Thetheory behind these approaches has been

discussed in depth in Ref. [11]. We also extend the SSO

modelfromRef.[11]bytaking into account theimpact

of detection noise through two separate mechanisms and

show that the resultingfrequency ﬂuctuations areequiv-

alenttothe PLLO, including thedetection-noise-limited

regime.

Thereare various designs forthe implementation of the

positive-feedback mechanism in the SSO conﬁguration.

BEŠI ´

Cet al. PHYS. REV. APPLIED 20, 024023 (2023)

The most common is to simply amplify andadjust the

phaseofthe resonator response signaltogeneratethe feed-

backdrive, resulting in asine-driven SSO. Alternatively,

amplitude-and duration-adjustable timed pulses can be

used in forming the drivefor the resonatorwith apositive-

feedbackmechanism,resulting in apulse-driven SSO [13–

18]. We have designed andrealized an SSOconﬁguration

basedonthe pulsed-feedback mechanism.Toexperimen-

tally investigate the SSO and the PLLO approaches, we

compare ourSSO implementation to aDSP- or FPGA-

basedlock-in ampliﬁer setup with an integrated PLL

system.

II. THEORY

A. Noise in nanomechanical resonators

We consider two noise sources,the thermomechanical

noise of the resonator and the detection noise generated in

the transduction and detection hardware.

The thermomechanical noise is the most fundamental

noise source of theresonator. It can be modeledasawhite-

noise force nth(t)at theinput of the resonator [11,19]and

it has aone-sided spectraldensity with units ofN

/Hz:

SF=4mω0kBT

Q,(1)

where kBis Boltzmann’s constant, Tis the temperature, m

is theeﬀective mass, ω0is theeigenfrequency, and Qis

the qualityfactor of the resonator.For aslightly damped

resonator,the eigenfrequency is approximately equal to

the resonance frequency ωr≈ω0.The white-noise forceat

theinput of the resonator is shaped by the complex force

susceptibility of the resonator,

χ(s)=1/m

s2+ω

Qs+ω2

,(2)

resultinginasteady-state thermomechanical-amplitude

noise at its output with apower spectraldensity

Sth(ω) =SF|χ(jω)|2.(3)

Operating at resonance, the thermomechanical-amplitude

noise reduces to

Sth =Sth(ω0)=SFQ

mω2

02

=4kBTQ

mω3

.(4)

The detection noise nd(t)is produced during the conver-

sion of the mechanical motion into an electrical signal. It

includes thenoise generated by thereadoutand electron-

ics. The detection noisecan be modeled as awhite-noise

source with respect to thethermomechanical-amplitude

noise:

Sd=K2Sth,(5)

where Kis adimensionless factorthat corresponds to

theratio between the detection-noise background and the

height of the thermomechanical-noise peak.IfK>1, the

thermomechanical-noise peak is buried in detection noise.

If K<1, the thermomechanical noise is resolved above

thedetection-noise background.

Theresonator is driven by acoherentforce F(t),which

results in asteady-state amplitude response at resonance of

x(t)=F(t)Q/(mω2

0)=Arcos (ω0t)with an amplitude Ar.

Both the thermomechanical andthe detection-amplitude

noise then translate into corresponding phase noise with

thefollowing power spectral densities [20]:

Sθth =2

Sth,

Sθd=2

Sd=K2Sθth .(6)

This transformation clearly shows that alargeoscillation

amplitude Ardilutes the eﬀect of thermomechanical and

detection noise. Themaximization of Aris limited by the

onset of nonlinearities in the resonator system.

Theconversion of the phase noise [see Eq. (6)]into fre-

quency noise Sω(ω) depends on the trackingscheme and

will be derived in the following sections.

B. Feedback-free scheme

The feedback-free (FF) scheme is asimple and well-

knownmethod wherethe resonator is driven at or close

to resonance. Figure 1showsaschematic representation

of the FF frequency-trackingscheme. It consists of asig-

nalgenerator that is drivingthe resonator with aconstant

frequencyclose to its resonance frequency.The resonator

motionisthen transduced andthe phase diﬀerence of the

resonator response with respect to the reference signal pro-

vided by the driving signalgenerator is obtained by aphase

detector (PD).Any suddenchange in theresonance fre-

quency, ωr,will cause acorresponding change in the

detected phase diﬀerence θ(t),asderived in Ref. [11]as

follows:

θ(t)=τrωr1−e−t/τr,(7)

where

τr=2Q

ωr

(8)

is the mechanical resonator time constant. The resonance-

frequency change ωrcan be easily extracted from this

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Resonator

noise

Detection

noise

Resonator

Phase-to-frequencyconverter

FIG.1.Ablock diagram of theFFfrequency-tracking scheme,

featuring asignalgenerator (SG) that is driving amicro- or

nanomechanicalresonator.The resonatorresponse is fedintoa

phase-diﬀerencedetector (PD). The phase-diﬀerencesignal is

then mappedtoafrequency signal by thephase-to-frequency

conversion mechanism. The thermomechanical noise is added at

theinput of theresonator andthe detection noise is added at the

input of thephase detector.

phase response. The drawback of the FFschemeisthat

thedrive frequency hastobewithin the line width of the

resonator,wherethe phase response is linear.This makes

thisscheme susceptibletothermaldrift [11,21]. Here, we

assume that the drive signal indeedsatisﬁes this condition

for the FF scheme.

The standardmap from the phase diﬀerencetothe

resonance-frequency changeisobtainedsimplybyadivi-

sion with τr,asdescribed in Ref.[11]and implemented in

Ref. [12]. The frequencyresponse in thiscase,which we

call the slow response,isgiven by

ωsr(t)=θ(t)

τr=ωr1−e−t/τr.(9)

The slow response contains low-frequency information for

resonance-frequency deviations.The speed of the above

response is limited by τr.For resonators with high quality

factors, this response time can become very long.

Alternatively,one canextract the frequency information

from the phase θ(t)via diﬀerentiation(with respect to

time), as hasbeenshown in Refs. [22,23]. This results in a

fast but transient response as follows:

ωfr(t)=dθ(t)

dt =ωre−t/τr.(10)

Thefast response contains high-frequency information

for resonance-frequency deviations but suppresses low-

frequencyphenomena such as thermal drift, due to dif-

ferentiation. By combining(adding) theslow and fast

responses, we obtain

ω(t)=θ(t)

τr+dθ(t)

dt =ωr(11)

as an instantaneous and nontransient frequency response,

which contains both low- and high-frequency information

for resonance-frequency deviations. This instantaneous

response will only be smoothed andslowed down by any

band-limiting mechanism, e.g., alow-pass ﬁlter,inthe

phase detector. Forinstance, when phase-diﬀerencedetec-

tion is performed with an in-phase andquadrature(I/Q)

demodulator,asinalock-in-ampliﬁer setup, the response

speed will be determined by the low-pass ﬁltersinthe

demodulator, represented by atransfer function HL(s).

Thebandwidth for these ﬁltersneedstobesmaller than

(twice) theresonance frequency in ordertoﬁlter out the

high-frequency(at twice the resonance frequency)signal

componentsthatare produced by the multipliersinthe

demodulator. By transformingEq. (11) into the Laplace

domain andincluding HL(s)to represent the band-limited

natureofphase detection, we obtain

HFF(s)=1

τr+sHL(s),(12)

which can be also written as

HFF(s)=1

τr

Hr(s)HL(s),(13)

where Hr(s)is asingle-pole low-pass ﬁlter with the time

constant of the resonator

Hr(s)=1

1+sτr

,(14)

capturing the input-output frequency-domain response of

the resonator. HFF(s)can be used in computing the

frequency-domain frequency response of theFFscheme

(with combined fast andslow responses) to step changes

in the resonance frequency,aswell as forcharacterizing

frequencyﬂuctuations due to noise.

Thefrequency ﬂuctuations caused by noise in the FF

scheme aredirectly determined by the phase ﬂuctuations

024023-3

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Cet al. PHYS. REV. APPLIED 20, 024023 (2023)

of the input signal to the phase detector.The phaseﬂuctu-

ations detected by the phase detector canbeexpressed in

the Laplacedomain as

θ(s)=θth(s)Hr(s)+θd(s).(15)

The thermomechanical phase ﬂuctuationsare shaped by

the resonator characteristicsbeforetheyare detected, while

the detection noise is fedintothe phasedetector unal-

tered [11]. The frequencyﬂuctuations of the FF tracking

scheme arethenobtained by simply multiplyingthe phase

ﬂuctuationsbythe transfer function of theFFscheme,

ω(s)=θ(s)HFF,(16)

yielding the transfer functions for the two noise sources:

HFF

θth (s)=1

τr

HL(s),

HFF

θd(s)=1

τr

Hr(s)HL(s).(17)

C. Self-sustaining oscillator

The standardSSO conﬁguration as showninFig. 2is

based on the Barkhausen criterion, which requires that the

closed-loopgain is equal to one. This can be achieved

by introducing anonlinearity (a saturatingampliﬁer) h(·)

in the loop, which stabilizes theamplitude. This nonlin-

earityalso generates higher-order harmonics, which are

Saturating

amplifier Phase shifter

SSO

Frequency

detector

Resonator

noise

Detection

noise

FIG. 2. Ablockdiagram showing astandardSSO scheme,

with aphase shifter andanonlinear saturating element in the

positive-feedback path. Thethermomechanical-noiseforce nth(t)

is added at theinputofthe resonator.The detection noise nd(t)

is added in thetransduction of thedisplacement signal xr(t)into

an electricalsignal xd(t).The electrical signal is then ﬁltered and

acquired by thefrequency detector.

well ﬁltered if the resonator has alarge quality factor. The

Barkhausen criterion also imposes aphase condition for

stable andsustained oscillations:the phase around the loop

needstoben2π,where n=1, 2 ··· is much smaller than

the quality factor of theresonator.The phase condition is

realized by introducingaphase-shifting element that phase

shifts or delays the signalatthe output of the resonator to

generate thefeedback drive.

Theworking principle of astandard SSOisdescribed in

Ref. [11], whereamodel wasderived. We start with the

following equation from Ref. [11]:

dt θr(t)=1

τr

Arss

mω2

[h(Arss)nd(t)+nth(t)]

τrQ

mω2

h(Arss)θd(t)+θth(t),(18)

with the steady-state andnoiseless amplitude Arss.The

above equation describes theﬂuctuationscaused by noise

at theresonator outputphase and hence in the signal xr(t)

in Fig. 2.Due to the feedback path,detectionnoise nd(t)

contributes to the phase ﬂuctuations of xr(t).However,

phase ﬂuctuations in thedetected resonator output[x

(t)

in Fig. 2]have an additional contribution due to detection

noise. Equation (18) can be augmented as follows to derive

theequation for phase ﬂuctuations in xd(t):

dt θ(t)=1

τrQ

mω2

h(Arss)θd(t)+θth(t)+d

dt θd(t).

(19)

As stated in Ref. [11], the gain condition for aself-

sustaining oscillator,

mω2

h(Arss)=1, (20)

needs to be met, which yields

dt θ(t)=1

τr

[θd(t)+θth(t)]+d

dt θd(t).(21)

Thetransfer functions from thermomechanical- and

detection-noise sources to the(frequency of the) output

xd(t)can be derived based on theabove equation:

HSSO

θth (s)=1

τr

HL(s),

HSSO

θd(s)=1

τr

Hr(s)HL(s),(22)

where Hris asingle-pole low-pass ﬁlter with the time

constant of the resonator. HLhas low-pass characteris-

tics andrepresents thebandwidth-limiting(noise-ﬁltering)

mechanism in the frequency-detectiondevice.

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D. Phase-locked loop oscillator

The PLLO is essentially aclosed-loopextensionofthe

FF scheme,which continually updates thedrive frequency

to matchthe resonance frequency of theresonator. The

block diagraminFig. 3shows the standardPLLO setup

with asimpliﬁed representation of its internal structure.

PLLO is usually realized digitally,e.g., by usingalock-in

ampliﬁer with an integrated PLL. The internal oscillator,

also called acontrolled or local oscillator (LO), of the

PLLO drives the resonator.The thermomechanical noise

nth is added to the drive at the input of the resonator.The

motion of the resonator xr(t)is transduced into an electri-

cal signal y(t).The electrical signal then goes into aphase

detector (PD) and is compared to the internal LO signal.

PD is realized with an I/Q demodulatorand hasinternal

low-pass ﬁlters(LPF) with acutoﬀ frequency fL(and time

constant τL=1/2πfL). This ﬁlterremoves high-frequency

signal components,aswellashigh-frequency noise, but

limits thebandwidthofphase detection. The output of PD

(minus aset point)isused as theerror signal θethat

represents the diﬀerence in phase betweenthe resonator

responseand the LO drive signal. The PI controller pro-

duces the control signalthat sets the frequency of the LO.

The negative-feedbackloop maintainsthe desired phase

diﬀerencebetweenthe resonatorresponseand the LO. The

speed with which thePIcontroller regulates theoscillator

is characterizedinRef.[19]using thesystem bandwidth of

the PLLO or fPLL [12]. ThePIcoeﬃcients arecalculated

PLL

LO PD

Resonator

noise

Detection

noise

Resonator

FIG. 3. The phase detector (PD) computesthe phase diﬀerence

betweenthe resonatorresponse andthe local oscillator (LO),

which (minus aset point) is fedinto the PI controller as an error

signal. The PI controllertunes thefrequency of theLO, whichis

used to drive theresonator.

from the desired system bandwidth as follows [11]:

kp=2πfPLL =1

τPLL

ki=kp

τr

,(23)

where kpis the proportional coeﬃcient and kiis the integral

coeﬃcient of the PI controller.The noise-transfer functions

for the PLLO can be derived based on the loop dynamics

as described in Ref. [11]:

HPLL

θth (s)=1

τr

(skp+ki)HL(s)

s2+s

τr+(skp+ki)HL(s),

HPLL

θd(s)=1

τr

Hr(s)

(skp+ki)HL(s)

s2+s

τr+(skp+ki)HL(s).(24)

With the parameters in Eq. (23),the above noise-transfer

functions take the following simpler forms [11]:

HPLL

θth (s)=1

τr

HL(s)

HL(s)+sτPLL

HPLL

θd(s)=1

τr

Hr(s)

HL(s)

HL(s)+sτPLL

.(25)

E. Allan deviation

Thestandardand well-established methodfor character-

izing frequency ﬂuctuations is theAllan deviation σy(τ )

[11,19,24]. It is the squareroot of the Allan variance,

which can be computed with

σ2

y(τ ) =1

2(N−1)



i=1

(yi+1,τ−yi,τ)2,(26)

where yiis the ith sample of the averaged frequency over

the averaging time τ,i.e.,

yi,τ=1

τiτ

(i−1)τ

y(t)dt.(27)

Thefrequency values whencomputing an Allan deviation

need to be normalized,resultinginafractional frequency

y(t)=ω(t)

ω0

.(28)

TheAllan variance can also be computed in the frequency

domain if the power spectral density of the fractional

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Cet al. PHYS. REV. APPLIED 20, 024023 (2023)

frequency ﬂuctuations is known:

σ2

y(τ ) =1

2π

τ2∞

0sin ωτ

24

ω2Sy(ω)dω.(29)

For white frequency ﬂuctuations with Sy(ω) =constant,

Eq. (29) reduces to

σ2

y=Sy(0)

2τ.(30)

Hence, in asystem limited by,e.g., thermal white noise,

the resulting Allan deviationexhibits a σy∝1/√τdepen-

dence withthe averaging time τ.

The power spectral density of the frequency noise

Sω(ω) can be computed as asuperpositionofthe power

spectral densities of the thermomechanical anddetection

phase noise[see Eq. (6)]multiplied with their correspond-

ingtransfer functions (magnitude squared)derived above

[11], which then readily yields the fractional frequency

noise required to compute the Allandeviation:

Sy(ω) =Sω (ω)

ω2

=Sθth (ω)|Hθth (jω)|2+Sθd(ω)|Hθd(jω)|2

ω2

=Sθth (ω)

ω2

0|Hθth (jω)|2+K2|Hθd(jω)|2.(31)

III. METHODS

The models for the three frequency-tracking schemes

were tested experimentallywith ananoelectromechani-

cal system (NEMS)resonator.While theFFand PLLO

trackingschemes were implemented withacommercial

lock-in ampliﬁer(HF2LI 50-MHz lock-in ampliﬁer from

Zurich Instruments), the SSO scheme was realized with a

frequency-trackingprototype devicefromInvisible-Light

Labs GmbH.

A. NEMS resonator

TheNEMS resonator used in this work (see Fig. 4)

consists of a1018-µm-sized square-membrane-shaped res-

onator made of 50-nm-thick low-stresssilicon-rich silicon

nitride (fabricated by low-pressurechemicalvapordeposi-

tion). Theelectricaltransduction is realized bytwo 5-µm-

wide Au traces passing on the resonator. Themembrane

is placed in the center of astatic magnetic ﬁeld of about

0.8 T, created by aHalbach arrayofneodymium magnets,

with the traces oriented perpendicular to the magnetic ﬁeld.

Exploiting the resulting Lorentz force, the metal traces can

be used both to drive withanaccurrent and, in return,

to detect the motionofthe resonator through themag-

netomotively induced voltage.The signal from the metal

NEMS resonator

Preampliﬁer

+–

FF, PLL, or SSO

FIG.4.Aschematic of the NEMS resonator used forall exper-

iments featuring electrodynamic transduction. The circular area

in the centerwherethe SiNdrumhead is perforatedhas no

relevance forthis work.

trace for detection is ampliﬁed with alow-noise diﬀer-

ential preampliﬁer with again factorof10

.The NEMS

chip is placed in avacuum chamber featuringarotaryvane

pump, reaching avacuum of 5.2 ×10−3mbar.With ares-

onance frequency of ωr=82.3 kHZand aquality factor of

Q=97 000, the NEMS resonator has aresponse time [see

Eq. (8)]ofτ

r=2Q/ωr=0.4s.

B. SSO with pulsed drive

TheSSO frequency-trackingsystem from Invisible-

Light Labs that wasused in this work is based on pulsed

positivefeedback [25,26], producing pulses of width Tw

with an adjustable delay Tdwith respect to the signalphase

θ0,and afeedback-controlled amplitude to sustain acon-

stantvibrational amplitude of the resonator.Aschematic

of the timing of the wave forms at the outputs produced by

the pulsed positive feedback is shown in Fig. 5.

Anarrow pulse with width Twin the time domain cor-

responds to asinc functioninthe frequency domain with

zeropoints at integer multiples of fb=1/Tw.The ﬁrst zero

point can be considered as the bandwidth of the pulse. The

frequencycontent above fbis small in comparison to the

onesbelow fband does not contributemuchtothe signal.

Hence, when actuating aresonator, apulsed signal can be

used instead of thetypical sinusoidalsignal, as is common

for,e.g., the characterizationoftuning forks[27].

Figure 6showsthe block-diagram representation of the

positive-feedback system used in this work. The signal

generated by the NEMS resonator is ﬁrstampliﬁed with

apreampliﬁer beforeitentersthe positive-feedback path.

The feedback signal is ﬁrst passed through aband-pass

ﬁlter,which serves twopurposes. First, it reduces detec-

tionnoise and, second, it attenuates unwanted modes of

the resonator.Afterward,the signal passes through aphase

detector, which is able to detect aphase at 0◦or 180◦.Phase

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θ0

Amplitude (V)

−1.0

−0.5

0.0

0.5

1.0

05×10−5 10−4 1.5×10−4

(a)

θ0

Amplitude (V)

0.0

0.5

1.0

1.5

Time (s)

05×10−5 10−4 1.5×10−4

(b)

FIG. 5. The timing of thepositive-feedback system. (a)The

sinusoidal signal at theresonator output is fed into thepositive-

feedback mechanism. Themechanism detects thephase θ0.(b)

The pulse-generation mechanism generates apulse with atime

delay Tdwith respect to θ0.

shifts introduced by components in theloop arecompen-

sated by an adjustabletime-delay element, which induces a

delay Tdto theoutputpulse. Tdneeds to be large enough to

cover aphase delay between 0◦and 180◦,which requires

amaximum delay of Td,max ≥π/ωr.The driving pulse is

generated via apulse-generation mechanism that is trig-

gered by the phase detector.The positive-feedbacksystem

generatesapulse with adjustable width Tw.Modiﬁcation

of the resonance frequency andthereforethe pulse fre-

quency at constant pulse width will result in achangeinthe

energy pumped intothe system andhence theamplitude of

the resonator will change. This behavioriscompensated

by regulating theamplitude of the pulse. The amplitude

regulation adjusts the pulse height in accordancewith the

measuredinput signal level and desired setpoint.

The amplitude controlisperformed by the amplitude-

regulation block. It measures the amplitude of thesignal

at the output of the band-pass ﬁlter andcompares it to the

desired setpoint.

To adjust the output voltagetothe required NEMS volt-

age values, a100-dB attenuator is placed at the output

of the positive feedback. Theband-passﬁlter,amplitude

regulation, Td,and Twareadjustableparameters.

C. Tracking-schemes setup

Forthe PLLO scheme,the desired systemtarget band-

width is determined by the PI coeﬃcients withthe relations

in Eq. (23).The low-pass ﬁlterbandwidth fLof the PLLO

demodulator is set 5timeslarger thanthe target bandwidth.

The sampling rate is set at least 10 times larger than the

LPF bandwidth.

BPF

Pulsed feedback

PLLFD

NEMS

FIG. 6. Ablock diagram of thepulsed positive-feedback sys-

tem provided by Invisible-Light Labsused for theSSO scheme.

The NEMS resonator responseisampliﬁed by apreampliﬁer

(PA) and ﬁltered by aband-passﬁlter (BPF). Thephase is

then detected by aphase detector (PD).The pulsegenerated

by the pulse generator is delayed by the delay line. The pulses

go through afeedback-controlled amplitude regulation (AR) to

drive theNEMS resonator at aﬁxedvibrational amplitude.

In order to comparethe tracking schemes, the sensor

needstobedriven at its resonance. In the case of theSSO,

thedelay of the pulse wasadjusted until the phase of the

pulses matched thephase of the resonator.This is obtained

when themaximum oscillationamplitude is reachedfor

aﬁxed pulse amplitude.Inthe caseofthe PLLO, to lock

ontothe phase that corresponds to the resonancefrequency,

afrequency sweep needstobeperformed.This sweep is

also needed to determine the resonance andhence the drive

frequency forthe FF scheme. To make sure that the res-

onator has the samevibrationalamplitude forall tracking

schemes, the drive amplitude was setsothat in allschemes,

the output of the preampliﬁer hadthe sameamplitude of

22.1 mV.The PLLO dynamics aredetermined by the cho-

sen target bandwidth. The dynamics of theFFapproach are

limited by the demodulation-ﬁlter bandwidth in the phase

detector.While the SSO core itself exhibits an instanta-

neousresponse to sudden resonance-frequency changes,

the dynamics arelimited by the frequency-detection device

that is used in conjunctionwith the SSO [11].

While the pulsed positive-feedback device comes with a

built-in frequency counter,inthis work to provide aone-

to-one comparison between the diﬀerent schemes, aPLL-

basedfrequency detector (PLLFD)was also used forthe

SSO scheme. Thecontrollingsignal forthe local oscillator

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Cet al. PHYS. REV. APPLIED 20, 024023 (2023)

of the PLL wasthenused to measure the frequency of

oscillationofthe SSOcore. This is in contrast to the useof

aPLL in the PLLOscheme, where thelocal oscillator sig-

nal in fact drives the resonator andits frequency is locked

to the resonance frequencybymaintainingthe appropri-

atephase diﬀerencebetweenthe resonator output and the

drive. Using the same PLL frequency detection allows us

to setthe system response time to be equalfor all three of

the tracking schemes.

IV.RESULTS ANDDISCUSSION

A. Analytical results

Figure 7shows the theoretical Allan deviations forthe

three frequency-trackingschemes. The Allan deviation for

aNEMS resonator that is limited by thermomechanical and

detectionnoise exhibits tworegimes with slopes propor-

tionalto1/τ and 1/√τ.A1/τ slope arises from (ampli-

ﬁed) detection noise, whereas themorefundamental ther-

momechanical noise results in a1/

√

τslope, as expected

for awhite-noise source [see Eq. (30)]. Equation (31) indi-

cates that the functional form of the Allan deviation is

determinedbytwo factors.The ﬁrst is the ratio between the

detection and the thermomechanical noise, K.Asshown

in Fig. 7(a),increasing Kincreasesthe relativeimpact of

detectionnoise, resultinginthe change of theslope from

1/√τto the 1/τ .The second factor that aﬀectsthe Allan

deviation is the system bandwidth fL,which acts as alow-

pass ﬁlter forsignal variations. In alltracking schemes,

which alluse an I/Qdemodulatorfor phase detection, sig-

nalvariations areﬁltered by the low-pass ﬁlter inside the

demodulator.Inthe PLLO scheme and in the PLLFD used

in conjunction with the SSO, thereisadditional ﬁltering

due to the negative-feedback-loopdynamics withaband-

width fPLL.Inpractice,the demodulator bandwidth is set

larger than theloopbandwidth(f

L≥5f

PLL). For the cal-

culations presented in Fig. 7,the bandwidths werechosen

equal, at fL=fPLL,toobtain an equal overall bandwidth

for allschemes. Since aﬁrst-orderdemodulator ﬁlter in

PLLO and SSO PLLFD eﬀectively results in an overall

second-order ﬁlter due to theloop dynamics, asecond-

order demodulator ﬁlter is used in the FF scheme to obtain

thesame eﬀectiveﬁlterorder for allthree schemes.

Figure 7(b) showsthe eﬀect of changing theﬁlter band-

width fLon the Allan deviation.Decreasing fLimproves the

ﬁltering of detection noise andresults in asmaller Allan

deviation–atthe expense, however, of alarger response

time[11,12].

It is important to pointout that even though thermome-

chanical noise is resolved above the detection-noise back-

ground (K<1) for allcalculations presented in Fig. 7,

detectionnoise can nonetheless aﬀect theresulting Allan

deviations.Asisshown in Fig. 7(a),frequency ﬂuctu-

ations do not start to reach thethermomechanical-noise

limit with a1/

√

τslope until K≤0.001. This behavior

can be ascribed to the long response time of the high-Q

NEMSresonator.This ﬁnding underlines the importance

of alow-noise readout that provides ahighly resolved

thermomechanical-noise peak to obtain minimalfrequency

ﬂuctuations with high-Qresonators.

B. Experimental testing

Thefrequency ﬂuctuations forthe diﬀerent track-

ing schemes were studied by collectingthe steady-state

frequencydata over 1min. The corresponding Allan-

deviationcurvescalculated from the frequency data are

presented in Figs. 8(a)–8(c).The integrated electronic

transduction of the NEMS resonator could not resolve its

thermomechanical-noise peak,resultinginK1. The

corresponding theoretical Allan-deviationcurvesare plot-

ted in Figs. 8(d)–8(f).

TheFFmeasurements shown in Fig. 8(a) wereper-

formed with asecond-orderdemodulation ﬁlter with the

Integration time τ(s)

(a)

Increasing

PLLO

SSO

Allan deviation σy

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−5 10−4 10−3 10−2 10−1 100101

(b)

Increasing fL

PLLO

SSO

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−5 10−4 10−3 10−2 10−1 100101

FIG. 7. Theoretical Allandeviations for thethree frequency-tracking schemes for (a)increasing detection noise (K=

{10−4,10

−3

,10

−2

,10

−1

})for aﬁxed bandwidth fL=1kHz and (b)increasing systembandwidth (fL={1Hz, 10 Hz, 100 Hz, 1kHz})

foraﬁxed K=0.1. All calculations wereperformed for aresonator time constant τr=0.4 s.

024023-8

SCHEMES FORTRACKING RESONANCEFREQUENCY... PHYS. REV. APPLIED 20, 024023 (2023)

Integration time τ(s)

Measurements

(a)

Slow

Fast

Sum

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−6 10−5 10−4 10−3 10−2 10−1 100101

Theory

(d)

Slow

Fast

Sum

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−6 10−5 10−4 10−3 10−2 10−1 100101

(b)

PLLO

SSO

Allan deviation σy

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−6 10−5 10−4 10−3 10−2 10−1 100101

(e)

PLLO

SSO

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−6 10−5 10−4 10−3 10−2 10−1 100101

(c)

PLLO

SSO

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−6 10−5 10−4 10−3 10−2 10−1 100101

(f)

PLLO

SSO

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−6 10−5 10−4 10−3 10−2 10−1 100101

FIG. 8. Acomparison of theexperimental Allan deviations with theory. (a)The Allan deviations for threeFFtracking modes with

second-order demodulation ﬁlters of bandwidth fL={100 Hz, 1kHz}.(b) Acomparison of thePLLO and SSO tracking schemes for

fPLL ={10 Hz, 50 Hz,500 Hz, 5kHz},whereaPLLFD is used as the frequency detector for theSSO scheme. (c)Acomparison of

PLLO and SSO with fPLL =500 Hz and theFFscheme (combined or summode) with fL=1kHz. (d)–(f) Theoretical model-based

computations using Eqs. (29) and (31) for thesame settings as in the measurements (a)–(c) andwith calibration forthe detection-noise

level.

cutoﬀ frequencies fL={100 Hz,1 kHz}.The experimen-

tal results show that the FF scheme in slow mode has a

slope proportionalto1/

√

τ.Changing the ﬁlter bandwidth

does not aﬀect the precisionperformance of this scheme,

because,inthe slowmode, the limiting and determining

factor is the mechanical time constant of the resonator.On

theother hand,inthe fast mode, the Allan deviation has

a1/τ slope, pointing to the fact that precision is deter-

minedby(ampliﬁed) detection noise. With alarger ﬁlter

bandwidth, the response becomesfaster but at the expense

of less precision, indicated by alarger Allan deviation. It

can also be observed that due to its high-pass but low-stop

nature, the fast responsedoesnot contain information

about slow processes such as thermal drift. Combining the

slow and fast responses, according to Eq. (11),results in an

Allan deviation that exhibits low-frequency phenomena,

while theresponse speed is limited only by thedemod-

ulationﬁlter.The experimental Allan deviations can be

recreated with high accuracy by thetheoretical model,

which is plotted in Fig. 8(d).

For the PLLO andSSO-PLLFD schemes shown in

Fig. 8(b),measurements were performed forfour diﬀerent

loop bandwidths, fPLL ={10 Hz,50Hz, 500Hz, 5kHz},

with ademodulationlow-pass ﬁlter cutoﬀ fL=5fPLL.

024023-9

BEŠI ´

Cet al. PHYS. REV. APPLIED 20, 024023 (2023)

The PI-controller parameterswerechosen according to

Eq. (23).Inthiscase,the closed-loop dynamics for the

PLLFD of the SSOand the PLLO arealmost equiva-

lent, especiallywhen fPLL τr1. Having the same ﬁl-

tering characteristicsand responsetimes for both tracking

schemes, afair comparison can be made. We observe

thatthe Allan deviations for SSO PLLFD andPLLOare

almost identical for allsystem bandwidths. As the theory

indicates, withincreasingbandwidth,the ampliﬁcation of

detectionnoisebecomesmoresevereand Allan deviations

with 1/τ dependenceinthe detection-noise-limitedregime

become worse. For fPLL =5kHz (fL=25 kHz) andasam-

pling rate of 230 kSa/s, apremature1/

√

τdependence

can be observed withaneven worse Allan deviation. This

behavior is aresult of the practical limitation of thedig-

ital PLL. Thisarises if thesampling rate is not at least

10–20times larger than the demodulator low-pass ﬁlter

bandwidth, to prevent aliasing.Itshouldbenotedthat, in

the model-based theoretical computations, it is assumed

that the sampling rateislarge enough to prevent aliasing

in Allan-deviation computations. Thereisalmost perfect

correspondence between themeasurementsand the model-

based computations, presented in Fig. 8(e),for smaller

bandwidths, when thereisnoaliasing.

Figure 8(c) shows the Allan deviations for the FF (com-

bined fast andslow response),PLLO, andSSO-PLLFD

trackingschemes forasimilar system bandwidth. Clearly,

all of the measured Allan-deviation curvesare practically

identical.The theoretical model, shown in Fig. 8(f),con-

ﬁrms that thereisnodiﬀerence in terms of performance

betweenthe three tracking schemes.

All experimental Allan deviations (exceptfor fast-mode

FF) exhibitathermal-drift-related rise for large τ.This is

not present in thetheoretical computations since thermal

drift is not modeled.

V. CONCLUSIONS

In this work, we have extendedthe existing models

for the FF and SSOtracking schemes. We have shown

that when the FF tracking scheme is operated in the sum

mode (combining the fast andslowresponse)proposed in

thiswork, it oﬀers speed versusaccuracy trade-oﬀchar-

acteristics equivalenttothoseofthe closed-loop SSO and

PLLO schemes. Thisisachieved by combining(adding)

theslow andfast frequencyresponses that can be obtained

by simple processingofthe phase outputfromthe demodu-

lator.Wehave also demonstrated that the SSO scheme has

the same frequency ﬂuctuation performance as the PLLO

scheme.Finally,wehave compared theFF, SSO, and

PLLO tracking schemes andshown that allthe tracking

schemeshave equivalent steady-state frequency ﬂuctua-

tion performance. Thereare twomainparameters that

aﬀect theperformance of all tracking schemes:(1) the

ratio between the thermomechanical-noise peak andthe

detection-noise ﬂoor, K;and (2) the ﬁltering properties of

the detection device. We have further shown that aself-

sustaining oscillator trackingschemewith apulsed drive

performs perfectly according to the theoretical model of

sinusoidal positive feedback.These results give the user an

optiontochoose the trackingschemebased on cost, robust-

ness, ease of implementation, and usability in practice

instead of fundamental diﬀerences in performance.

ACKNOWLEDGMENTS

We would like to thankAndreas Kainz and Franz

Keplinger forconstructivediscussions that gave us moti-

vationfor designing thesystem described in the paper.

This work received funding from the European Innovation

Council underthe European Union Horizon Europe Tran-

sition Open program (GrantAgreementNo. 101058711-

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024023-11

8094 IEEE SENSORS JOURNAL, VOL. 24, NO.6,15MARCH 2024

Adaptable FrequencyCounter With Phase

Filtering forResonance Frequency Monitoring

in Nanomechanical Sensing

Hajrudin Beši´

c,Alper Demir , Fellow, IEEE,VeljkoVuki´

cevi´

c, Johannes Steurer,and Silvan Schmid

Abstract—Nanomechanical sensorsbased on detecting

and tracking resonance frequencyshifts are to be used in

manyapplications.Various open- and closed-loop tracking

schemes, all offering atrade-off between speed and preci-

sion, have been studied both theoretically and experimen-

tally. In this work, we advocate the use of afrequencycounter

(FC) as afrequency shiftmonitor in conjunction with aself-

sustaining oscillator (SSO) nanoelectromechanical system

(NEMS) conﬁguration.Wederive atheoretical model forchar-

acterizing the speed andprecision of frequencymeasure-

ments with state-of-the-artFCs. Based on the understanding

provided by this model,weintroduce novelenhancements

to FCs that result in atrade-off characteristics whichisona

par with the other tracking schemes. We describe alow-cost

ﬁeld-programmable-gate array(FPGA)-based implementation

forthe proposed FC and use it withthe SSO-NEMSdevicein

order to studyits frequencytracking performance.Wecom-

pare theproposedapproachwith thephase-locked-loop-basedscheme both in theoryand experimentally.Our results

showthat similar or better performance can be achievedatasubstantiallylower cost andimproved ease of use.Weobtain

almost perfect correspondence betweenthe theoretical model predictions andthe experimental measurements.

IndexTerms—Field-programmable-gate array(FPGA), frequencycounter (FC), nanoelectromechanical system

(NEMS), phase-lockedloop (PLL),resonance frequency, self-sustaining oscillator (SSO).

I. INTRODUCTION

FREQUENCY counters areastandard equipment to char-

acterize thefrequencyﬂuctuations of oscillators and

clocks,especially in estimatingthe well-established time-

domainmeasureoffrequencystability,namely theAllan

Deviation(AD) [1],[2].The averagingofinstantaneous fre-

quency over acertain observation (gate) time, which forms the

basisfor calculatingAD, is naturally performed with astan-

Manuscriptreceived 29 November 2023; revised 12 January2024;

accepted14January 2024. Date of publication 23 January2024;

date of currentversion 14 March 2024. Thisworkwas supported by

the European Innovation Council under the European Union’sHorizon

Europe Transition Open Programunder Grant101058711-NEMILIES.

Theassociate editor coordinating the reviewofthis articleand approving

itfor publication wasDr. Chung-Chih Hung. (Corresponding author:

Hajrudin Beši´

c.)

Hajrudin Beši´

c, VeljkoVuki´

cevi´

c, Johannes Steurer,and Silvan

Schmid are with the InstituteofSensor and Actuator Systems,

TU Wien, 1040 Vienna, Austria (e-mail: hajrudin.besic@tuwien.ac.at;

veljko.vukicevic@tuwien.ac.at;johannes.steurer@tuwien.ac.at; silvan.

schmid@tuwien.ac.at).

Alper Demir is with theDepartmentofElectrical Engineering, Koç

University,34450Istanbul, Turkey (e-mail: aldemir@ku.edu.tr).

Digital Object Identiﬁer 10.1109/JSEN.2024.3355026

dard frequencycounter (FC). In this work, we propose using

an improvedFCwith high resolution and accuracy [3] as afre-

quencyshift monitor for an oscillatory signal that is generated

by aself-sustaining oscillator (SSO)-nanoelectromechanical

system (NEMS) device, as opposed to simply using it as atool

for characterizing its rawfrequencystability in thepresence

of thermomechanical and detection noise. The goal is to detect

small frequencyshiftsdue to events of interest, arising, e.g.,

from the interactionofthe nanomechanical resonator with a

mass, temperature, or force stimulus,asfastand preciseas

possible.Wedevelop atheoretical model for characterizing

the FC measurements, and showthat the averaging (gate) time

of astandard counter can be used to balance the trade-off

between the speed of detection and measurement precision.

Based on the understanding provided by this model,wepro-

pose anovel counter architecture where the simple averaging

of frequencyoveragate time that spansacrossmultiple

signal cycles is replaced by adigital ﬁlter with adjustable

bandwidththat operates on theresampled timestamps of the

signal edges.The ﬁltered timestampsare subsequently mapped

to frequencymeasurements. We show that it is crucial to

Formore information, see https://creativecommons.org/licenses/by/4.0/

BEŠI ´

Cetal.: ADAPTABLE FREQUENCYCOUNTER WITH PHASE FILTERING 8095

perform theﬁltering before the conversionofthe timestamps to

frequencyvalues, especially in cases where transduction noise

is dominant. Whilethe proposedcounter is not suitablefor

directly estimating the rawADofthe signalsource anymore,

itoffersbetter trade-offcharacteristicsasafrequencyshift

monitor.Conceptually,the output of theFCcouldbeusedto

synthesize acleaneroscillatory signal thattracksthe frequency

shifts of interest butwith subdued unwanted frequencyﬂuc-

tuations. We characterize theprecision of thecounter output

by computing the AD of thisconceptually synthesized signal.

Furthermore, we addressanissue that relates to input signal

dictated samplingrateinFCs.Our approach introduces a

robustresampling techniquethatresults in aconsistent,ﬁxed

sampling frequency. This facilitates subsequent digital signal

processing (DSP)onthe outputofthe FC, enhancing its

versatility and application scope ascompared with standard FC

designs. This method marks an improvement overconventional

implementations as documented in [3],[4],and [5].

The standardand well-established technique fortracking

the frequencychanges of an oscillatory signal source is

aphase-locked loop (PLL),where thesignalgenerated by

aclean controlled-oscillator (CO)isphase- andfrequency

locked to the noisy signal sourcewith aclosed-loopfeedback

system [6].The feedback loop is designedsothatthe CO

tracksthe frequencyshifts of interest whilesuppressing rapid

ﬂuctuations duetonoise, with theloopbandwidthserving

as thecontrol knob for trading offtrackingspeed versus

precision. The precisionofthe PLL output is characterized by

computing the AD of the CO signal. In aPLL implementation,

in additiontothe CO, aphase difference (between thesignal

source and theCOoutput)detectorisneeded to generate the

error signal in thefeedback loop. Inthe contextofNEMS-

basedsensors, PLLs areusually realized usingalock-in

ampliﬁer-basedsetup [7],[8],[9].Instead of lockingaCO

to the sensor signal to trackits frequency, we recommend a

newdesign using an FC to directly measure the resonance

frequencyofthe sensor.The sensoritself is excited by narrow

pulses withlow energyand oscillates freely [9].Weuse

areciprocal FC in acontinuousmeasurementmode where

the counterhardwareisnot reset between measurements.

This technique wasﬁrst usedinthe HP 5371 frequency

analyser [4].Itgreatly increases thenumberofsamples. The

useofcontinuous timeintervalmeasurements makesiteasier

to study the dynamic frequencybehaviorofasignal.

We compare the proposed self-sustainingoscillator(SSO)

with FC schemetothe PLL approach bothintheory and

experiment, and showthat similarorbetter performance can

be achievedwith respect to frequencyresolutionand stability

of operation.Wedescribe alow-cost ﬁeld-programmable-

gate array (FPGA)-based implementationofthe proposed FC.

Whilethe DSP in alock-in ampliﬁer canalso be implemented

on an FPGA,considerablymoreresources are needed to

implement the CO, the phase demodulators, andthe rest of the

PLL functionality.Furthermore, only alow-Q bandpassﬁlter

is used to condition thesignal for theFC. Thus, theproposed

FC-based scheme offers similar or better performance butat

asubstantially lower-cost and improved ease-of-use.

II. THEORY

A. Interpolating ReciprocalFCWithContinuous

Timestamping

We consider astate-of-the-art counter,namely an interpolat-

ing reciprocal counter with continuous timestamping [3],[5].

In order to understand the speed and precision properties of

such acounter used as afrequencyshift monitor,wedevelop

asimple model that captures its characteristics. Let fs(t)

denote theinstantaneous frequency(measured in units of Hz)

of the signal source, which includes anyﬂuctuations due to

noiseaswell as shiftsdue to events of interest. We deﬁne

φ(t)=fs(t)dt as the signal phase(unitless, equal to phase

in radians divided by 2π). In the FC, timestampsfor the

boundaries of fullsignal cycles, i.e., at the rising signal edges,

are generated using ahigh-frequency, high-precision internal

clock and an interpolator.That is,time tnwhere φ(tn)=n

(nis an integer) is measured with aclock counter forfull

clock cycles and an interpolator between twoclock edges that

precede and succeed asignal edge [5].

In thetypical settingwhere an FC is used to characterize

the frequencystability of ahigh-quality signal source, the

resolution of thetimestampsmay be limited by theclock

frequencyand thequality of theinterpolating circuitry.Inthe

application, we considerhere, the signal source exhibits rel-

atively large frequencyﬂuctuations, resulting in deviations

in thetimestampsthat are much larger than this resolution

limit. In the model,wethus assume that timestamps tncan

be measured precisely. In areciprocal counter,the frequency

of the source is estimated from the timestamps for one signal

cycle with

fc(tn)=tn

tn−1fs(t)dt

tn−tn−1=φ(tn)−φ(tn−1)

tn−tn−1=1

tn−tn−1

.(1)

Thus, fc(tn)representsthe average of the instantaneous

frequency fs(t)over one signal cycle between tn−1and tn.

The highestrate at which an FC can generateanoutput is

limited by thesignal frequency(or twice the signal frequency

if fallingsignal edges are also usedwith a50% duty cycle).

With (1),the gate time of the counter is set to thecycle time

of the signal source. Afrequencyestimate with gate timeof

kcycles can be computed with

f(k)

c(tn)=k

tn−tn−k

.(2)

If asudden frequencyshift occurs in the signal source between

tn−1and tn,its effect will be fully reﬂected in the frequency

estimate in (1) at tn+1(within twocycles), whereas it will be

at tn+k(within k+1cycles) for the one in (2).However,the

precisionofthe estimatein(2) is higher since rapid frequency

ﬂuctuations aresuppresseddue to the inherent averaging over

kcycles instead of just one. Thus, gate time can be usedas

acontrol knob for tradingoff responsespeed versus precision

in an FC that is used as afrequencyshift monitor.

The averaging inherent in (2) corresponds to asimple mov-

ingaverage ﬁlter (MAV GF). Instead, anyﬁlter that may offera

better response speed versus noise ﬁlteringcharacteristics can

be used. In order to pursue this idea, we ﬁrstneed to better

8096 IEEE SENSORS JOURNAL, VOL. 24, NO.6,15MARCH 2024

understand howfrequencyﬂuctuations affect thefrequency

estimates computed with anFC. Foraconstantnominal signal

frequency fo,weconsider

φ(t)=fo(t+α(t)) (3)

where α(t)representsthe time (phase) noiseofthe source.

Ideally,the instantaneous frequency fs(t)and thefractional

frequency ys(t)canbecomputedfrom φ(t)with atime-

derivative as follows:

fs(t)=˙

φ(t)=fo1+˙α

(

)

,y

(

)=f

(

)

o=1+˙α

(

)

(4)

where ˙α(t)represents the fractional frequencynoise.

Let us nowderivethe fractional frequencyestimate for a

reciprocal counter.Based on (3),the timestamps tnsatisfy

φ(tn)=n=fo(tn+α(tn)),which yields tn=nTo−α(tn)

where To=(1/fo)is the nominal cycle time ofthe signal.

We substitute this expression for tnin (1) to derive the

following:

yc(tn)=fc(tn)

fo=1

1−α(tn)−α(tn−1)

(5)

for the fractional frequency estimatecomputed in areciprocal

counter. The operation ((α(tn)−α(tn−1))/ To)in (5) above

corresponds to the (discrete-time)derivative of α(t)over one

cycle. Ideally,asin(4),the conversionofphase to frequency

isalineartransformation.However,inareciprocal counter,

this conversioninvolves anonlinear operation, as seen in (5).

If z(fractional frequency noise) denotesthe timederivative

of α(t),the (ideal) linear transformationtofractional fre-

quency can be represented by 1+zas in (4),whereas it is given

by the nonlinear function (1/(1−z)) in (5) for areciprocal

counter. The powerseries expansion

1−z=1+z+z2+··· (6)

indicatesthat the twotransformations are (approximately)

equal only when zis small.

Detectionnoise (generated in thetransductionofmechanical

motion into an electrical signal) in aNEMS device results

in whitephase noise [7],[8],[9].Thiscorresponds to a

frequencyﬂuctuation spectrum thatincreaseswith frequency.

Thus, zmay contain strong high-frequencycomponents.Inthis

case, thequadratic z2terminthe powerseries expansion

needs to be taken intoaccount toaccurately characterize the

ﬂuctuationsinthe frequencyestimated by areciprocal counter.

The high-frequencyspectral componentsinzmix with each

other through z2to produce low-frequencyﬂuctuations,known

as intermodulation noise.Inorder to prevent or minimize the

degrading impactofthisnonlinear phenomenon on the accu-

racy of the frequencyestimates computed by an FC, alow-pass

digital ﬁlter canbeapplied to the timestamps tn,and hence to

phase noisesamples α(tn),before thetimestamps are converted

to frequencyestimates. Ideallywhenphase to frequencycon-

version is linear,a(linear)ﬁlter maybeappliedtothe phase,

or equivalently,tothe frequencydata, since theordering of lin-

eartransformations does notchange theﬁnaloutcome. In the

case of areciprocal FC, thelow-frequencyintermodulation

noise generated inherently in the conversion of timestamps

to frequencyestimates cannot be removedwith subsequent

low-pass ﬁltering. While timestamp-to-frequencyconversion

always generates intermodulation noise,its effect will be

minimal if high-frequencyﬂuctuations are suppressedﬁrst,

before theconversion, with adigital ﬁlter.The bandwidth

and the characteristicsofthis ﬁlter can be chosen to trade

offspeed versus precision when the proposed counter is used

as afrequencyshift monitor.

B. Sampling Rate andDecimation

In areciprocal counter,the sampling rateatthe output

is determined directly by the frequencyofthe input signal,

givenby f

rate =fs/k,where kis thenumber of cycles

counted within one gate time. This input dependencyresults in

problems when subsequent DSPisperformed on the sampled

FC output.For instance, if anysort of ﬁlteringisperformed,

achange in theinput signal frequencywill consequently alter

thesampling rate, thereby affecting the dynamics of the ﬁlter.

To mitigate this issue,the input signal-dependent sampling rate

should be converted into aﬁxedone.

One approach to achieving aﬁxedsampling rate is to

implement continuous event-triggered timestamp counting as

proposed in [5].This involves using adedicated counter that

generates triggerevents at regular intervals of Tint,which

determines thesampling rate. However, thesamples cannot be

takenprecisely at multiples of Tint.Instead, they are generated

at the next rising edge of the input signal.Asaresult, there

is an inherent uncertaintyinthe sampling time, up to one

period of theinput signal. The impact of this uncertainty on

theoverallmeasurement varies depending on thenumber of

signal periods encompassedwithin one interval. Whenthere

are numerous signal periods,the uncertaintyhas alesser effect.

However, if thereare onlyafewsignal periods,the irregular

sampling interval introduceserrors that can affect the accuracy

of the measurement.

Although the sampling rate fs/kis directly linked to the

input frequency fs,the sampling instants always alignwith

theinternal clock (with frequency fCLK)edges of theFC. The

sampling rate can be transformed up to fCLK by simply inter-

polating the acquired datathrough the use of azero-orderhold.

However, this introduces high-frequencyharmonics due to the

abrupt transitions between the samples.Toaddressthis, alow-

pass ﬁlter (LPF) can be applied to attenuatethe harmonics.

The combined process of low-passﬁltering anddecimation

(to aﬁxedfraction of fCLK)after thezero-order hold can

be efﬁciently achieved using acascaded integrator-comb ﬁlter

(CIC), as describedin[10].Fig. 1(a) depictsthe second-order

CIC ﬁlter employed in this study,featuring twointegrator

sections and twocomb sections. Thecomb sectionsintroduce

adelay of N=2. The downsampler,with avalue of R=213,

is positioned between the integration and comb sections.Thus,

thesampling rate of the ﬁnal output is givenby f

new =

fCLK/R,independent of the input frequency fs.Fig. 1(b)

shows the transfer function of theCIC ﬁlter after decimation.

Notably,itdoes not exhibitadistinct separationbetween the

passband and stopband, thusnecessitating further ﬁltering with

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Cetal.: ADAPTABLE FREQUENCYCOUNTER WITH PHASE FILTERING 8097

Fig. 1. (a) Blockdiagramofthe second-order CICdecimator with the

decimation factor Rand the comb section delay N.(b) Transfer function

of the decimated output.

aﬁnite impulse response (FIR)orinﬁnite impulse response

(IIR) ﬁlter.

The speed of theresponse to afrequencyjump is intrin-

sicallylimited by theinput signal frequencyasdescribed

in (1).When the data is resampledusingthe CIC ﬁlter,

it results in additional ﬁltering. This additionalﬁltering could

potentially slowdownthe response,particularly if thenew

sampling frequency(f

new)isless than the original sampling

rate ( frate). However, therangeoffrequencystepsthat this

methodcan handle is theoretically unlimited.Itiscapable

of tracking frequencystepsofany magnitude,making it

exceptionally suitablefor gatheringdata from devices that

requiremonitoring across abroad range of frequencies.

C. Allan Deviation

AD σy(τ ) is awidely used and well-established method

for characterizing frequencyﬂuctuations [7],[11],[12].For

AD,the frequencyvalues need to be normalized,resulting in

afractionalfrequency

y(t)=ω(t)

ω0

.(7)

AD is thesquare rootofthe AllanVariance, which can be

computed from sampledfrequency data usingthe following

equation:

σ2

y(τ)=1

2(N−1)



i=1yi+1,τ −yi,τ 2.(8)

Here, yirepresents the ithsample of the averaged frequency

overthe averaging time τ.Itiscalculated as

yi,τ =1

τiτ

(i−1)τ

y(t)dt.(9)

Thus, the averaging operation above has to be part of the

frequencymeasurement and data acquisition process. An FC

naturally performs this operation when it is used to char-

acterize the rawfrequencyﬂuctuations of its input signal.

On the other hand, (8) is routinely used in practice on

measured frequencydata, also in cases where an FC is not

used and/or theaveraging operation is not inherently included

in themeasurement process. Such use is justiﬁed only when

thesampling rate is sufﬁciently large when compared with the

system bandwidth, i.e., the smallesttime τthat determinesthe

samplinginterval is small enough. In this case, the frequency

that is being measured is (almost) constant overthe sampling

interval(smallest τ)resulting in

yi,τ =1

τiτ

(i−1)τ

y(t)dt ≈y(iτ).(10)

Averaging when τis equal to an integer mmultiple of

thesampling interval is performed by simply averaging m

consecutive samples of the acquired rawdata.

When an FC is employed as afrequencyshift monitor,aswe

propose in this work, in contrastwithits useincharacterizing

thefrequencyﬂuctuations of its input signal, AD should be

computed for the conceptually synthesized oscillatory signal

based on thecounter output.Thus, theaveraging in (9)

needs to be performed in addition to theinherent averaging

performed by thecounter itself.While theinherent averaging

of the counter is over atime intervaldetermined by its gate

time speciﬁcation,the average in (9) is computed for all τthat

is of interestinthe AD characterization.

The Allan Variance can alternatively be computed in the

frequencydomainifthe power spectral density of fractional

frequencyﬂuctuations, denoted by Sy(ω),isknown.The

equation for this computation, shown below,

σ2

y(τ)=1

2π

τ2∞

0sinωτ

24

ω2Sy(ω)dω(11)

includes the frequencydomain equivalent of the averaging

operation in (9),shouldthus be used on Sy(ω) which was

measured or computed without inherent averaging. Forwhite

frequencyﬂuctuations,where Sy(ω) is constant, (11) simpliﬁes

σ2

y=Sy(0)

2τ.(12)

Hence, in systems limited by thermal white noise, the resulting

AD exhibits a σy∝1/√τdependence on the averaging

time τ.

This study focuses on twoprimarynoisesources,ther-

momechanical noiseand detection noise. Thermomechanical

noise is regarded as thefundamental noise source in NEMS

resonators, resulting from the random movement of resonator

molecules.Onthe other hand, detection noise arises from

the transduction of the resonator’smechanical motion into

an electrical signal,aswell as from electroniccomponents

involved in thedetection process.

The power spectral density of frequencynoise, Sω(ω),can

be computed as asuperpositionofthe power spectral densities

of thermomechanical anddetection phasenoise,multiplied by

8098 IEEE SENSORS JOURNAL, VOL. 24, NO.6,15MARCH 2024

their corresponding transfer functions (magnitude squared) as

derivedin[7] and [9].Thisyieldsthe spectral density of

fractional frequencynoise requiredfor computingthe AD

Sy(ω)=Sω(ω)

ω2

=Sθth (ω)|Hθth (jω)|2+Sθd(ω)|Hθd(jω)|2

ω2

=Sθth (ω)

ω2

0|Hθth (jω)|2+K2|Hθd(jω)|2(13)

where K=(Sθd/Sθth )1/2is theratio between the thermal and

detection noise and Hθth and Hθdarethe transfer function of

thethermomechanical and detectionnoise to thefrequency

output, respectively.For theSSO frequencytrackingscheme,

the transfer functionsare givenby[9]

HSSO

θth (s)=1

τr

HL(s)

HSSO

θd(s)=1

τr

Hr(s)HL(s)(14)

where Hris asingle-pole LPF withthe time constant of

theresonator. HLhaslow-passcharacteristics and represents

the bandwidth limiting(noise ﬁltering)mechanisms in the

frequencydetectiondevice. The theoretical computation of

AD is performedwiththe frequencydomain approach in (11),

where thespectral density of fractionalfrequencynoiseisﬁrst

computed usingafrequency domain model of theSSO and the

FC, as in (13) and (14),that includesintermodulationnoise

generatedinthe timestamp to frequencyconversion.

III. METHODS

We describe the experimentalsetup forthe comparison of

the twomethods, theproposed FC andthe PLLfrequency

detector(PLLFD),for frequencyshift monitoring of an

oscillator.Our measureofassessment for precisionisthe AD.

Central to ourinvestigationisthe SSO, as detailed in [9],

drivenbynarrowpulses and oscillatingfreely.The pulse

duration andtiming is automatically adjustedtoattain the

desiredresonance frequencywithin aclosed-loop setup

involving the resonator and thepulsegenerationmechanisms.

Notably, the frequencymeasurementand detectionoperate

outsideofthis loop. Our study involves resonancefrequency

measurements using theproposedFCand theestablished

PLLFD, as depictedinFig. 2(b).Wecomputethe AD in two

experimental conditions, using theFCoutput andthe PLLFD

output. Forthe FC, weexploreﬁlterconﬁgurations,cut-off

frequencies, and timestamp versus frequencyﬁltering.

A. NEMS Resonator Setup

In thisstudy,weutilized aNEMSresonatorconsistingof

asquare 50-nm thick silicon nitridemembranemeasuring

1018 µmoneachside, as introduced in [9] andshown in

Fig. 2(a).Toachieve electrical transduction, we incorporated

two5µm-wide gold (Au) electrodes spanningoverthe res-

onator.The membrane wasplaced within astatic magnetic

ﬁeld of approximately 0.8T,generated by aHalbach array

composedofneodymium magnets. The orientationofthe

Fig. 2. (a) Picture of the NEMS resonatorusedinthe experimentsand

its connectiontothe setup. (b) Blockrepresentation of the SSOtracking

scheme with the FC and the PLLFD.

traces wasperpendicular to the magnetic ﬁeld.Bycapitalizing

on the resulting Lorentz force, one metal trace is served for

the purposeofdriving theresonator with an ac current and

the second electrode for detecting its motion through the

magnetomotively induced voltage.

To amplify the detected signal from themetal trace,

we employed acustom-made, low-noise differential

pre-ampliﬁer with again factor of 104.The NEMS was

operated in vacuum with apressure of 8.2·10−6mbar. The

NEMS resonator had aresonance frequencyof f

r=119 kHz

and aquality factor of Q=57.5k.Consequently,the

response time of theNEMS resonator is calculated with

τr=2Q/ωr,is154 ms.

B. Self-SustainingOscillator

The resonator utilized in this study operates as an SSO

(implemented in PHILL from Invisible-Light Labs GmbH),

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Cetal.: ADAPTABLE FREQUENCYCOUNTER WITH PHASE FILTERING 8099

Fig. 3. Blockdiagramofthe proposedFCarchitecture,where sin is the input signal, fis the frequencyoutput.

as described in [9] anddepicted in Fig. 2(b).The transduced

output of the NEMS resonator isconnected to apreampliﬁer

(PA). The ampliﬁed signal is thendirectedtoabandpass

ﬁlter(BPF) (with gain 1and bandwidth 5or20kHz for

differentmeasurements). The BPFinsidethe SSO loop is not

needed to improve theSSO performance, andinsome cases,

it is notnecessary.However,itgives thesystemthe ability

of mode selectionbysuppressingthe buildup of unwanted

modes. On the otherhand, the BPF bandwidthlimits the

systemresponse speed. Therefore,itisimportanttoensure

that the bandwidth of the BPF islarge enough in order to

prevent it from becomingalimitingfactorinthe system’s

overall response speed. In the context of this work,where the

NEMS is employed as an infrareddetectorwith athermal

response time constant (τresp)rangingfrom50to200 ms,

aBPF bandwidth exceeding 1kHz is deemed adequate.The

output of thebandpassﬁlter is linked to acomparator(COMP)

with a50mVhysteresis,whichtransforms thesinusoidal

signal intoarectangular waveform,triggeringthe pulse gen-

eration mechanismthat drives theNEMS resonator. Thepulse

generation mechanism iscomprised of twocomponents:one

generates apulsewith awidth of Tw,while the other delays

thepulse generated at the feedbackoutput by atime of Td.

Since theNEMS resonator can only toleratelow currents, the

generated pulseneeds to be attenuated (ATT) by afactor of

105before being applied to theNEMS.

Frequencydetectionisaccomplished usingtwo different

methods, the PLLFD and our proposedFC. In principle, the

PLLFD andFCcan be connectedanywhereinthe SSO loop.

Beforethe dataisacquired by both frequencydetectors, anti-

alias ﬁltering has to be performed.The anti-aliasingﬁlter will

limit the noise goingintothe frequencydetectorand will take

partinthe HLﬁltering termof(14).Itwill also prevent noise

foldingwhich would degrade the resultingADasshown in [9].

Forthe PLLFD, this anti-aliasing functionisserved by theLPF

in its phase detector,and the PLLFDisout of convenience

connected after thePA. On theother hand, to avoid aliasing

in the FC, the signal must ﬁrst pass throughabandpassﬁlter

with abandwidth less than half of the input signalfrequency,

which also coincides with the maximum sampling rate of the

FC. Fordevices similar to the one usedinthiswork, BPF

bandwidthsbelow50kHz will fullﬁll thecriteria. To eliminate

the need for twobandpass ﬁlters (one in theSSO loop and

one at the input of theFC),the FC can be connected at any

position in betweenthe bandpass ﬁlterand theresonator,and

thein-loop BPF will act as an anti-aliasing ﬁlter. Thus, the

FC is connected to theoutput of thebandpass ﬁlter.

With the proposed technique, trackingthe resonance fre-

quencyofadevice does notrequire theconstructionofa

complexcontrol system. Knowledge of thereadout method,

arough estimation of the resonancefrequencyand the

response speed of thedevice aresufﬁcient to ﬁnd an oscillation

and track its frequency. The devices utilized in this study are

characterized by the resonant frequency105 kHz <fr<

125 kHz and responsetimes 50 ms <τ

resp <200 ms.Notably,

the magnetomotive readout employed measures the velocity,

rather than theposition,ofthe resonator.This method induces

a(π/2)phaseshift relative to theposition.Consequently,the

−(π/2)phaseshift (from thedrive to position)intrinsic to the

resonator is neutralized by the readout’s phaseshift, resulting

in anet in-loop phase of 0. This analysis assumes minimal

electrical delays at lower frequencies, although suchdelays

may warrant consideration at higher frequencies.Hence, in this

context, the pulsedelay can be set to Td=0. If an optical

readout that does not cancel out thephase delay caused by the

resonator is used, thepulse delay shouldbeset to

Td=3π/2

2πfBPF

(15)

where fBPF is the bandpass central frequency. The pulse delay

element will generate a −3(π/2)phaseshift (time delay

corresponds to anegative phaseshift)at f

BPF.The BPF

exhibits aphase shift of 0onlyatthiscentral frequency.

By sweeping thecentral frequencyofthe BPF,the loop phase

condition will only be met if fBPF =frand an oscillation will

be excited.

C. FrequencyCounter

Fig. 3illustrates theblock diagram of the FC archi-

tecture. The FC employed in this study is an enhanced

version of astate-of-the-art interpolating reciprocal counter

with continuoustrigger events (implemented in PHILL from

Invisible-Light Labs GmbH).Ithas been modiﬁed to better

suitthe taskoftracking resonance frequencychangeswith

improved usability.

The front-end of theFCisafrequencydivider,which

enables counting kcycles of the input signal,where kcan

be anyinteger and speciﬁes thenumber of counted periods

in one timestamp.Toavoid aliasingand noise folding, it is

recommended to keep kas small as possible, i.e., setitto

k=1, since the sampling rateofthe counter, frate,isdirectly

dependent on the input signal frequency, givenby f

rate =fs/k.

The subsequent stage of the system is the main counter

with an interpolator.Itfunctionsasastandardreciprocal

counter,tallying the number of rising edges of the internal

clock that occur in an interval of kperiods of the input signal

and generating the timestamps for the interval boundaries.

However, due to thepotential error of up to one clock cycle

in such acounter,aninterpolation mechanism is employed

to enhance the precisionofthe timestampsupto100 ps at

100 kHz.

8100 IEEE SENSORS JOURNAL, VOL. 24, NO.6,15MARCH 2024

The output rate of themain counterisdirectly linkedtothe

frequency of its input, necessitatingresamplingtoachieve a

ﬁxed sampling rate. First, the timestamp seriesisinterpolated

ontothe internal clock transitions time grid usingazero-

orderhold. Second, it is decimated to thedesired sampling

rate throughaCIC decimator.Third, theresampled timestamp

series is low-pass ﬁltered,whichdeﬁnesthe ﬁnal system

bandwidth as speciﬁed by HLin (14).Finally,the ﬁltered

timestamp series is converted to the frequencyoutput using (1)

or (2).

Constructingadigital PLL entails threecritical components:

an analog-to-digital converter (ADC),adigital-to-analog con-

verter (DAC), and ﬁeld-programmable gate array (FPGA)

circuitry. Forminimizingtimequantizationerrors, the ADC

and DAC typically operate at high samplingrates, often

exceeding 40 MHz, and generally requirearesolution of

14 bits.Itisimportant to notethatboththe ADC and the

DAC,particularly thosewith such speciﬁcations, are expen-

sive components.Onthe FPGA, implementingdemodulators

and proportional-integral-derivative (PID)controllers for the

PLLinvolves multiplicative operations. This necessitatesusing

FPGAs equipped with DSPslices forefﬁcientexecution.

However, FPGAswith DSPslices areconsiderably more

expensive than their non-DSP counterparts.

Conversely,anFC’sarchitectureincludesaCOMP,

an FPGA, and an interpolator.COMPs are relatively inex-

pensivecomponents and can even be omittedifthe counter

is connected post theSSO’sCOMP, as utilized in this study.

Theinterpolator can be integrated internally viaatime-to-

digital converter (TDC), as discussed in [13],orimplemented

using cost-effective integrated circuits like theTDC7200.

As illustrated in Fig. 3,all componentsofthe FC can be

incorporated intoaneconomicalFPGA without DSP slices.

Thisaspect renders theFCamorebudget-friendlyalternative

compared to the digital PLL.

IV.RESULTS AND DISCUSSION

We conducted10smeasurements(as showninFig. 4)

to analyze theoutput frequencyﬂuctuations due to ther-

momechanicaland detection noisefor various frequency

shift detection scenariosand systemparameter choices.The

experimental setup, which incorporatedmagneto-motive read-

out, introduced signiﬁcant detectionnoise. Consequently,the

thermomechanical noise peak couldnot be resolved at the

resonatoroutput, resulting in avalue of K>1. As discussed

in [9] and [11],this leads to an ADwith a1/τ dependencyif

the system time constant is shorterthanthe resonator time

constant.When K<1, the sensorisintrinsicallylimited

by the thermomechanical noise of theresonator, notbythe

noisegeneratedinthe transductionand thefrequencydetec-

tion mechanisms. Also in such thermomechanically limited

systems, ﬁltering in thefrequencydetectorasrepresented by

HLin (14) can be used to trade offspeed versusaccuracy.

However, the enhancement techniques proposedand that are

to be demonstrated below,while beneﬁcialfor considerably

alleviating detection noise inﬂuence, will nothavethe same

levelofimpact on systems where thermomechanical noiseis

Fig. 4. Time-domainsteady-state measurements acquired by the

main counter (with interpolator block), using twodifferent bandpass ﬁlter

bandwidths: 20 and 5kHz.

the dominant factor.Inthe following,weexamine theFC

under four different conditions.

A. FilteringofTimestamp Data VersusFrequency Data

In the ﬁrst experiment,shown in Fig. 5,wecompare two

alternativesfor an FC, with conversion of timestamp seriesto

frequencydata after or before low-pass ﬁltering.Asarticulated

in Section II,the expected 1/τ dependence of the AD is

altered to 1/√τfor large τ.This is due to themixing of

thehigh-frequencynoise componentsbythe nonlinearity of

thetime-to-frequencyconversionprocess, resulting in inter-

modulation noiseatlower frequencies. Thiscan be alleviated

by removing or reducing high-frequencynoisebefore time-to-

frequencyconversion by low-passﬁlteringthe counteroutput

in the form of atimestamp series. The ADs of thesemeasure-

ments are shown in Fig. 5showing that ﬁlteringthe timestamp

data ﬁrst and then conversiontofrequencyyields better AD

than conversion before ﬁltering,aspredicted by theory.

B. Sweeping theGateTimeofthe Counter

In the second experiment, we investigate the impact of

varying the number of counted periods kof thesignal, i.e.,

thegate time of the counter,onthe performance and the

ﬁltering process in theFC. As seen in Fig. 6,increasing k

seemingly does not lead to an improvement in the AD without

suppressingit(raw).Incontrast, low-passﬁltering the FC

output results in improvedADs. The bestADisobserved for

k=1with theLPF.

ADs for the unﬁltered rawdata at the counter output fall

exactly on top of each other when kis varied, albeitstarting at

larger τvalues for larger kdue to the larger sampling interval

of kperiods. Thisresult is puzzling at ﬁrst thought, since

theoretical considerationsindicate that we shouldbeable to

use thegate time of the counter as acontrol knob for trading

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Cetal.: ADAPTABLE FREQUENCYCOUNTER WITH PHASE FILTERING 8101

Fig. 5. Effect of the position (before or aftertime-to-frequencyconver-

sion)ofaﬁrst-order LPF withacut-off frequency of200 Hz on the AD.

The “Raw” curve is forthe rawunprocessedoutput of the FC without any

subsequent ﬁltering. The“LPF time”curve corresponds to the output

of the proposedFCwithlow-pass ﬁltering before time-to-frequency

conversion. The “LPF frequency” curverepresents the use-caseof

ﬁltering the output of aconventional reciprocal counter where time-to-

frequency conversionhas already been performed. The SSO loop BPF

with a20-kHz bandwidth wasused forall experiments.

Fig. 6. Comparisonbetween the ADs of rawand ﬁltered (in the FC with

aﬁrst-order LPF with acut-off frequency of 200Hz) datafor varying gate

time as set with kcounted periods.The SSO loop bandpass ﬁlter (BPF)

used in this experiment hadabandwidth of 5kHz with the corresponding

time constant τBPF.

offresponsespeed versusprecision. While theresponsetime

of the counter is deﬁnitely prolongedwith increased gate time

(sudden frequency shiftatthe input will be fully reﬂected in

the counter output after kperiodsofthe input,asdiscussed

before), results in Fig. 6suggestthat thereisnoimprovement

in theprecisionofthe output.

The seemingly unexpected result in Fig. 6can be resolved

and deciphered as follows.The sampling rateatthe output of

the counter is inverselyproportional to thegate time, with

the counter producing ameasurement for every kperiods

of theinput signal. When AD is computed based on this

sampled frequencydata for the smallest τof kperiods, the

only averaging in the sense of (9) reﬂected in the data is

theone that is inherently performed by the counter front-end,

with no additional averaging in the actual AD computation

as discussed before. This means that the AD computed as

such is actually acharacterization of thefrequencyﬂuctuations

of thecounter input signal,asopposed to the conceptually

synthesized signal based on the counter output. Thus, it makes

perfect sense that computedADs for varying kfall exactly on

topofeach other,since theyall represent acharacterization of

the input signal, not the counter output.Another perspective

on theseemingly puzzling results in Fig. 6is as follows. With

increased gate time, the sampling rateatthe outputofthe

counter is not large enough to justify the use of the frequency

data to compute the AD (for the counter output) with (8),

since (10) does not hold in this case, and the additional

averaging that needs to be part of the AD computation is

missing.

While we resolved theresults in Fig. 6based on theoret-

ical arguments, it is desirable to experimentally observethe

precision improvement one can obtain in an FC by increasing

thegate time. This requires somehowincreasing the sampling

rateatthe counter output. With agate time of kinput

periods,one can attainak-foldincreased sampling rate by

employing asequence of kparallel counter front-ends, where

each front-end counts over kperiods of the input signal,

butwith one period delay relative to the previous one in an

interleavedmanner.However,this would increase thehardware

cost andcomplexity considerably.Instead, we simply emulate

kparallel counter front-ends by ﬁrst producing output fromthe

counterwith k=1for every period of the signal,and then

processing this output with an MAVGFwith awindowlength

of k.Apart from alarger latencyatthe output, this emulation

is hardware-equivalenttohaving kparallel counter front-ends

and does not involveany approximations. One can obtain the

frequencydata at the k-fold lower sampling rate(output of

only one of the kfront-ends,used to generate Fig. 6)bysimply

downsampling the MAVGFoutput with adownsampling ratio

of k.

The ADs computed from the rawcounter output with k=1,

the MAVGFoutput with k=121, thedownsampled moving

average output, as well as LPF processed versions of the mov-

ingaverage output and its downsampled version, are shown

in Fig. 7.Asexpected, we observethe perfect coincidence of

the AD curvefor thedownsampled moving average output

with the curvefor theraw counter output.Wenow also

observethe theoretically claimed precisionimprovement at

the counter output with increased gate time (emulated with

the moving average output) even when there is no extra low-

pass ﬁltering. With increased samplingrate, computation of

AD with (8) is justiﬁed since (10) nowholds.The additional

8102 IEEE SENSORS JOURNAL, VOL. 24, NO.6,15MARCH 2024

Fig. 7. Measurement-based model ofthe FC acquisition mechanism

emulating alarge gatetime with k=121 by applying an MAVGFtodata

acquired with k=1and downsampling, with comparisonoflow-pass

ﬁltering at different stages.(Raw: no ﬁltering; MAVGF: moving average

ﬁlter; LPF:low-pass ﬁlter.)

averaging required in ADcomputationfor τvalues larger than

the sampling intervalisperformedusingthe data samples

available at the higher rate.

Further,low-pass ﬁltering thedownsampledmovingaverage

output results in aprecision improvement as indicated by

Fig. 7,but not as much as the one weobservefor the

moving average output at the highersamplingrate and its

ﬁlteredversion.Onthe other hand, it seems strange that the

moving average output has higherprecisionwhencompared

with itsdownsampledversion.Inthe end, speciﬁc frequency

measurement values in the downsampleddata are exactly

equal to asubset of the values in thehigherrate data, and

therefore should have thesame precision. In fact, use of AD

in ordertocharacterize the precisionofthe downsampleddata

isnot appropriate since (10) does not hold. However, one can

simply compute the standard deviationofthe downsampled

data, which is in fact equal to thestandard deviationofthe

higher-rate date assuming statisticalstationarity. AnyDSP

with memory,that involves dynamics over time (including AD

computation), on the downsampled data is not meaningful,and

introduces aliasing andnoise foldingdue to thelow sampling

rate.When gate time is set to thelowestvalue, theinput signal

period with k=1, the counterfront-endaveragesoverthe

signalperiod Ts,which is alsoset to thesampling interval

at the counter output. Therefore, afront-endbandpassﬁlter

with amaximum bandwidthset tohalf thesignalfrequency

is neededtoprevent aliasing and noise folding, even when

k=1.

We thus conclude that, even though gate time can be used

as acontrol knobfor tradingoff precisionversusresponse

speed whenanFCisused as afrequencyshift monitor,

it is better to set thegate time tothe smallest valuewith

Fig. 8. ADs showing that resampling and ﬁltering the signal at

aﬁxedsampling rate achieve the same performance, whereas the

event-triggered timestamp method results in adegradation. (Raw: no

ﬁltering; LPF: low-pass ﬁlter.)

k=1and generateoutput at the largest ratepossible,with

an appropriatefront-end bandpassﬁlter to prevent aliasing.

Then, instead of emulating alarger gate time with asimple

MAVGF(aspeciﬁc type of FIR ﬁlter), it is better to usean

appropriate, bandwidth adjustable FIR or IIR digital ﬁlter that

can offer abetter precision versus speed trade-off. Forthe

results we present in this work, we used aﬁrst-order IIR LPF.

C. Resampling foraFixed Sampling Rate

In the third experiment, we consider resamplingofthe FC

output,which has an input-dependent sampling rate, to aﬁxed

samplingfrequency. We compare ﬁlteringofthe rawoutput

and the resampled version. We alsocompare our proposed

resampling technique with the continuous event-triggered

timestamp method described in [5].

We consider and compute sixversions of the frequencydata

as follows.

1) The rawmain counter output (for k=1) with an

input-dependent sampling rateisgenerated.

2) The rawoutput is passed through aﬁrst-order Butter-

worthLPF with acut-offfrequencyof200 Hz.

3) The rawoutput is processed using theevent-triggered

timestamp methodwith Tint =100 µs.

4) The output processedwith theevent-triggered timestamp

method is also passed through the LPF.

5) The rawoutput is ﬁrst processed with azero-orderhold

to increasethe sampling frequencytothe frequency

fCLK =76.92 MHz of the internal clock. Subsequently,

it is decimated using aCIC decimator withadecimation

factor of R=8192, resulting in aﬁnal samplingrate of

9.4kHz.

6) The resampled dataispassed through the LPF.Wenote

thatthe digitalLPF is implemented at the respective

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Cetal.: ADAPTABLE FREQUENCYCOUNTER WITH PHASE FILTERING 8103

Fig. 9. ADs of raw(unﬁltered) and low-pass ﬁltered FCoutput in

comparison with aPLLFD withthe same bandwidth. Resultsshowthat

there is no difference in performance between acommercial PLLFD with

the FC proposed in this work.

sampling rate of thedataitisappliedto, corresponding

to thesame cut-offfrequency in each case.

TheADs for the sixversionsoffrequencydata described

above areshown in Fig. 8.The event-triggered timestamp

method produces an uncertaintyinthe samplinginstantsup

to oneperiod of the input signal,whichmanifests itselfinthe

form of additional frequencynoise resultinginalarger AD,

discernible in Fig. 8right before thermal drift kicksin. The

additional frequencynoise produced by this technique, which

cannot be suppressed, is even morenoticeable after thedatais

processed with the LPF.Onthe other hand, thedata produced

by ourproposed resamplingtechnique exhibits aslightly

improvedADcompared to the rawmain counter output.

This improvement can be attributedtothe inherentlow-pass

ﬁltering performedbythe CICdecimator.Furthermore, if the

resampled signal is further processed with an LPF, the AD

obtained is identical to that when theraw counter output is

low-pass ﬁltered with thesame cut-offfrequency.

D. Proposed FC Versus PLLFD

In theﬁnal experiment, we compare theproposed FC with

acommercial, lock-in-based PLLFD.The PIDcoefﬁcients of

the PLLFD are generated by the software that comeswith the

equipment, targeting aloop bandwidth of 200 Hz,resultingin

the values kp=2.92 Hz/degand ki=13.34 Hz/deg/sec. The

LPF in the PLL demodulatorisaﬁrst-orderﬁlter with acut-off

frequency of 1kHz, and the PLL operates at asampling rate

of 27 kHz. The transducedNEMS output after thepreampliﬁer

is fed to the PLLFD, whereas it is processed with abandpass

ﬁlter(with abandwidth of 5kHz)toproducethe input for the

FC. The FC rawoutput (with k=1) is processed with aCIC

decimator with adecimationfactor of R=8192, resulting in

aﬁnal sampling rateof9.4 kHz. The output of thedecimator

is processedwith aLPF with acut-offfrequencyof200 Hz.

Fig. 9illustrates the comparisonbetween the state-of-the-

art PLLFDmethod for frequencytrackingand the proposed

FC-based technique. It can be observed that both methods

exhibit almostthe same performance in bothmeasurementand

theory.

V. CONCLUSION

In this study,weinvestigated various aspects of frequency

shift monitoring mechanisms based on FCs for resonant

sensors. We characterized their precision, both in theory

and experimentally,inthe presence of thermomechanical and

detection noise. Through theoretical models,analyses, and

articulate arguments, combined with aseries of experiments

andelucidated results,wehavegained valuable insights into

various scenarios,system architectureand parameter choices.

We proposed anovel and cost-effective FC-based frequency

shiftmonitoringscheme, which wasoverlooked in the NEMS

literature. Our FC-based architecture featureswide bandpass

ﬁltering for signal conditioning combined with digital low-pass

ﬁltering in the sampled data domain before timestamp series

for thesignal transitions are convertedtofrequencydata.

This architecture not only alleviates the detrimentaleffect

of intermodulation noise generated by thenonlinearity of

time-to-frequencyconversion, butalsoenables aﬂexible and

practical platform for real-world applications where all DSP

is performed at aﬁxed, input-independent sampling rate.

We investigated mechanisms for trading-offresponse speed

versus precisioninour proposed FC-based scheme. Although

we have shown that the gate time of acounter can be used

as acontrol knob for this purpose, it is moreeffective and

efﬁcient if thegate time is settothe smallest possible value,

i.e., thecycletime of the input signal,togenerate an output

at the largest sampling ratepossible. Then, speed versus

precisiontrade-offcan be conveniently achieved by modifying

the digital ﬁltering characteristicsinamore ﬂexible manner.

The output of astandard reciprocal counter,where gate

time is set to thecycle time of the input signal, has an

input-dependent sampling rate. Thisisinconvenient forthe

subsequent DSP. We developed aresampling schemethat

involves azero-order hold and aCIC decimator to convert the

output to aﬁxed, input-independent sampling rate. We have

shown experimentally that the resampled and ﬁltered output

has thesame (or slightly better)precision as the ﬁltered raw

output of the FC as characterized by ADs.

Finally,wecompared the precisionofthe proposed

FC-based scheme with acommercial implementation of the

common andstandard PLLFD techniqueinterms of ADs.

Our results showed that both methods achievethe same

performance, indicating that theproposed FC-based frequency

trackingmethod can serveasaviable and cost-effective

alternative to the state-of-the-art PLLFD method.

Overall, our experimental results, theoretical models, anal-

yses,and ﬁndings contribute to abetter understanding of

frequencydetection mechanisms.Thisunderstandingcom-

bined with our FPGA-based ﬂexibleand cost-effective

platform pavesthe waytodeveloping newfrequencyshift

8104 IEEE SENSORS JOURNAL, VOL. 24, NO.6,15MARCH 2024

monitoring schemes with enhanced precision, responsespeed,

and/or reliability.

Looking forward, theinsightsgained fromour study open

up several avenues for further research and development in the

ﬁeld of precision frequencymeasurementfor sensingapplica-

tions. There is potential foradvancing theDSP techniques

used in ourarchitecture. Machine learning-aided algorithms,

for instance,could be employed todynamically optimize and

adaptﬁltering parametersinreal time,potentially leading to

even moreprecise and responsivefrequencytracking. Our

research lays thegroundwork forexcitingdevelopments,and

we anticipatethat the concepts and techniquespresented in

this work will contribute to innovations in sensorsbased on

frequencyshift monitoringinthe near future.

ACKNOWLEDGMENT

The authors acknowledge TU Wien Bibliothek forﬁnancial

support throughits OpenAccess Funding Programme.

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[10] R. G. Lyons, Understanding Digital Signal Processing.London, U.K.:

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[13] F. Dadouche, T. Turko, W. Uhring, I. Malass, J. Bartringer,and

J.-P.LeNormand, “Design methodologyofTDC on lowcost FPGA

targets,” Proc. 9thInt.SENSORCOMM,pp. 29–34, 2015.

30264 IEEE SENSORS JOURNAL, VOL. 24, NO.19, 1OCTOBER 2024

OptimizedSignal Estimation in Nanomechanical

PhotothermalSensing via Thermal Response

Modeling andKalman Filtering

Hajrudin Beši´

c,Andreas Deutschmann-Olek ,Kenan Meši´

c, Kostas Kanellopulos ,

and Silvan Schmid

Abstract—Wepresent an advanced thermal response

model formicro- andnanomechanical systems in photother-

mal sensing, designed to balance speed and precision. Our

model considersthe twotime constants of the nanome-

chanical element and thesupportingchip, triggeredby

photothermal heating, enabling precisephotothermalinput

signalestimation through Kalmanﬁltering. By integrating

heat transfer and noise models, we applyanadaptive Kalman

ﬁlter (AKF) optimized forﬁeld-programmable gate array

(FPGA) systems in real time or ofﬂine. This method, used

forphotothermal infrared (IR) spectroscopywith nanome-

chanical resonatorsand aquantum cascade laser (QCL) in

step-scanmode,enhances response speed beyond standard

low-pass ﬁlters(LPFs),enabling faster data acquisition and

reducingthe effects of drift and random walk. Measurements

showthe signiﬁcantimpact of the thermal expansion coefﬁ-

cients’ratio on the frequencyresponse.The AKF,informedbythe QCL’sinput characteristics, accelerates the system’s

response,allowingrapid and precise IRspectrum generation. The use of the Levenberg–Marquardt algorithm and power

spectral density (PSD) analysis forsystem identiﬁcation further reﬁnes our approach, promising fast and accurate

nanomechanical photothermal sensing.

IndexTerms—Frequency counter,infrared (IR) spectroscopy, Kalman ﬁlter,nanoelectromechanical system (NEMS),

resonance frequency, self-sustaining oscillator(SSO).

I. INTRODUCTION

NANOMECHANICAL resonators, pivotal forphotother-

mal sensing through resonancefrequencytracking,

ﬁndextensive utility in ﬁeldssuchasinfrared (IR) spec-

troscopy [1],[2] and broadband thermalIRdetection [3],

[4],[5],[6],[7],[8],[9].Demonstrated by these studies,

Manuscript received 29 July 2024; accepted15August 2024. Date

of publication29August 2024; dateofcurrentversion 2October 2024.

Thisworkwas supported in partbythe European InnovationCouncil

throughthe European Union’sHorizon Europe Transition Open Program

under Grant 101058711-NEMILIES,inpartbythe Defense Advanced

ResearchProjectsAgency (DARPA) Optomechanical Thermal Imaging

(OpTIm) Technical Area (TA) 1BroadAgency Announcement (BAA)

underGrant HR001122S0055, and in partbyTUWienBibliothek

through the Open Access Funding Program. The associateeditor coor-

dinating the review of thisarticle andapprovingitfor publication was

Prof.Benoit Gosselin. (Corresponding author:Hajrudin Beši´

c.)

Hajrudin Beši´

c, Kenan Meši´

c, Kostas Kanellopulos,and

SilvanSchmid arewiththe Institute of Sensor and Actuator Systems,

TU Wien, 1040 Vienna, Austria (e-mail: hajrudin.besic@tuwien.ac.at;

e12031882@student.tuwien.ac.at; kostas.kanellopulos@tuwien.ac.at;

silvan.schmid@tuwien.ac.at).

Andreas Deutschmann-Olek is with the Automation andControl

Institute, TU Wien, 1040 Vienna, Austria (e-mail: Deutschmann@

acin.tuwien.ac.at).

Digital Object Identiﬁer 10.1109/JSEN.2024.3446369

nanomechanical photothermal sensing boasts lowpicogram

and lowpicowatt sensitivity,respectively,without the need for

cryogeniccooling.Nanomechanical photothermal sensing is

applicable to diversesamples such as nanoparticles [10],[11],

pharmaceutical compounds [2],and polymer thin ﬁlms [12].

However, nanomechanical resonators face twoprimary chal-

lenges: 1) aresponsespeed typically ranging from 5to100 ms,

creating an intrinsic tradeoffbetween speed and precision

and 2) astep response dominated by twodifferent transient

responses,with the slower time constant often exceeding

100 s [13].Toaddressthese challenges and enhance both the

speed and precision of nanomechanical photothermal sensors,

by motivationofthe recent works on Kalman ﬁltering in nano-

electromechanical system (NEMS) and other applications [14],

[15],[16],[17],wedevelop acomprehensivethermal heat

transfer andnoise model, suited for optimal (Kalman) ﬁltering.

We testand applythe model and the Kalman-ﬁlter-based

signal estimation by means of twoexperiments:one for

modelvalidation on asimple nanomechanical silicon-nitride

stringresonator,and another for conducting IR spectroscopy

(NEMS-IR) on amore complexnanoelectromechanicaldrum-

head resonator.

Formore information, see https://creativecommons.org/licenses/by/4.0/

BEŠI ´

Cetal.: OPTIMIZED SIGNAL ESTIMAT ION IN NANOMECHANICAL PHOTOTHERMAL SENSING 30265

Fig.1. Microscope image of the nanomechanical resonators. (a) Silicon

nitride string resonator. (b) Nanoelectromechanical drumhead resonator

employedfor IR spectroscopy.

In the twoexperiments, we employ pulsed laser light in

the visible and mid-IRrangeofthe electromagnetic spectrum,

respectively. Theprecise knowledge of thetimingand shape

of theinput signal allows for adaptiveKalman ﬁltering,

signiﬁcantly improvingresponsespeed compared to conven-

tional low-pass ﬁlters(LPFs). The designedKalman ﬁlter

estimatesthe input signal—speciﬁcally,the absorbed power—

effectively bypassing the slowtimeconstantcharacteristic

of the measured signal.Additionally,byincorporating the

system’s noise proﬁle, the ﬁltering process is optimized for

greater accuracy. System identiﬁcationwas achievedthrough

aparameter ﬁttingalgorithm appliedtoastep response,

adjusting initial parameters to align themodeled responsewith

the actual measurement data.

We showthat the presented modeland Kalmanﬁltering

effectively minimize the inﬂuence of slow,undesirable pro-

cesses such as drift and random walk,pavingthe wayfor fast,

precise nanomechanicalphotothermal sensing.

II. METHODS

A. Experimental Setups

1) String ResonatorSetup:To validate our model,

we employananomechanical silicon nitridestringresonator,

as shown in Fig. 1(a) anddetailedin[18].The resonator has

athickness of 56 nm, awidth of 5 µm,and alength of 1mm.

It operates under atensile stress of approximately 350 MPa,

ensuringoptimal performancewithin avacuum environment

maintainedatapressurebelow 10−5mbar to minimize air

damping effects. Theresonance frequencyofthe deviceis

recorded at around 170 kHz, accompaniedbyahigh quality

factor (Q)of1.6×106and aresonator time constant of

2.5s.These parametersare integral to our experimentalsetup,

providing afoundation for assessingthe efﬁcacyand accuracy

of ourproposed thermal heat transferand noise model in a

controlled environment.

The correspondingexperimental setup is depicted in

Fig. 2(a).Itutilizesalaser diode (LPS-635-FC from Thor-

labs GmbH) withawavelength of 638 nm as theheating

source. The laser diode is interconnected viaanoptical

cabletoanelectro-optical modulator(EOM), which acts as

aswitch to create ideally shaped lightpulses. The EOM

is controlled by a“Trigger” signal, which is connected to

our ﬁeld-programmable gate array (FPGA)-based electronics

(PHILL from Invisible-Light Labs GmbH). From the EOM,

thelaser is directed through an optical cable to alaser-Doppler

vibrometer (LDV) (MSA-500 from Polytec GmbH). The LDV

focuses the probing laserbeam ontothe nanomechanicalstring

resonator and simultaneously measures its motionusing an

additional readout laser with awavelengthof633 nm.The

readout signal is thentransmittedvia the “Receive”line back

to PHILL. The string resonator is actuated usingapiezo

shaker,which is drivenbyPHILL through the “Drive” line.

2) NEMS-IR Setup:TheNEMS-IR spectroscopyinvesti-

gation employed ananoelectromechanical resonator,which

consists of asilicon nitride drumhead that is 50 nm thick

andmeasures 1018 µminlength on each side, as detailed

in [19] and [20] and depicted in Fig. 1(b).For the purpose of

electrical transduction, the setup includes twogold electrodes,

each being 5 µminwidth, placed across theresonator.The

system waspositioned in astatic magnetic ﬁeldofabout0.8T,

produced by aHalbach array using neodymium magnets,

with theelectrode traces aligned perpendicularlytothe ﬁeld

lines.Thisarrangement uses theLorentzforce, enabling one

electrode to act as adriverbyapplying an alternating current,

while the other electrode measured the resonator’smovement

viathe induced voltage from magnetomotiveforces.For signal

detection enhancement,adifferential preampliﬁer with alow

noise proﬁle and again of 104wasutilized.The drumhead

resonator operated in avacuum environment, maintaining

apressure of 1.6 ×10−5mbar. It exhibited aresonance

frequencyof f

r=134 kHz, aquality factor of 13 k, and

aresonator time constant of 31 ms.

The experimental setup used to perform NEMS-IR is shown

in Fig. 2(b).Inside the chamber,the chipisplaced between

themagnets of the Halbach array and above the ZnSe window

that is located at thebottom of thevacuum chamber.The

chipelectrodes are contacted with spring-loaded contacts. The

readout electrode pins are connected to the differential pins of

the preampliﬁer.The output of the preampliﬁer is connected to

the“Receive”line. One pin of thedrive electrode is connected

to the ground. To limit thehigh current ﬂowing through the

chip andprevent damage, the other pinisconnected overan

attenuator,with gain 10−3,tothe “Drive”line of thesystem.

To conduct NEMS-IR, we employatunable quantum cas-

cade laser (QCL) as the laser source [1],[2].Speciﬁcally,the

MIRcat from Daylight Solutions laser is utilized, capable of

generating up to 500 mW of average laser powerand scanning

the laser wavenumber within arange of 1790–1122.5cm

−1

The laser’semission is focused ontothe NEMS chip through

aparabolic mirror and aZnSe window. By sweeping the

wavenumber,the laser induces frequencydetuning of varying

magnitudes at different wavenumbers, thereby generating the

IR spectrum.

The laser supportsbothcontinuous and step scan modes.

In continuous mode, thelaser remains activated while its

wavenumber is swept across the entire spectral range. Con-

versely, in step scan mode, the laser is set to aspeciﬁc

wavenumber,activated, and after apredetermined hold time,

deactivated andadjusted to anew wavenumber.This sequence

is reiterated until thefullspectrum is covered. The stepscan

mode is preferred for its ability to generateaspectral sample

30266 IEEE SENSORS JOURNAL, VOL. 24, NO.19, 1OCTOBER 2024

Fig. 2. Schematic representation of thetwo measurement setups.Photothermal sensing with (a) nanomechanical string resonator in the visible

and (b) nanoelectromechanical drumheadresonator to performIRspectroscopyNEMS-IR. Both setupsare based on self-sustained frequency

trackingwith PHILL.

by subtracting the “laser-off” frequency valuefromthe “laser-

on” value. Employing shortsamplingintervals allows for

thesuppression of slowprocessessuchasthermal drift and

random walk. The laser’s state is synchronized with thePHILL

frequencytracking device viathe “Trigger” line, as illustrated

in Fig. 2(b).

3) Frequency Tracking:The nanomechanical resonators are

drivenattheirresonance frequencyinaself-sustainingoscil-

lator (SSO) resonance trackingschemedescribed in [19]

and [21].The feedback of theSSO is realized with thePHILL,

as schematically depictedinFig. 2.The feedback of the

closed-loop SSO scheme is realizedwith phase andamplitude

controlling elements to satisfythe Barkhausencriterion. The

feedback consists of abandpass ﬁlter(BPF)whose main task

is mode selection of themeasurement modeand suppression

of unwantedmodes. After the BPF,acomparatorwith50-mV

hysteresis is placed to detect edgesand act as a0

◦phase

detector. The output of the comparator isconnected to the

pulse-generating mechanismthat generates timed pulses with

apulsewidth Tw,delay relativetothe detectedphase of Td,

and an amplitude A,which can be between 3and −3V.

Afrequencycounter developed in [20] isused to acquire the

frequencywithaﬁxed sampling rateofaround 20 kHz.

B. Thermal Response Model

The resonance frequencyofaprestressed resonator is pri-

marily inﬂuencedbythe initial tension [22].Variations in

thermal expansion between the resonatorand its supporting

frameresult in the resonator’s strain beingtemperature-

dependent, thereby inducing atemperature-dependent tensile

stress. Consequently,the resonancefrequencyisalsochanged

by temperature variations. Assumingalinear thermal expan-

sionrelationshipfor both the resonatorand its frame, the

temperature-induced strain in each of thetwo in-planedirec-

tionscan be linearly approximated as [22],[23]

ϵ=ϵ0−αrTr+αfTf(1)

where ϵ0is the strain at the initialtemperature T0without laser

illumination, αr/fare the coefﬁcients of thermal expansion of

the resonator/frame, and Tr/f=Tr/f−T0are thedifference

between theresonator/frame and the initial temperature.Solv-

ing (1) for thetemperature-dependent stressineach of the two

in-plane directions yields [24]

σ(T)=σ

0−E

1−να

rT

r−αfTf(2)

where σ0is the stressatT

,Edenotes theYoung’smodulus,

and νis Poisson’sratio. The fundamental mode eigenfre-

quencyofaprestressed resonator can be calculated fromthe

stress with the relation [22]

ω0=π

Lσ0

ρ−E(αrTr−αfTf)

ρ(1−ν) (3)

where Lis the length and ρis the mass density

of theresonator.For smallchanges in temperature,the

equation can be approximated by the ﬁrst-orderTaylor

approximation

ω0(T)≈ω0(T0)1−1

E(αrTr−αfTf)

σ0(1−ν) .(4)

In this simpliﬁed model,weignore howtemperature affects

Young’smodulus,thermal expansioncoefﬁcients, and mass

density.However,inreality,aresonator’slengthand ther-

mal expansion rates usually increase with temperature,while

Young’smodulus and mass density tendtodecrease. However,

theseeffects are negligible compared to the stress-change-

induced frequencydetuning. From theprevious equation, the

changeinfrequencycaused by heating due to achange in

temperaturecan be expressed as

ω0=−ω

α

T

r−α

T

(5)

where gis aconstant factor,which for asquare drumhead is

g=E/[2σ0(1−ν)]and for astring with auni-axialstress

ﬁeldreduces to g=E/[2σ0].

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Fig.3.(a) Schematic of differentparts of theresonator andframe

contributing to thethermal circuit model. (b) Linearized thermalequiva-

lent circuit model of the resonator andframe and its thermal connection

tothe environment. (The ground symbol represents the environmental

temperature T0.)

To utilize the static relation (5),the dynamics of Trand

Tfneed to be described. Fig. 3(a) shows theschematic

representation of theheat ﬂowofthe chip.Operating in

vacuum, there are twodominanttypesofheat ﬂowinthe

system: conduction and radiation [25].The resonatorradiates

heatonbothsides andconducts heat over theframe into

the environment. Theheat ﬂowcan be approximated with

asimple heat-transfer circuitdiagram showninFig. 3(b).

The thermal equivalentcircuit modelconsists of the heat

source Paandthermal resistances and capacitances of the

resonator,frame,and chipholder. The absorbed power Pa

is calculated by multiplyingthe impinging power P0by

thethermal absorption coefﬁcient β,asexpressed by the

equation

Pa=P0β. (6)

Theheatﬂow due to radiationcan be modeled as aheat source

described by theStefan–Boltzmann law [22],[25]

qrad =2ηArσSB(Tr+T0)4−T4

0.(7)

Forsmall temperaturechanges Tr,the radiationcan be

linearized around T0and modeled as athermal resistance to

the environment with aconstant thermal resistance

Rrad =Tr

qrad =1

8ηArσSBT3

(8)

where ηis theemissivity, Aris thesurface of theresonator,

and T0is the ambient temperature. From thethermal equivalent

circuit model, alinear state-space model can be extracted as

˙

˙

Tf

 

=





−Rr+Rrad

RrRradCr

RrCr

RrCf−Rr+Rf

RrRfCf







 

Tr

Tf

 

+



0



  

Pa+w

(9)

where Asis the dynamic matrix, Bsis theinput matrix, xsis

thestate vector, ˙

xsis its derivative,and wis theprocess noise

in thesystem. From (5),the corresponding output equation of

thestate-space model can be formed as follows:

ω0



=−ω0gαrω0gαf

 

Tr

Tf

 

+v(10)

where Csis the output matrix of the system, yis the measured

quantity,and vis themeasurement noise. The state-space

model from (9) and (10) can be discretized in time [26] fora

chosensamplingtime tsthat yields

xs(k+1)=eAsts



xs(k)+ts

eAsτBsdτ

 

Pa(k)+w(k)

y(k)=Cs



xs(k)+v(k)(11)

where Fsis the discretedynamic matrix, Gsis the discrete

input matrix, and Hsis the discrete output matrix. The discrete

state-space model (11) is particularly well-suited for signal

processing on digitalsystems.

III. NOISE MODELINGAND KALMAN FILTERING

Noisemodeling is acrucial aspect of Kalman ﬁltering,

an optimal estimation technique widely used in control sys-

tems, navigation,and signal processing [27],[28].The primary

goal of Kalman ﬁlteringistoestimate thestate of adynamic

system from aseries of noisymeasurements.Toachieve this,

theKalman ﬁlter relies on accuratemodels of both the system

dynamics and the noise affecting thesystem. Speciﬁcally,the

noiseiscategorized into twotypes:measurement noise and

processnoise,described by wand vin (9) and (10).Measure-

ment noise represents the errors in the observations [29],[30],

while process noise accounts for the uncertainties in the system

model itself [31].The measurement variance(R)quantiﬁesthe

expected errors in the measurementswith theexpectedvalue

R(t−τ) =E[v(t)v(τ )],and theprocesscovariancematrix

(Q)characterizes the uncertainties in thesystem dynamics

with the expected value Q(t−τ) =E[w(t)w

T(τ )].These

covariance matrices are essential for theKalman ﬁlter to weigh

thereliability of the predictions versus the newmeasurements

correctly.Byaccurately modeling the noise, the Kalman

ﬁlter can optimally combineinformation from the model

and the measurements,resulting in preciseand reliable state

estimates.Understanding and implementingnoise modelsare,

therefore, fundamental to the effectiveapplicationofKalman

ﬁltering [32],[33].Inthe following,wewill describethe

30268 IEEE SENSORS JOURNAL, VOL. 24, NO.19, 1OCTOBER 2024

Fig. 4. Noise contributionsinthe SSO frequency trackingscheme,

represented in phase space.

measurementand processnoise of thesystem,disregarding

slowprocesses suchasdrift andrandom walk.Theseslow

noise processesare irrelevant because samplingoccurs on

much smaller timescales.

1) Measurement Noise:The measurement noise refers to

the disturbances generated duringthe measurementofaspe-

ciﬁc physical quantity,inthisinstance, thetemperature of the

resonator. Forameasurementperformed in an SSOconﬁgu-

ration, twomainnoise contributors canbeidentiﬁedasshown

in Fig. 4:thermomechanical noiseand detectionnoise. The

thermomechanical noise is fundamental to nanomechanical

resonators and can be modeled as white force noise actingon

the resonator.Thiscauses an amplitude noise to theposition

of the resonator at resonance withthe powerspectral density

(PSD) [19],[21],[22] of

Sthm =4kBTrQ

mω3

(12)

where kBis Boltzmann’sconstant, Tristhe temperature, m

is theeffective mass, ω0is theeigenfrequency, and Qis the

quality factor of the resonator.

Detection noise is produced by transducing the mechanical

motion of theresonator into an electrical signal andiscaused

mostly by transduction and electronic noise. Beingtypically

whiteGaussian noise, detection noisePSD can be described

with respect to the thermomechanical noise

Sd=K2Sthm (13)

where Kis adimensionless factor that corresponds tothe ratio

betweenthe detection noise background andthe height of the

thermomechanical noisepeak.The amplitude domainPSDs

canbetransformed into phasespace by [21],[34]

Sθthm =2

Sthm

Sθd=2

Sd=K2Sθth (14)

with thevibrational amplitude at resonance Ar.

The measurement noise sources contribute to themeasuring

noiseintwo different ways. The ﬁrst is caused by phase

detection noisepassing through the BPF,then through the LPF

of thefrequencycounter,and subsequently being transformed

into frequencythrough differentiation. The second contributor

is theclosed-loop noise caused by the thermomechanicaland

detection noise combined. Thistype of noise is described by

the Leesoneffect [19],[21],[34],which assertsthat within

an SSO tracking scheme, phase noiseisintegrated and is

proportional to the reciprocal of the resonator’stime constant

τr,provided that it is signiﬁcantly larger than theinverse of the

BPF’sbandwidth. This integrated noise, after passing through

the BPF and frequencycounter ﬁlter,isthen differentiated

andtransformed into frequency. It can be described by white

Gaussian thermomechanical noisepassingthrough the transfer

function Gvyielding thePSD

Sv(s)=|G

(s)|

thm (s)(15)

with

Gv(s)=sK+√K2+1

τr

(1+s·τBP)(1+s·τFC)(16)

where τBP representsthe time constant of the BPF,and τFC is

thetime constant of thefrequencycounter.Considering that

themeasurement noisefollows aGaussiandistribution (even

though it does not have aconstant PSD), astraightforward

approach would be to empirically measure the noise variance

and use

R=σ2

v.(17)

Brown [29] and Popescu and Zeljkovic [35] suggest that

abetter approach for treating colored noiseistobuild a

noise model based on its transfer functionand augment it to

thesystem in (9).The noise state space can be describedin

controllable canonical form as

xv(t)=



−

BPτFC −1

τBP +1

τFC 



 

xv(t)+0

1



uv(t)

(18a)

v(t)=√K2+1

τBPτFC

τBPτFC 

 

xv(t)(18b)

where uvis the white measurement noise sequence with its

auto-correlation E[uv(t)uv(τ )]=S

thm δ(t−τ),where δ(·)is

theDirac delta function. The noise model can be discretized

in time as shown in (11) yielding thediscrete-time matrices

Fv,Gv,and Hv.

2) Process Noise:The processnoise represents the dis-

turbances acting on themodeled states. In this model, it is

epitomized by the temperature ﬂuctuations exertedonthe

resonator.Temperature ﬂuctuations are afundamentalphe-

nomenon affecting all structures,primarily arisingfromthe

statistical and quantum aspectsofheat exchange [36].This

processinvolvesthe absorption and emission of photons at

the resonator surfaces,alongsidethermal conduction to and

from thesurrounding environment via phonons. The resultant

noise from this heat exchange manifests as shot noise, which

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exhibits white frequencynoise behaviorﬁltered by theres-

onator thermal response.

In thecontext of thesystem described in (9),two temper-

ature statesare modeled: theresonatorand theframe. The

frame,due to itslarge size and very high thermal capac-

ity,isassumedtohaveastatic temperature with negligible

ﬂuctuations. Conversely,the resonatorexperiences signiﬁcant

temperature ﬂuctuations,describedbythe spectral density [36]

STr(s)=¯

STr|Gth(s)|2=4kBT2

0Rth|Gth(s)|2(19)

where Rth =Rr∥Rrad is theequivalentthermal resistance,

and

Gth(s)=1

1+s·RthCr

(20)

is aﬁrst-order LPF determined by the resonator’sthermal

properties. This behavior,when expressed in thecontrollable

canonical state-space form, can be articulated as follows:

˙xw(t)=1

τth 



xw(t)+1



uw(t)(21a)

w(t)=



τth

0



 

xw(t)(21b)

where uwis the white measurementnoise sequencewith its

auto-correlation E[uw(t)uw(τ )]=¯

T

δ(t−τ).The state

space canagain be discretized asshown in (11),yielding

the discrete state-space model Fw,Gw,and Hw.Combining

the system dynamics (11) with thediscretized colored noise

models (18) and (21) yields

xs(k+1)=Fsxs(k)+GsPa(k)+Hwxw(k)(22a)

xw(k+1)=Fwxw(k)+Gwuw(k)(22b)

xv(k+1)=Fvxv(k)+Gvuv(k)(22c)

y(k)=Hsxs(k)+Hvxv(k)(22d)

resultinginamorecomprehensive modelthataccounts for

bothmeasurement and process noiseproﬁles.

A. AdaptiveKalmanFilter

The fundamental speed versusaccuracylimit of nanome-

chanical resonators is intrinsic. The accuracycan be improved

by noise ﬁltering, at the cost of speed reduction [19],[21].

However, this limit can be broken usingadditionalinformation

about thesystem. In the case of NEMS-IR usingQCL step

scan mode, it is knownthat thepower input hasastepshape

with unknownheightand known positionintime (overthe

triggerinFig. 2). It can be modeled as an unknown but

constant power with asmall disturbance term wPato introduce

aform of uncertainty

Pa(k+1)=Pa(k)+wPa(k). (23)

wPadescribes all the noise processesofthe laser intensity,

including ﬂicker noise, random walk,and shot noise [37],

[38],[39].The ﬁrst twodominate at lowfrequenciesand are

considered colored noise.Anexample is modehopping in

diode lasers, which is caused by environmental temperature

changes [39].Athighfrequencies,shot noisedominates and

is characterized as white Gaussian noise. It is important to note

that wPachanges with input conditionslikethe laser power.

If the noise is not modeled in wPa,itwill appear in the power

estimate of Pa.Inthis work, wPais modeled as white Gaussian

noise and slow processesare disregardeddue to thefast sample

acquisition.

Through thethermal equivalentcircuit model and (9)

and (10),the dynamics of the system arealsowell-known.For

problems of this type, an adaptive Kalman ﬁlter (AKF) can be

used [14],[40].Given the knowledge of the model, inputs, and

stochastic properties of the system, it is capable of estimating

the system in astatistically optimal manner [40].Despite the

knowledge of the input timing and shape, theheight of the

absorbed laser power is unknown. Therefore, the modelhas

to be extended andthe Kalman ﬁlter needs to estimatein

addition to the states the valueofP

.Thiscan be achievedby

augmenting the model [40],byadding (23) to (22) yielding







xs(k+1)

xv(k+1)

xw(k+1)

Pa(k+1)







 

x(k+1)

=





Fs0HwGs

0Fv00

00F

00 01







 







xs(k)

xv(k)

xw(k)

Pa(k)







 

x(k)

+





000

001







 





uv(k)

uw(k)

wPa(k)



 

(24a)

y(k)=HsHv00



 

x(k). (24b)

Notice that the measurement noise has been transferred from

theoutput equation to thestate equation. As aresult, the

systemdoes not have ameasurement noisecovariance matrix

and all noise processes are described by theprocess noise

covariance matrix. Thismatrix, described in the discrete-time

domain, is givenby

Q=E[¯

]=



s00

T

00σ



(25)

where σ2

Pais theprocesscovariance matrix of the estimated

absorbed power.The variance σ2

Pais atuning factor and has

to be determined empirically.The Kalman ﬁlter algorithm is

described in [14] and consists of the following steps.

1) Predict:

x(k|k−1)=Fˆ

x(k−1|k−1)(26a)

P(k|k−1)=FP(k−1|k−1)FT+GQGT.(26b)

2) Observe:

˜y(k)=y(k)−Hˆ

x(k|k−1)(26c)

V(k)=HP(k|k−1)HT+R.(26d)

30270 IEEE SENSORS JOURNAL, VOL. 24, NO.19, 1OCTOBER 2024

3) Kalman Gain:

K(k)=P(k|k−1)HTV−1.(26e)

4) Estimate:

x(k|k)=ˆ

x(k|k−1)+K(k)˜y(k). (26f)

5) EstimateCovariance:

P(k|k)=(I−K(k)H)P(k|k−1)(26g)

where ˆ

xis theestimated state, ˜yisthe difference between

themeasured and predicted statesalsocalled theinnovation

or residual, Pis the covariancematrix of the state estimate,

and Kis the Kalman gain.The notation (n|m)meansthat the

corresponding quantity is estimated at atime nfromthe time

mthat is located in thepast, i.e., m≤n.The Kalman ﬁlter is

arecursive algorithmused to estimatethe state of adynamic

system from aseries of noisy measurements.Itcombines

predictions from amodel with newmeasurements to produce

an estimatethat minimizes the meanofthe squarederror.

Thecovariance matrix Prepresents the uncertaintyinthe state

estimate. It is updated at each step to reﬂectthe conﬁdence in

the estimate.

Anynoise process not incorporated into theKalman ﬁlter

model willmanifestinthe estimated output.This occurs

because theKalman ﬁlter must reconcile themeasurements

with themodel. If aparticularnoisesource is not accounted

for in the model, the ﬁlter will interpret this noise as part

of the input (power estimate) signal.For example, unmodeled

random walk, caused by laser intensity ﬂuctuations or NEMS

thermaldrift,willappear in theestimated powerdata.Sim-

ilarly, dynamic environmentalchanges, such as temperature

ﬂuctuations, vibrations, and external noise sources, will also

affect the system if not properly modeled.

Whenever thelaserisswitched on or off, estimates of Pa

fromprevious data areclearlybecoming invalid.This can

easilybeincludedinthe AKF by manually increasingthe

covariance of the estimate ˆ

Pa(or even thewholecovariance

matrix P). This way, the AKF adaptsquickly to newmeasure-

ment information at theexpense of more noisyestimates in

the beginning. After awhile, theﬁlter will converge back to

its originalﬁlteringstrength. The estimate of theimpinging

power ˆ

P0is obtained, bysimply dividing theabsorbedpower

estimate ˆ

Pawiththe absorptioncoefﬁcient of thestructure β.

Forreal-timeapplications, the Kalmanﬁltercan be imple-

mented on an FPGA. The FPGAcan be programmed using

hardwaredescription languages (HDLs) such as Verilog or

VHDL, or through high-levelsynthesis (HLS) tools like

MATLAB HDLCoder.When usingMATLAB, it cangen-

erateHDL code fromMAT LABcode,which can then be

ﬂashedontoanFPGA, such as theXilinxZynq-7010 on a

STEMLab 125-14 (RedPitaya) system. Asix-stateKalman

ﬁlter,aspresented in this work,utilizes approximately 42 of

thedigital signal processing(DSP) slices whencompiled onto

theFPGA. One of the constraints of FPGA implementation

is the necessityofusingﬁxed-point arithmetic, which is

limited by the 25 ×18 bitDSP slices. This limitation can

lead to quantization errors. To improve numeric precision,

onecan usetwo DSP slices overtwo clock cycles for a

single multiplication instead of usingone DSP slice in one

clock cycle. This approach consumes more FPGAresources

and increases the calculation latencybut enhances numeric

precision.

B. SystemIdentiﬁcation

Mathematical modelsare often derivedfrom idealized sce-

narios andneed to be calibrated to reproduceexperimental

data. In real-world applications, factors such as chip fabrica-

tion, laser alignment,orspot size of the laser illuminating

themembrane can vary widely.Provided the system is linear,

astep responsesystem identiﬁcation can be apowerfultool

to properlycharacterize different parameters of adynamical

system. The method of least squares is widely regarded as

asimple andeffective technique for extractinginformation

from adataset [41].The method is optimal in the sensethat

theparameters determined by the least-squares analysis are

normally distributed about the true parameters with the least

possiblestandarddeviations.

The method of leastsquares is aparameter estimation

method in regression analysis based on minimizing the sum

of thesquares of theresiduals (a residual being the difference

between an observed valueand the ﬁt value provided by

amodel) made in the results of each individual equation.

However, problems with this method can ariseifthe numerical

algorithm gets stuck in alocal minimum of theerror function,

resultinginsuboptimal parameters. To prevent this from hap-

pening, in this case, initialparameters are roughlycalculated

analytically.Theyensure that thenumerical algorithm does

not start too farfrom theglobal minimum, thus increasing

its chances of ﬁnding theoptimal solution. In this work,the

Levenberg–Marquardt algorithm wasused due to its robust-

ness.

IV.RESULTS AND DISCUSSION

Thermal systems that operate in alinear regime usually do

not exhibitovershoots. It is mentioned in Section II-B that

it is possiblethat thedeformation of the frame can cause

this effect due to its thermal expansioncoefﬁcient αf.The

magnitude of the resulting frame-induced frequencyshift has

manycontributors: αf,g,and all parameters contributing to

Tf.Inreality,these parameters are hard to characterize,

andany combination of parameter values that yields the same

input–output behavior can be mathematically considered valid.

Therefore, it is unnecessary to determine theexact parameters

of the system; it sufﬁces to ﬁnd those that yield the correct

input–output behavior to estimate theabsorbed or impinging

power of thelaser.More generally,linear state-space models

like (9) are not unique [40].Manystate-space representations

can capture thesame input–output dynamics.Thus, some

model parameters that are difﬁcult to model or measure, such

as αf,Rf,and Cf,can be estimated and ﬁt to describethe

system dynamics correctly even though theymay not represent

the physically correct values.

A. StringResonator Setup

In the ﬁrst experiment, theperformance of the model is eval-

uated. The thermal model is identiﬁed by ﬁtting the parameters

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Cetal.: OPTIMIZED SIGNAL ESTIMAT ION IN NANOMECHANICAL PHOTOTHERMAL SENSING 30271

TABLEI

COMPARISON BETWEEN ANALYTICALLY CALCULATEDAND FIT PARAMETERS

Fig. 5. Results of theexperiments performedonthe stringresonator. (a) Systemidentiﬁcation through calculation and subsequent ﬁttingofthe

step response to the measured signal. (b) Characterization of measurementnoise via thePSD of themeasured signal, alongside acomparison with

thecalculated process noise.(Values are displayedinfractionalfrequency.) (c) Visualization of theAKF estimates forameasured step response,

contrasted against theinput laser powertodemonstrate stableand precise input powermeasurement. (d) Detailed view of thetransient behavior

in the powerestimate,highlighting thesigniﬁcantinitial estimate error and itsrapid convergence to asteady state. (e) Reproducibilityand linearity

testconducted by executing ﬁvesuccessivemeasurementsacross three powerlevels,including analyses of both“on” and “off”laser transients.

(f) Application of theAKF to astaircase function,showcasing immunity to thesystem’ssecond slowtime constant.

to astep response generated by ajump in laserpower of

P0=5.8µW. To ﬁt thedata, theinitial parametersmust be

determined. In this work,weadoptedathree-step approach for

data ﬁtting.Initially, we calculated theparameters analytically

from theresonator geometryand material properties. Subse-

quently,amanual ﬁt of the parameterswas performed,and

in theﬁnalstep, we ranthe Levenberg–Marquardt algorithm

to obtain theoptimal parameters. Table Idisplaysthe values

of the parametersobtainedthrough data ﬁtting, and Fig. 5(a)

shows the step responses for both theanalytically calculated

andﬁtdata.

The system’smeasurement noise is characterized by per-

forming a30-s measurementatasteadystate with thelaser

turned off. The BPF 3-dBbandwidthand theLPF in the

frequencycounter are bothset to 1kHz, chosen to be at least

one order of magnitudelarger than anytransient phenomena

in thesystem. The samplingrate of the frequencycounter is

set to approximately20kHz. From the acquired data, aPSD is

generated, as shown in Fig. 5(b).For better visualization, all

PSDsare displayed in fractional frequencies. By ﬁtting (15)

to thePSD,the measurement noise is calibrated to be SV=

5×10−16 Hz−1,while the thermal noise ﬂoor is calculated

to be ¯

STr=6×10−17 Hz−1.

From the PSD analysis, it can be observed that random

walk noise dominates at lowfrequencies. This random walk

signiﬁcantly impacts the accuracyofthe ﬁt model, which

in turn directly affectsthe precision of the estimatedsignal.

To minimize model ﬁttingerrors caused by random walk,

30272 IEEE SENSORS JOURNAL, VOL. 24, NO.19, 1OCTOBER 2024

Fig. 6. Results from theNEMS-IRexperiments. (a) System identiﬁcation forthe NEMS-IR setup. (b) AdaptiveKalman ﬁltering applied to a

laser powerstep,illustrating arapid powerestimate,acomparatively slower measurement estimate, and an even slowerresponse from an LPF

possessingequivalent ﬁltering strength to the AKF. (c) Detailed view of asegment fromthe QCL step scan,comparing AKF and LPF data.

(d) Comparison of thepolystyrene spectrumobtained using theAKF estimate and theLPF,withred lines highlighting thecharacteristic peaks.

(Some axes areinverted to provide an easiervisualization.)

the ﬁtting should be performedovershortertime scales,

ensuring that these scales arestilllongerthanthe second time

constant.

After the system ischaracterized by thestep response

andthe noise by thePSD,the AKF can be employed.

Fig. 5(c) illustrates the AKF’sresponsetoalaserpower

jump of P0=5.8µW, induced by controlling the EOM.

At the moment of the step, the covariance matrixofthe

AKFisincreased by afactorgreater than 104,signiﬁcantly

expanding its bandwidth. Thismodiﬁcation renders the sys-

tem more sensitive to noise, resulting inahighlyinaccurate

initial estimate. Subsequently,the AKF swiftly decreases the

bandwidth, allowing it to rapidly convergetoasteady state.

In this state, the measurementestimate aligns with the mea-

surement signal trajectory,and theinput estimate accurately

reﬂects thestep applied to the resonator. Fig. 5(d) presents

acloser viewofthe input step estimate’s transientbehavior,

demonstrating that approximately100–200 ms are required

for theinput estimatetostabilize at theinput signal power

value.

In the subsequent experiment, thereproducibility of the

system is assessed. Forthree distinctpower levels (5.8, 15, and

25.6µW), thelaser is alternately turned on and offﬁve times,

with a50-s intervalbetween each sample. An estimate of the

input power is collected 200 ms after the laser event. Fig. 5(e)

displays the results, revealing that the system exhibits almost

perfect linear behavior in both“ON”- and “OFF”-states. The

standard deviation among the samples is approximately 1% of

theinput power,underscoring the system’shigh reproducibil-

ity.This is particularly noteworthy consideringthe signiﬁcant

potential for drift and laser power ﬂuctuations due to the

lengthyintervals between sample acquisitions.

In theﬁnal experiment,the bias of the laser diode is adjusted

to incrementally increase its output power in astepwise man-

ner.Altering thepower in such amanner results in degradation

of the step edges,attributabletothe internal powerregulation

circuitryofthe laser diode, which causes slow edgesand

edges with overshoots. As observed from Fig. 5(f),despite the

transients not being perfectly consistent, the power estimation

closely resembles astaircase function.

BEŠI ´

Cetal.: OPTIMIZED SIGNAL ESTIMAT ION IN NANOMECHANICAL PHOTOTHERMAL SENSING 30273

B. NEMS-IR Setup

At the outset of the experiment, it is imperative to charac-

terize the model of the chip. Initially,the QCL is activated.

Employing techniques described in Section III-B,the tran-

sientstep response is ﬁt tothe measured data,asdepicted

in Fig. 6(a).Analysisofthe data reveals twodistinct time

constants: arapid transient in thepositive direction and a

markedly slower transient in thenegative direction. To account

forvaryingabsorbance of the structures at different laser

wavenumbers, all laser powersinthe subsequent experiments

will be normalized as

Pa=ˆ

P1300

(27)

where P1300 is themeasured power at awavenumber of k=

1300cm

−1

,and ˜

Pais thenormalized absorbed power estimate.

In the initialexperiment, theQCL is set to awavenumber of

1300cm

−1with apower output of 0.45 mW. Upon activation,

thelaser illuminatesthe NEMS device, andthe ensuing data

areprocessed using boththe AKF andanLPF,with the

cutofffrequencyof1Hz,possessingacomparable ﬁltering

strength (approximately 0.4-Hzstandard deviationovera

300-ms recording period). Asillustrated in Fig. 6(b),the data

reﬁned by theAKF exhibitasigniﬁcantly quicker response

compared to that processed by theLPF.Upon detecting a

jump, theKalman ﬁlter’sbandwidthexpands substantially,

facilitating arapid response. Subsequently,the bandwidth

narrowsswiftly,and the data convergetoasmoothercurve

that representsthe optimal estimate. Fig. 6(b) also reveals that

the estimated absorbed power reactsthe most promptly.For

models that are both stableand reliable, thepower estimate is

deemed suitable. Notethat anymismatch between the model

and the actual experimental system maybemappedinto wrong

estimates by theAKF tryingtoreconcile themodeland

measurements.

In the second experiment, thechipiscoated with

polystyrene, and data collectionspans adurationof3min.

The QCL operatesinstep scan mode,systematically covering

the wavenumber rangefrom 1789 to 1122 cm−1in increments

of 1cm

−1

.For each wavenumber,the laserisactivated for

100msand subsequently deactivatedfor 150 ms.

Fig. 6(c) presents asection of the time-seriesdata obtained

from the scan. Notably, the input estimates rapidly converge

to astable, ﬂatline. Incontrast, themeasurementestimate

closelyand smoothly tracksthe measurementdata. Although

theLPF succeeds in attenuating noise, it concurrentlydampens

the signal as well. Thisobservationunderscores theAKF’s

signiﬁcant enhancement of the system’s responsiveness. Uti-

lizing aconventional LPF necessitatesawaitingperiodthat

is ﬁve–ten timeslonger than theresonator’stime constant to

allowthe signal to stabilize.Afurther complicationarisesdue

to framestretching,which, inﬂuenced by anotably sluggish

time constant,alters the signal.This effect complicates the

determination of an optimal sampling pointfor thesignal.

To generate the spectrum, dataare collecteddifferentially

by subtracting consecutivemeasurements corresponding to

the laser’s ON-and OFF-states.Thismethod ensures that the

resultingspectrum remains unaffectedbyslownoise processes,

such as thermal drift and random walk processes. Fig. 6(d)

displays the resultant spectra for the AKF and LPF,which

prominently feature the characteristic peaks of polystyrene.

It can be observed thatthe spectrum generated by the AKF has

less noiseand is cleaner,incomparisontothe one generated

by theLPF data.

V. CONCLUSION

In this study,weaimed to enhance thespeed and precision

of nanomechanical photothermal sensors by developing a

comprehensivethermal heat transfer andnoisemodelsuited

for optimal Kalman ﬁltering. We validated our model through

experiments on asilicon-nitridestringresonator and applied

it to photothermal IR spectroscopyusingamore complex

nanoelectromechanical drumhead resonator.

Our thermal model accurately captures system dynamics,

including thesigniﬁcant inﬂuence of the thermal expansion

coefﬁcients ratio on the frequencyresponseand taking into

account the dual time constant of the resonator.The adaptable

Kalman ﬁlterwedeveloped provides rapid and preciselaser

power estimation, suitable for real-time FPGAimplementation

or ofﬂine use. The experimental results demonstrated the

model’saccuracyand the AKF’s effectivenessinmanaging

measurement noise and maintaining stablepower estimation.

In IR spectroscopyapplications, theAKF enabled fast data

sampling and improved precision by eliminating slow time

constant issues and slow noise processes likerandomwalk,

thus successfully identifying characteristic spectral peaks.

Overall, our comprehensive thermal model and Kalman-ﬁlter-

based approach signiﬁcantly advance the performance of

nanomechanical photothermal sensors.The enhanced speed

and precision achievedthrough this work offerpromising

avenues for further research and practical applications in the

ﬁeldofnanomechanical sensing. Future effortswill focus on

reﬁning the model andexpanding its applicability to other

resonator types and sensing modalities.

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Dissertation

AdvancedTechniquesfor Resonance

FrequencyMonitoringin

Nanomechanical Sensing: Integrative

Approaches andApplications

Submittedinpartial fulﬁllmentofthe requirements forthe

academic degree Doktor derTechnischenWissenschaften (Dr.

Techn.), at

InstituteofSensorand Actuator Systems

Technische UniversitätWien

By:

Dipl.-Ing. Hajrudin Bešić

01226946

Supervisedby:

Univ.Prof. Dr.sc. Silvan Schmid

Vienna,November19, 2024

We cannot solve problems with the same thinkingweusedtocreate them.

-AlbertEinstein

Contents
Contents
Contents I
Abstract III
Acknowledgment V
ListofPublications VII
ListofSymbols IX
Contributions to Original Knowledge 1
Introduction 3
1TheoreticalBackground andMethods 7
1.1Nanomechanical Resonators ............................. 7
1.1.1DampedLinear Resonator .......................... 7
1.1.2FreeUndampedVibration ..........................9
1.1.3FreeDampedVibration ........................... 9
1.1.4DrivenDampedVibration .......................... 9
1.1.5QualityFactor .................................12
1.2Noise in Nanomechanical Resonators ........................ 14
1.2.1Noise Fundamentals .............................. 14
1.2.2Relationbetween Amplitude,Phase andFrequency Noise ....... 18
1.2.3Thermomechanical Noise ........................... 32
1.2.4Detection (Transduction)Noise ....................... 33
1.2.5Temperature FluctuationNoise .......................36
1.3AllanVariance ..................................... 37
1.4Noise in Closed-LoopSystems ............................ 40
1.5Transduction andActuation .............................42
1.5.1Magneto-motive Transduction andActuation ..............43
1.5.2Optical Transduction WithLaser-DopplerVibrometer ......... 44
2Schemes forTrackingResonance Frequency 45
3Adaptable FrequencyCounter with PhaseFiltering 51
I

Contents

4ThermalResponseModellingand Kalman Filtering 57

5Conclusionand Outlook 63

5.1Conclusion ........................................ 63

5.2Outlook ......................................... 64

Abstract

This dissertation exploresadvanced techniquesfor resonancefrequency monitoring in

nanomechanical sensing,withafocusonintegrative approachesand theirapplications.The

researchpresentsacomprehensivestudy on variousfrequency tracking schemesfor NEMS

(nanoelectromechanical systems) resonators,comparing open-loopand closed-loopmethods.

The ﬁndings demonstrate that alltrackingschemes canachieve comparableperformance in

termsofaccuracy andspeed,suggesting that theperformance limitsare intrinsic to the

NEMS resonatoritself. This allows thechoiceoftracking scheme to be basedonpractical

considerationssuchascostand ease of implementation.

Additionally,anadaptable frequency counterarchitectureisalsodeveloped,oﬀering

enhanced usabilityfor frequencymonitoring taskswithinthe NEMS community.

Further contributionsinclude thedevelopment of athermal modelfor NEMS infrared spec-

troscopy, accurately capturing thetransient response of NEMS to infraredlight absorption.

This modelaccountsfor both theintrinsic response of theresonator andthe frame-induced

response. Theintroduction of an adaptable Kalman ﬁlter, integratingthe NEMS response

modelwithinherent noiseproﬁles, enablesfasterand more precisemeasurementsthrough

optimaladaptiveﬁltering.

These advancements contributetothe ﬁeld of nanomechanical sensing,providing new

toolsand methodologies for more accurate andeﬃcient monitoring of resonancefrequencies

in NEMS devices.

III

Acknowledgment

Iwould liketoexpress my deepestgratitude to thosewho have supportedmethroughout

thejourney of completing this dissertation.

First andforemost, Iwanttothank my parents, Mujo and Elbira,for theirunconditional

love,support, andencouragement.You have always believedinme, andyourguidancehas

been invaluable in helping me reachthispoint.Tomybrother Haris,thank youfor always

beingthere andfor your unwavering support.

Iamimmenselygratefultomywife, Andrea,for herendless patience,love, and

understanding throughout this process. Your supporthas been my pillarofstrength.To

my belovedchildren, Nadia and Hana,you have been my inspirationand motivation to

pushthrough thetoughest moments.

Iwould also liketoextend my sincerethankstomyprofessorand mentor,Univ.Prof. Dr.sc.

Silvan Schmid.Yourguidance, expertise,and encouragement have been instrumental in

shaping this work. Iamdeeply thankful foryourmentorship andfor theopportunitiesyou

have providedme.

IamalsogratefultoProf. AlperDemir forhis valuablecontributions as acoauthor

andtoJohannesSteurer,whose expertise in electronics hasgreatly enriched my research.

Your insightsand assistance have been crucialtothe success of this work.

Special thanks go to my colleagues Kostas,Robert, Niklas,and Paolo.Your

collaboration,discussions, andcamaraderie have made this journeynot only productive

but also enjoyable.Iam truly fortunate to have hadthe opportunitytoworkalongside

such talented andsupportive individuals.

Lastly,Iwant to thank everyone whohas contributed,directly or indirectly,tothis

dissertation.Yoursupporthas been indispensable,and Iamdeeply appreciativeofall the

help Ihavereceived.

List of Publications

ListofPublications

Belowisthe listofpublications that resulted from theresearchworkinthis thesis:

H. Bešić, A. Demir, J. Steurer, N. Luhmann, andS.Schmid, "Schemes for Tracking

ResonanceFrequency forMicro-and Nanomechanical Resonators," Physical Review

Applied,vol.20, no.2,August2023, Art. no.024023.

DOI: https://doi.org/10.1103/PhysRevApplied.20.024023.

H. Bešić, A. Demir, V. Vukićević,J.Steurer,and S. Schmid, "Adaptable Frequency

Counter With PhaseFiltering for ResonanceFrequency Monitoring in Nanomechanical

Sensing," IEEE SensorsJournal,vol.24, no.6,March 2024, pp. 8094-8104.

DOI: https://doi.org/10.1109/JSEN.2024.3355026.

H. Bešić, A. Deutschmann-Olek,K.Mešić, K. Kanellopulos,and S. Schmid, "Optimized

Signal EstimationinNanomechanical Photothermal Sensing viaThermal Response

Modellingand Kalman Filtering," IEEE SensorsJournal,vol.24, no.19, October

2024.

DOI: https://doi.org/10.1109/JSEN.2024.3446369.

K. Schmerling,H.Bešić, A. Kugi,S.Schmid, A. Deutschmann-Olek, "Optimal

Sensing of MomentumKicks with aFeedback-Controlled Nanomechanical Resonator,"

arXiv:2411.0221.

DOI: https://doi.org/10.48550/arXiv.2411.02215.

VII

List of Symbols

ListofSymbols

Symbol Description

AArea

ADynamic matrixofastate-space

αThermalexpansion coeﬃcient

BMagnetic ﬂux density

BInput matrixofastate-space

BW Bandwidth

cDamping coeﬃcient

COutput matrixofastate-space

fFrequencyinhertz (Hz)

fxxProbabilitydensity function of x

FtTime-domaindrivingforce

HsTransferfunction of alinear system

kSpring constant

kBBoltzmannconstant

LLength

mMass in kilogram (kg)

meﬀ Eﬀective mass in kilogram (kg)

ntNoisesignal

NNoiselevel in whitenoise processes

PPower

QQualityfactor

RxτAuto-correlation function of x

SxfOne-sidedPSD of x,asafunction of linear frequency[V

/(Hz)]

SxωOne-sidedPSD of x,asafunction of angularfrequency[V2/(Hz)]

SII

xfTwo-sidedPSD of x,asafunction of linear frequency[V

/(Hz)]

SII

xωTwo-sidedPSD of x,asafunction of angularfrequency [V2/(Hz)]

xωTwo-sidedPSD of x,inthe unit [V2/(rad/s)]

TTime period

tTime

Temperature

τrResonatortime constant

uInput quantity of asystem

WEnergy

List of Symbols

Symbol Description

XfFrequency-domainrepresentationofasignal xtin linear frequency

XωFrequency-domainrepresentationofasignal xtin angularfrequency

Xrms Root mean square value of variable x

xtArbitrary signal

x0Amplitude of x

XVector

xIndicates that xis acomplexvariable

ˆxEstimateofx

∗Conjugatecomplexvalue of x

Electromotive force

Resonatorsusceptibility

yOutput quantity of asystem

ωAngularfrequency in radians persecond (rad/s)

ω0Eigenfrequency

ωrResonancefrequency

ΦMagneticﬂux

φDimensionless phase or angle

σStandard deviationofasignal

σ2Variance or normalized AC power

µMean value of asignal

µ2Mean or normalized DC power

ϕ, θ Phase, Angle

ϑPhaseintimeunits

ζDamping ratio

zDisplacement

Contributions to Original Knowledge

This thesispresentsthe following contributions to theoriginalknowledge in theﬁeldof

nanoelectromechanical systems (NEMS) andoptomechanics:

•

Arigorousand comprehensive study on frequencytracking schemesfor NEMS res-

onators, demonstrating throughbothexperimental andtheoretical resultsthatall

schemes(open-and closed-loop) canachieve comparable performance. Theﬁndings

suggest that theperformance limit is intrinsic to theNEMSresonator itself, allowing

thechoiceofschemetobebased on practicalconsiderationssuchascost, ease of

implementation,and application-speciﬁc requirements.

•

Thedevelopment of auniqueSelf-SustainingOscillatordesignwithpulsedfeedback,

notable forits ease of implementation, cost-eﬀectiveness, andhighconﬁgurability.

•

Thecreationofanadvanced frequencycounter architecture that is both adaptable

andcost-eﬀective, oﬀering enhanced usabilityfor frequencymonitoring taskswithin

theNEMScommunity.

•

Anovel thermalmodel forNEMSinfraredspectroscopy that accurately captures the

transient response of NEMS to infrared light absorption.Thismodel accounts forboth

keytimetransients: theintrinsic response of theresonatorand theframe-induced

response.

•

Theintroduction of an adaptableKalmanﬁlter that integrates theNEMSresponse

modeland thenoise proﬁlesinherent to NEMS devices, enablingfasterand more

precise measurements throughoptimal adaptive ﬁltering.

Introduction
Introduction
Background andMotivation
Nanoelectromechanical (NEMS) sensorshaveemerged as apowerfultoolfor awiderange of
applications due to theirhighsensitivity andthe abilitytodetect minute physical changes.
These sensorsoperate by monitoring resonancefrequency shifts,which canindicate various
environmental changesbythatinduceashiftinfrequency by changing parameters such
as change in mass[1
–
5], change in damping [6,7], or change in stiﬀness [8
–
10]. The
abilitytoaccurately track these frequencyshifts is crucialfor theeﬀectiveapplicationof
nanomechanical sensorsinbothresearchand industry.Itiscommonly perceivedwithin
thecommunitythatopen-loop frequency tracking schemesexhibit slower performance
comparedtotheir closed-loopcounterparts[11,12]. Moreover,literature[12]indicates that
thereisaninherent trade-oﬀbetween speed andaccuracy amongvarious conﬁgurations of
closed-loopfrequency tracking schemes. It is essential to thoroughly investigatebothopen-
andclosed-loopconﬁgurations andtoexperimentally validate whetherthe performance
limitations in speed andaccuracy of aNEMSsystemare due to thespeciﬁc trackingschemes
employed,orifthese limitations arefundamentally intrinsic to theNEMSitself.
Traditionalmethods formonitoring resonancefrequencies,suchasphase-locked loops
(PLL),havebeen widelyuseddue to theirprecision andstability. However, thesemethods
oftencomewithsigniﬁcantcostand complexity,makingthemless accessiblefor broader
applications.Frequency counters (FC) have been acost-eﬀectiveand simpleralternative,
oﬀeringcompetitive performancefor frequencyshift detectioninnanomechanicalsystems.
IncorporatingFCs as an alternativetoPLLs forfrequency detectioncould oﬀer signiﬁcant
beneﬁtstovarious research groups in this ﬁeld.
Whilevarious models existthatdescribethe behavior of NEMS in infrared sensing
(NEMS-IR)[10,13
–
20], thereremains aneed foracomprehensive modelthatincorporates
both transients—thoseinduced by theheating of themembraneand theheating of the
frame—intothe estimation of theabsorbedpower of theNEMSdevice.
3

Introduction

This dissertation exploresadvanced techniquesfor resonancefrequency monitoring in

nanomechanical sensing,novel tracking schemes, focusing on thedevelopment andapplica-

tion of adaptable frequencycounterswithphase ﬁltering,their integrationwithinfrared

spectroscopy,thermal modeling forinfraredspectroscopyand implementation of Kalman

ﬁlteringfor infraredspectroscopy.

Research Objectives

Theprimary objectives of this dissertationare:

To compare variousschemes fortracking resonancefrequency in micro-and nanome-

chanical resonators,evaluating theirtrade-oﬀs between speed andprecision.

To developanadaptable frequencycounter with phaseﬁlteringfor enhanced resonance

frequencymonitoring in nanomechanical sensing as an alternativetoPLL.

3. To modelthe thermalbehaviorofNEMSresonatorsininfraredspectroscopy.

To integrateKalmanﬁlteringtechniquesfor optimalsignalestimation in infrared

spectroscopy.

Structureofthe Dissertation

This dissertation is organized as follows:

•

Chapter 1:Theoretical Background andMethods -Oﬀers theoreticalfounda-

tionsthatare notextensivelycovered andmethods used in theexisting publications.

•

Chapter 2:Schemes forTrackingResonance Frequency -Exploresdiﬀerent

tracking methodsand comparestheir performanceboththeoretically andexperimen-

tally.

•

Chapter 3:Adaptable Frequency Counter withPhase Filtering -Presents the

design,implementation, andexperimentalresultsofthe proposed frequencycounter

forNEMSapplications.

•

Chapter 4:ThermalResponseModellingand KalmanFiltering -Proposes a

thermalmodel forNEMS-IR andexaminesthe useofKalmanﬁltering forimproving

frequencytracking accuracy,including experimental validation.

Introduction

•

Chapter 5:Conclusion andOutlook -Summarizes thekey ﬁndings, contributions,

andpotential future research directions in theﬁeldofnanomechanical sensing.

Throughthese chapters,thisdissertation aims to provideacomprehensive understanding

ofadvancedresonancefrequency monitoring techniquesand theirapplications,contributing

to theadvancementofnanomechanical sensing technology.

Chapter1

TheoreticalBackground andMethods

In this chapter, thetheoretical fundamentalsand methodsnecessary forunderstanding the

subsequent chapters arediscussed. Thechapter begins with an explorationofthe lumped

elementmodel of NEMS resonators,deﬁning theprimary quantities that will be used

throughoutthe dissertation. Following this,the main noisesources in NEMS resonators

aredescribed,providing afoundation forlater discussionsonnoise characterization.

Next, theAllanvariance, avaluable tool formeasuring frequencystabilityinresonators,

is explainedindetail. This section aims to equipthe reader with theknowledge required to

understand themethods used forassessing frequencystability.

Finally,the transduction andactuation schemes, used in this work,are brieﬂy described.

Most of thephysical quantities andanalyticalcalculationsinthis chapterare based on the

calculationsprovidedin[21].

1.1Nanomechanical Resonators

1.1.1DampedLinearResonator

Mechanical vibrations in physical systemsare associated with theperiodic conversion of

kineticenergytopotential energy andviceversa.Elastic elements such as beams, strings,

andplatesstore potentialenergyinthe form of deformationenergy, typically modeled

as aspring (Figure 1.1). Whenthe system is displaced from itsrest position,restoring

forces acceleratethe mass back toward equilibrium, converting potentialenergyintokinetic

energy.Asthe system traverses therestposition, kineticenergyistransformed back into

potentialenergy. This periodic energy exchange wouldcontinueindeﬁnitely if notfor the

presence of dissipative forces.

Chapter1 TheoreticalBackground andMethods

Figure1.1:Lumped elementmodel of aresonator

Forthe analysis of free vibration(eigenfrequency)ofcontinuum mechanical resonators,

dissipative forces areoftenneglected,assuming constant totalenergy. However, in real

systems, energy is dissipatedthrough mechanisms such as viscous damping, acoustic

transmission,surface losses, andinternaldissipation.Dissipative forces areapproximately

proportional to thevelocityofthe vibrationalmovementand areoften modeledasadashpot

(Figure 1.1).

Theforced vibrationofadamped system with asingledegreeoffreedom canberepresented

by thelumped elementresonatormodel.This system consists of alinearzero-mass spring,

alinear damping element, andamass. Whensubjected to aperiodic drivingforce

FtF

0cosωt

,the dynamics of thesystemare describedbyasecond-orderdiﬀerential

equation

m¨zc˙zkz Ft,(1.1)

where

is themass,

is thespring constant,and

is thedamping coeﬃcient. This

equation characterizes thesystem’sresponse to theappliedforce, taking into accountthe

eﬀectsofboththe spring anddamping elements.

1.1.2FreeUndamped Vibration

In thecaseofanondriven lumped-elementresonatorwithzerodamping (

c

0),the

totalenergyofthe system remains constant.The resonatorbehaves as an oscillator

withavibrationalamplitude expressed as

ztz

cosω0t

.During oscillation,energy

oscillatesbetween kineticand potentialforms.The eigenfrequency

ω0

of theundamped

free mechanical system is derivedfromthe equation

ω0k

m.(1.2)

1.1.3FreeDampedVibration

Foranondriven system,the equation of motion

(1.1)

reduces to thehomogeneousdiﬀerential

equation

¨z2ζω0˙zω2

0z0,(1.3)

wherethe dampingratio ζis deﬁnedas

ζc

km.(1.4)

Solving this equation with thetrial solution

ztz

γt

yields solutionsthatdescribethe

system’s behavior underdiﬀerentdamping conditions:

•Overdamped:ζ1.

•Critically damped:ζ1.

•Underdamped:ζ1.

1.1.4DrivenDampedVibration

Theresponse of alinear lumped-elementresonatortoaharmonic driving force

Ft

cosωt

combinestransient andsteady-stateresponses. Forthe steady-state response the

diﬀerential equation of motion is solved using complexvariablesfor theforce

FtF

iωt

anddisplacement zz0eiωt,resulting in expressionsfor thecomplexamplitude

z0F0m

ω2

0ω22iζω0ωz0eiϕ,(1.5)

Chapter1 TheoreticalBackground andMethods

where

is thedisplacementamplitude and

thephase diﬀerence between theforce

anddisplacement signals. The magnitude of thecomplexdisplacementamplitude canbe

calculatedas

0z

 F

m

ω

0ω

24ζ

2(1.6)

andthe phaseas

ϕargz0arctan 2ζω0ω

ω2ω2

.(1.7)

Theequations foramplitude andphase canbothberewrittenasfunctionsofrelative

frequency ωω0as

z0F0k

1ω

ω0224ζ2ω2

ω2

02(1.8)

and

ϕargz0arctan 2ζω

ω0

ω

ω021

.(1.9)

By deﬁningthe static deﬂectionas

sF

k

,whichisthe displacementamplitude that

wouldbecausedbyastatic or lowfrequency force, thedisplacementcan be writtenas

ztz

δz0cosωt ϕ,(1.10)

where δz0is therelativeamplitude response andisgiven by

δz01

1ω

ω0224ζ2ω2

ω2

02.(1.11)

Theamplitude andphase responses arecalculatedand plottedinFigure 1.2.

It canbeseenthatthe amplitudereaches amaximum near theeigenfrequency

ω0

,which

is calledthe resonancefrequency.Itcan be calculatedthrough therelation

∂z0∂ω 

whichyields

ωrω012ζ2.(1.12)

Forverysmall damping

ζ

1, theresonancefrequency is very closetothe eigenfrequency

ωrω0.The phase response of thesystembecomes

ϕrarctan 12ζ2

ζ.(1.13)

(a)

10−1100101

ω/ω0

10−2

10−1

100

101

δz0

ζ=1

ζ=0.1

ζ=0.01

(b)

10−1100101

ω/ω0

−1.0

−0.8

−0.6

−0.4

−0.2

0.0

ϕ/π

ζ=1

ζ=0.1

ζ=0.01

Figure1.2:

Frequencyresponseofthelineardampedresonator modelfor diﬀerentdamping

ratios

.(a) Shows theamplitude response of themodel accordingtoequation

(1.11)

.Atresonance frequency for

ζ

1the systemsexperiences maximum

ampliﬁcation, whichincreases with decreasing damping ratio. (b)Shows the

phase response obtainedbyequation

(1.9)

.Itshows that thephase hasa

π

phase shiftatresonance andanincreasingslope fordecreasing damping ration.

Chapter1 TheoreticalBackground andMethods

1.1.5QualityFactor

Thereare multiple deﬁnitions of thequalityfactor. It canbedirectly derivedfromthe

amplituderesponseequationbyexamining theheightofthe resonancefrequency peak.

Equation (1.8)atthe resonancefrequency yields

z0ω=ωrF0

2ζ1ζ2F0

kQ, (1.14)

with

being thequalityfactorthatalsorepresentsthe ampliﬁcationofthe displacement

amplitude at resonance. Forverysmall damping

ζ

1, thequalityfactorcan be simpliﬁed

Q1

2ζ.(1.15)

In termsofphase response,the phaseslope at resonanceincreases with thequalityfactor.

Theslope of thephase response is givenby

∂ϕ

∂ωω=ωr1

ζωr2Q

ωr

.(1.16)

Amorephysicalinterpretationofthe qualityfactoristhe ratiobetween thestoredand

dissipatedenergyofthe systeminone cycleatresonance

Q2πW

ΔW,(1.17)

where

is thestoredenergyand Δ

thelostenergyduring oneoscillation cycle. If the

displacementatresonance is givenby

zz

cosωrt

,the totalenergyofthe system canbe

calculatedas

Wmax









2m˙z2







1

2mz2

0ω2

r.(1.18)

Thelostenergyduring onecycle canbedescribed by theworkdonebythe dissipative force

Fdc˙zandisgiven by

ΔW2π/ω

d˙zdtπcz2

0ωr,(1.19)

whichyieldsthe qualityfactor

Qmωr

c.(1.20)

Incorporating thedeﬁnitionsprovidedinequations

(1.4)

and

(1.12)

,the qualityfactorcan

0.000 0.002 0.004 0.006 0.008 0.010

Time (s)

−1.0

−0.5

0.0

0.5

1.0

Amplitude

e−t/τr

e−t/τrcos(ωrt)

Figure 1.3: Qualityfactorcharacterizationbythe ring-downmethod

be rewritteninterms of thedamping ratio

Q12ζ2

2ζ,(1.21)

andcan be simpliﬁedtoequation(1.15)for slight damping (ζ1).

Amethodofmeasuring thequalityfactoristhe ring-downmethod. Oneway of doingit,

is to drivethe resonatoruntil an oscillation with steady-state amplitude is reached. After

turning oﬀ thedrive,the systemcan be modeledasafree damped vibrationdescribed by

theexponentially decaying oscillation

ztz

−t/τ

rcos ω01ζ2t,(1.22)

with thetime constant

τr

ζω0

.Equation

(1.15)

canbeexpressed in termsofthe time

constant as

Q1

2ωrτr.(1.23)

Thering-downmethodisparticularlyuseful forhigh

resonators andwill be employedin

this work fordetermining thequalityfactor.

Chapter1 TheoreticalBackground andMethods

1.2Noise in Nanomechanical Resonators

1.2.1Noise Fundamentals

Oneofthe main characteristicsofevery sensoristhe minimum quantity it candetect,also

knownassensitivity [21]. Thesensitivity is thedetectionlimit,which is usually constrained

by noisephenomena.Astationary ergodic noisesignal

xt

,observedinatime interval

is arandomprocesstypically describedbythe following quantities [22–26]:

1. Mean:The averagevalue of thenoise signal:

µlim

T→∞

TT/2

−T/2

xtdt. (1.24)

2. NoisePower:The mean square of thenoise signal:

Plim

T→∞

TT/2

−T/2

x2tdt. (1.25)

3. NoiseVariance:The spread or dispersionofthe noisesignal:

σ2lim

T→∞

TT/2

−T/2xtµt2dt. (1.26)

By taking thesquarerootofthe variance, thestandard deviation

of thenoise process

can be calculated.Inthis work, we will primarily deal with Gaussian noiseprocesses. A

Gaussian noiseprocessisarandomprocesswithaGaussian distributionasits probability

densityfunction (PDF). Figure 1.4 showsthe PDFofaPython-generated randomsequence

with µ1and σ1.

Noiseprocessescan also be describedinthe frequencydomain. Astandard tool to examine

noise signalsinthe frequencydomain is thepower spectraldensity (PSD)[22,24]. It describes

thepower distributionofastationary randomprocess across frequencies. According to

theWiener-Khinchin theorem, thetwo-sidedPSD

SII

xf

andthe auto-correlation function

Rxτof astationaryrandomprocess areaFourierpair, described by therelations:

SII

xf∞

−∞ Rxτe−i2πfτ dτ, (1.27)

Rxτ∞

−∞ SII

xfei2πfτ df. (1.28)

−2 0 2 4

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

fx(x)

Mean: 1.00

Std Dev: 1.00

Figure1.4:

Python simulation forvisualizationofthe probabilitydensity function of a

Gaussian noiseprocess, highlighting itsmeanand standard deviation.

Knowing oneofthese twopropertiesallows theother to be obtainedbythe Fourier

transform.

In theliterature, twoversionsofthe PowerSpectralDensity (PSD)are commonly used:

theone-sidedPSD,denoted as

Sf

,and thetwo-sided PSD, denotedas

IIf

.The

two-sidedPSD is symmetricaround

f

0and canbeconvertedintothe one-sidedPSD

using therelation:

Sf2S

IIffor f0.(1.29)

To clarifythis, consider thenormalized DC power, whichisthe square of themeanofthe

signal.For atwo-sided PSD, theDCpower is theareaunderthe impulseat

f

0and can

be expressedas[22]:

PDC µ20+

0−SIIfdf. (1.30)

In contrast,for aone-sidedPSD,the normalized DC powerisgiven by:

PDC µ20+

Sfdf. (1.31)

Thevariance, or thenormalized AC powerofasignal,can be obtainedbyintegrating

Chapter1 TheoreticalBackground andMethods

thetwo-sided PSDoverall frequencies, excluding theimpulseatf0[22]:

PAC σ20−

−∞ SIIfdf∞

0+SIIfdf. (1.32)

Similarly, forthe one-sidedPSD,the normalized AC poweris:

PAC σ2∞

0+Sfdf. (1.33)

Thepower spectraldensity (PSD)isdirectly relatedtothe magnitude spectrumofa

signal.The spectrumofasignal

xt

canbeobtainedbyperformingthe Fouriertransform,

yielding:

Xf∞

−∞ xte−i2πft dt, (1.34)

Xω∞

−∞ xte−iωt dt, (1.35)

where

Xf

and

Xω

representthe signal spectruminthe linear frequency

(Hz) and

angularfrequency ω(rad/s) domains,respectively.

Forexample,if

xt

hasunitsofVolt(V),both

Xf

and

Xω

will have unitsof

Volt-seconds (Vs) due to thetime integration. Thetwo spectra arerelated by substituting

ω2πf in Xω:

XfXω,where ω2πf. (1.36)

Althoughthe spectra areequivalent, they areexpressed on diﬀerentfrequency scales.

This distinction is crucial, particularlywhenintegrating over thefrequency axis,suchasin

theinverse Fouriertransform:

xt∞

−∞ Xfei2πft df, (1.37)

xt1

2π∞

−∞ Xωeiωt dω. (1.38)

In Eq.

(1.37)

,the integrationoverthe linear frequency

restores theoriginalsignal

with unitsof(V).However,inEq.

(1.38)

,the factor 1



π

is required to accountfor

therelationship d

ω

,ensuring theunitsremainconsistentand thetime signal is

recoveredin(V).

Thetwo-sidedPSD canbecalculatedfromthe corresponding magnitude spectra as

follows:

SII

xflim

T→∞

TXf2,(1.39)

SII

xωlim

T→∞

TXω2,(1.40)

wherebothversionsofthe PSDhaveunitsof(

Hz

)and diﬀeronlybythe frequency

scaling. Thetotal signal powerisobtainedbyintegrating thePSD over allfrequencies:

P∞

−∞ SII

xfdf1

2π∞

−∞ SII

xωdω. (1.41)

Again, when integrating over angularfrequency

,the correctionfactor1



π

is necessary

to maintain unit consistency, similartothe inverseFourier transform.

It is worth noting that in some literature[27], thePSD with respect to angularfrequency

is expressed in unitsof(

(rad/s)

). In such cases, we denote this PSDas

ω

,which

includesthe 12πscaling inherently in theunits. Thetotal poweristhencalculatedas:

P∞

−∞ Sc

xωdω. (1.42)

Onespecial type of noiseprocess that is very commonly used in many models is white

Gaussian noise(WGN).Ithas aGaussian-distributedPDF andaconstant PSD(

Sf

const.

). Theconstantvalue of thePSD forsuchaprocess, referredtoasthe noiselevel,is

typically denotedas

SfN

forasingle-sided spectrum, or

SIIfN

2for adouble-sided

spectrum.

When anoise process passes throughalinear system,spectralanalysiscan be useful.

Figure 1.5 showsalinear system with atransfer function

Hs

that hasanoisesignalwith

apower spectraldensity

Sxs

at itsinput (where

sαiω

is thecomplexfrequency).

Theresulting PSDatthe output of thesystemcan be calculated by multiplyingthe PSD

of theinput signal with thesquared magnitude response of thesystem[25]

SysHs2Sxs,(1.43)

where

Hs

is thetransfer function of thesystem. If thetransferfunction

Hs

is known

asimple wayofdeterminingthe square magnitude response is by using therelation

Chapter1 TheoreticalBackground andMethods

Property Description

SII

xfS

xfSII

xfhasevensymmetryaround f0.

SII

xfRS

xfis real.

∞

−∞ SII

xfdfP

Theareaunder

SII

is themeanvalue of the

powerofxt.

xtVS

xfV

Hz If xthasunitsV,then SII

xhasunits V2

Hz .

0+

0−SII

xfdfPDC µ2

Theareaunderanimpulsein

f

0is

thesquareofthe mean or thenormalized DC

powerofxt.

0−

−∞ SII

xfdf∞

0+SII

xfdfPAC σ2

Theareaunder

SII

excluding theimpulseat

f

0isthe variance or normalized AC power

of xt.

SII

xf0S

xis positive forall frequencies.

Table 1.1: Properties of thepower spectraldensity SII

xfof thesignal xt.

Hs2HsH∗s,where H∗sis theconjugatecomplexofHs.

Hs

sS

sHs2Sxs

Figure1.5:

Blockdiagram representation of theresponseofalinearsystemtonoise input.

In this work thenoise processeswill forthe most part been treated in thespectraldomain

thereforesomepropertiesofthe PSDwillbesummarized in Table 1.1 (takenfrom[22]).

1.2.2Relationbetween Amplitude, Phaseand FrequencyNoise

Noiseprocesses areacritical limiting factor in theperformance of NEMS sensors,making it

essential to understand howtheyimpact thesignalofinterest. High-

resonators,due to

theirwell-deﬁned resonancefrequency andsteep phaseresponse, areparticularlywell-suited

as frequencytransducers, whereaphysical quantity is mapped to aresonancefrequency

shift. To accurately model systemperformance,itiscrucial to comprehend theinterplay

between amplitude noise, phase noise, andfrequency noise.

Consideranarbitrary noiseless signal

xtAcosω0t

,withamplitude

andangular

frequency ω0.Various typesofﬂuctuations canbeintroduced as follows [26]:

•Amplitude ﬂuctuations:

xtAA

cosω0t(1.44)

where Anrepresentsthe amplitude noise.

•Phaseﬂuctuations:

xtAcosω0tϕn(1.45)

where ϕndenotesthe phase noise.

•Frequency ﬂuctuations:

xtAcosω0ωnt(1.46)

where ωnsigniﬁesthe frequencynoise.

Amplitude,phase, andfrequency noiseare generally notcompletelyindependent.Amore

comprehensive expression forthe signal canbewrittenas:

xtAcosω0tϕnt,(1.47)

where

xt

represents the signal,consistingofacosine wave with amplitude

,angular

frequency ω0andarbitrary phase ϕ,and ntdenotesthe additivenoise.

Quadrature (I/Q)Demodulator

Theamplitude andphase noiserelationcan be shownonastandard quadrature (I/Q)

demodulator[22,24,28]. This type of demodulator,shown in Figure 1.6,isalsoapartofmany

lock-inampliﬁers, such as theZurich InstrumentsMFLI. It takesasignal

xtAcosω0t

andmultipliesitwithits quadrature components

cosω0t

and

sinω0t

.Multiplying

thosevaluesby2will yieldthe quadrature components

sIt

and

sQt

in theamplitude

domain. Multiplying them by

2

will yieldtheir valuesinthe RMSdomain. We chosethe

ﬁrst becauseitismoreconvenient to work in theamplitude domain,eventhoughsome

commercialdevices likethe Zurich InstrumentsMFLIuse the 2normalization.

Chapter1 TheoreticalBackground andMethods



xref,It2cosω0t



xref,Qt2sinω0t

Low-pass

ﬁlter

Low-pass

ﬁlter

xt

x′

Figure1.6:

Quadrature demodulatorthatcreates an analytical signal from asinusoidat

frequency ω0

I/QDemodulatoion of aSinusoidalSignal

Firstly, anoiseless sinusoidal signal is appliedatthe input of theI/Q demodulator

xtx

0cosω0tϕ0.(1.48)

Theﬁrststepincalculating thein-phase componentistomultiply it by xref,I2cosω0t:

x′

Itx

0cosω0tϕ02cosωt(1.49)

x0cosϕ0cos2ω0tϕ0.(1.50)

Applyingalow-pass ﬁlterto

′

t

eliminatesthe doubled frequencycomponent (ifthe

low-pass cut-oﬀ frequencyislower than 2ω0), yielding thein-phase component:

xItx

0cosϕ0.(1.51)

Analogously, thequadraturecomponent is derivedas:

xQtx

0sinϕ0.(1.52)

Thein-phase andquadraturecomponentscreateacomplexanalyticalsignal:

xxIixQ.(1.53)

Figure1.7:

Eﬀect of quadrature demodulation on noiseinthe in-phase (I)and quadrature

(Q)components. Theinput noisespectrum

Sxf

is convolvedwiththe reference

spectrum, doubling thenoise PSDinbothcomponents:

Sx,IfS

x,Q

f

f

.The bandwidth

represents thelow-passﬁlter bandwidthinthe

demodulator.

If thephase of theinput signal

xt

nπ

,the wholesignalwill be containedinthe in-phase

component

xxI

.Onthe otherhand, if thephase of theinput signal is

nπ

2, thewhole

signal will be in thequadraturecomponent.For otherphases, thesignalwill be split

between thetwo.The complexsignal xcanberepresented in polarformas:

xtrte

iϕ0(t)(1.54)

with

rtx

tx

t,(1.55)

ϕ0tarctan xQt

xIt(1.56)

representing theamplitude andphase of theoriginalinput signal xt.

Chapter1 TheoreticalBackground andMethods

I/QDemodulationofaNoiseSignal

Secondly,awhiteGaussiannoise signal is appliedtothe input of thequadraturedemodulator,

representedby:

xtnt,(1.57)

where

nt

is thenoise signal.Whenanoisesignalwithapowerspectraldensity (PSD)

SII

xN

2isappliedtothe input,spectralanalysisbecomesauseful tool forpredicting the

noiseinthe quadrature components.Asdescribed in Section 1.2.1,where thePSD of the

noise at theoutput of alineartime-invariant(LTI) system is calculated by multiplyingit

by thesquared magnitude of thesystem’s frequencyresponse, themaindiﬀerence here is

that thenoise is instead multipliedbythe referencesignal. Thein-phase referencesignalin

thefrequency domaincan be describedby[22,28]

Xref,Ifδff

δff

,(1.58)

whichyields thesquaremagnitude

Xref,If2δff

δff

.(1.59)

Analogously, thesameresult is obtainedfor thequadraturereference

Xref,Qf2X

ref,If2

Multiplicationofthe noise at theinputwiththe reference in thetime domain resultsinthe

convolutionofthe twosignals in thefrequency domain. This is described by therelation

SII

x,I/QfS

xfX

ref,I/Qf2SII

xfδff

δff



SII

xfS

xf2S

xf.(1.60)

Thenoise at theinput,withaconstant PSD, is convolvedwithtwo Diracdelta impulses,

each with amagnitude of 1. Consequently,the noisePSD at theoutput of themultiplier

is twicethatofthe input noisePSD,asillustratedinFigure 1.7.The low-pass ﬁlter

following themultiplierisequivalent, forthe noisesignal, to aband-pass ﬁlter with the

same bandwidth

placed before themultiplier. Althoughthe low-pass ﬁltered signal

exhibitstwice thePSD of theinput,ithas thesametotal noisepower (variance) as the

band-pass ﬁltered signal centeredaround

with bandwidth

.This equivalence arises

becausethe band-passsignalcontains twospectralcomponentsatthe positive andnegative

frequencies, each with bandwidth

,matching thebandwidthofthe low-pass ﬁltered

signal.

Sincethe noiseisconvolved with diﬀerent signals, thenoise processes in thetwo arms of

thedemodulatorcan be treatedastwo uncorrelatedprocesses:

xItn

t,

tn

t,

tn

t,(1.61)

whereboth

nIt

and

nQt

have thesamestatistical properties likePDF (inthis case

Gaussian), PSDand variance described by therelations

fnInIf

n

 1

2πσ2exp n2

2σ2,(1.62)

SII

x,IfS

x,Qf2S

xf,(1.63)

σ2

Iσ2

Qσ2.(1.64)

Thequadraturepair nIand nQtogether form theanalyticalnoise signal

ntn

tinQt,(1.65)

whichisacomplexrandomvariable.The PDFofacomplexrandomvariablecan be deﬁned

asjoint PDFofits real andcomplex part [28]. Thejoint PDFof

and

canbewritten

as theproductofthe PDFs from equation (1.62)

fnI,nQnI,n

 1

2πσ2exp n2

In2

2σ2.(1.66)

Figure 1.8 showsathree-dimensionalplotofthe jointprobabilitydensity function (PDF),

describedbyatwo-dimensionalGaussian distribution. Sincethe quadrature components

have equalvariances, thecontoursofthe jointdistributionare concentriccircles.The joint

distributioncan be projectedontothe twocoordinate planes, yielding themarginalPDFs

(

fnInI,f

n



of thequadraturecomponents, as also showninFigure 1.8.Itisoften

usefultorepresent thethree-dimensionalplotintwo dimensions by taking thecontour at

radius

,whichcorresponds to thestandard deviationofthe quadrature noiseprocesses.

This is depictedinFigure 1.9a.

Chapter1 TheoreticalBackground andMethods

PDF

Figure1.8:

Thejoint probabilitydensity function (PDF)ofthe quadrature components

is representedbyatwo-dimensionalGaussian distribution. By projecting the

jointPDF onto the twocoordinateplanes, themarginaldistributions of the

quadrature components areobtained, each following aone-dimensionalGaussian

distribution.

It is oftenmorepractical to representsignals in termsofamplitude andphase (polar coor-

dinates) rather than in termsofthe real andimaginary components (Cartesian coordinates).

To achieve this,acoordinate transformationisrequired.For thepolar representation of

theamplitude andphase arecalculatedasshown in equations(1.67)and (1.68):

rtn

tn

t,(1.67)

ϕtarctan nQt

nIt.(1.68)

Thejoint PDFfromequation

(1.66)

canbetransformed from Cartesian to polarcoordi-

natesusing theJacobian[29]

fr,ϕ r, ϕf

,nQnI,n

J,(1.69)

wherethe Jacobiandeterminant Jris givenbythe transformation

nIrcosϕ,(1.70a)

nQrsinϕ.(1.70b)

This resultsinthe jointPDF in polarcoordinates

fr,ϕ r, ϕ r

2πσ2exp r2

2σ2.(1.71)

It is evident that thejoint PDFinpolar coordinateshas no dependenceon

,indicating

that

and

arestatistically independent.Therefore,the jointPDF canbefactorized as a

productofthe amplitude andphase PDFs

fr,ϕ r, ϕf

rf

ϕ.(1.72)

Foracircularly symmetric Gaussian distribution, thephase

is uniformly distributed

between πand π,asexpressed by

fϕϕ









2π

,πϕπ,

0,otherwise.

(1.73)

This leadstothe amplitude PDFfollowing a Rayleighdistribution [24,28]:

frrr

2exp r2

2σ2,r0.(1.74)

with themean

µrσπ2

andvariance

σ2

r(4−π)

2σ2

.Fromthe generalconservation of

powerlaw fornoise signals,if

and

arezeromeanprocesses, thefollowing relation can

be written

µ2

rσ2

rσ2

Iσ2

Q,(1.75)

wherethe variancesofthe I/Qarms contributetothe sumofthe mean squaredand variance

of theresulting amplitude noise. Theamplitude variance

σ2

is smallerthanthe sumofthe

initialinput variances,because some of thenoise powerisconvertedtothe mean value

µr

I/QDemodulationofaSinusoidalSignalWithAddedNoise

Nowthatthe demodulatoroutput is treated forsignals

xt

andnoise processes

nt

separately,their sumcan be considered as asuperposition.Assuming theinput signal is

givenas

xtx

0cosω0tϕ0nt,(1.76)

Chapter1 TheoreticalBackground andMethods

(a) (b)

Figure1.9:

Cartesian representation of thesignals in theI/Q arms.(a) When only noise

is presentatthe input of theI/Q demodulator(

xtnt

), thedistribution

is circularlysymmetric around theorigin, resulting in an evenly distributed

phasenoise as described in relation

(1.73)

.(b) Whenacombination of a

sinusoidalsignaland noiseispresentatthe input of theI/Q demodulator

(

xtx

cosωt ϕ0nt

), thesinusoidalsignalintroduces ameanoﬀset

µx0

in thejoint Gaussian distributiondue to thenoise. As theamplitude

of thesinusoidalsignalincreases, thedistributionshifts further away from the

origin, therebyreducing thephase noise ϕwhen µσ.

thequadraturecomponentsatthe output of thedemodulatorare givenby
x
Ix
0
cosϕ0n
I
,
x
Qx
0sinϕ0n
Q
.(1.77)
Theamplitude andphase can be calculatedasinequations (1.67)and (1.68), resulting in
themeanamplitude
µ2
rµ2
Iµ2
Qx2
0.(1.78)
Thequadraturepairforms an analytical signal
xtx
I
tixQt,(1.79)
with ajoint Gaussian distribution
fxI,xQxI,x
Q
 1
2πσ2exp xIx0cosϕ02x
Qx
0sinϕ02
2σ2,(1.80)
whichsharesthe same properties as thejoint PDFinequation
(1.66)
,but is shiftedby
x
0
cosϕ0
in thein-phase directionand
x0sinϕ0
in thequadraturedirection, as shownin
Figure 1.9b.
By applying aproceduresimilartothe onedescribed in theprevioussection forderiving
theRayleighdistribution, it canbeshown that theamplitude probabilitydensity function
forasinusoidalsignalwithaddednoise follows a Rice or Rician distribution[24,28,30]:
frrr
σ
2exp r2µ2
r
2σ2I0µrr
σ2,(1.81)
where
σ2σ2
Iσ2
Q
is thevarianceofthe quadrature components,and
I0
is themodiﬁed
Bessel function of theﬁrstkind of zeroorder.Equation(1.81)can be normalized as
fvvvexp v2a2
2I0av(1.82)
with
fvvσfrr,a
µ
r
σ
,v
r
σ
.(1.83)
Figure 1.10a showsthe Rice distributionfor diﬀerentvaluesof
a
.Fromthe curves, two
observations canbemade:
27

Chapter1 TheoreticalBackground andMethods

(a)

0.0 2.5 5.0 7.510.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

fv(v)

a=0

a=1

a=2

a=3

a=4

a=5

(b)

−2 0 2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

fϕ(ϕ)

µ=0

µ=1

µ=2

µ=3

Figure1.10:

TheRicedistribution.(a) Rice amplitude distributionfor diﬀerentvaluesof

(b)Ricephase distributionfor diﬀerentamplitudesofsignal µ

•

If only noiseispresent at theinput (

a

0),the Rice distributioncollapses to the

Rayleigh distribution.

•

a

1, thedistribution becomesGaussian in thevicinityof

va

.Thisoccurs when

thesinusoidamplitude is much larger than thenoise deviation

.Inthis case the

amplitude noiseisapproximately Gaussian with adeviation of σAσ.

Thephase probabilitydensity function (PDF)isdescribed by theRicephase distribution[30]

fϕϕ1

2πexp µ2

2σ21µ

σπ

2erfcx  µ

σ2cosϕϕ0,πϕπ, (1.84)

where

ϕ0arctanµQµI

is themeanphase determined by thephase of thesinusoid, and

erfcxisthe scaled complementary errorfunction.

Figure 1.10b showsthe phasedistributionfor diﬀerentsignalamplitudes. Thefollowing

observations canbemade

•

Whenthe mean amplitudeiszero(

µ

0),the phase distributionreduces to aconstant

12πbetween πand π.

•

As theamplitude of thesignalincreases, the

erfcx

function part of theequation

becomeslarger, leading to thephase anglebeing more concentratedaround themean

phase angle

ϕ0

.This concentration eﬀect causes thedistributiontobecome more

peakedand symmetric, resembling aGaussian distributioncenteredaround

ϕ0

for

µσ.

•

Withincreasing amplitude expression

(1.84)

asymptotes theGaussian distribution

yielding thelimiting case [29]

lim

µ→∞ fϕϕ 1

2πσ

exp 



ϕϕ02

2σ

µ2



.(1.85)

whichresultsinaphase variance σϕσµ.

In summary,fromequation

(1.85)

theamplitude-to-phase noiseconversionofasinusoid

signal with additive whiteGaussian noisecan be simpliﬁedfor alarge amplitude compared

to noise(x

0σ

)as

ϕσ

,(1.86)

where

σx

is thenoise deviationofthe noisecomponent of thesignal

xt

,and

is the

amplitude of thesinusoidalpartofthe signal.Ifoperating within thebandwidth

thedemodulator, an equivalentexpressioncan be derivedinthe frequencydomainfor the

phase noisePSD

Sϕω2S

ωω



,for πBW ωπBW. (1.87)

Thereisadirect relationship between phase noiseand frequencynoise,asdescribed by

equations

(1.45)

and

(1.46)

.Speciﬁcally,frequency noisecan be derivedfromphase noiseby

taking thederivativeofthe phase ﬂuctuations with respect to time [31]. This relationship

is expressedas

t∂ϕnt

∂t ,(1.88)

where

ωnt

represents thefrequency ﬂuctuations and

ϕnt

denotesthe phase ﬂuctuations.

Consequently,the powerspectraldensity (PSD)ofthe frequencynoise increases with

frequency, andisgiven by

Sωωω

ω,(1.89)

where Sωωis thefrequency noisePSD and Sϕωis thephase noisePSD.

Chapter1 TheoreticalBackground andMethods

Another wayofacquiring thefrequency in mechanical oscillatorismapping it to the

phase, resulting in phase noisebeing directly proportional to thephase noise[21,31,32]

Sωω1

ω.(1.90)

Both phase to frequencytechniqueswill be furtherdiscussed in Chapter 2.

NoiseRelations in aFrequencyCounter

Afrequency counterisaninstrument that measures thetime period of afullsignalcycle.

Letusassume that anoiseless sinusoidalsignalisapplied to theinput of thefrequency

counter:

xtx

0sinω0t,(1.91)

where

is theamplitude and

ω0

theangularfrequency.The frequencycounter typically

uses zerocrossingsonthe positive slopeofthe signal to measurethe time period,which

canbedeﬁnedbythe following conditions:

xt0,(1.92a)

˙xt0,(1.92b)

yielding samplesattimes:

tkk2π

ω0kT0,(1.93)

where

is thesignalperiodand

kN

is an integer. Thefrequency counterestimates the

period of onecycle as:

T0tktk−1,(1.94)

which, foranoiselesssine wave andaperfect time resolutionofthe counter, will result in

T0T0.The frequencyisthencalculatedasthe inverseofthe measured period.

Now, considerthe case wherethe frequencycounter measures asinusoidalsignalwith

addedwhiteGaussian noise:

xtx

0sinω0tnt,(1.95)

where

nt

is theadditivenoise.The detected zerocrossing will occurwhenthe measured

signal satisﬁes

xt

0. Duetothe presence of thenoise

nt

,the detected phase

ˆϕ

Figure1.11:

Schematicrepresentationofthe signal

xt

linearized around thevicinityofthe

zerocrossing.The sine signal is shownalong with theadditivenoise standard

deviationinthe verticaldirection. Increasing theamplitude

enhances the

slopeofthe signal, therebyreducingthe phase error, described by therelation

σϕσ

x0.

will deviatefromthe actual signal phase ϕω0t.This phase errorcan be expressedas:

ϕeˆϕϕ. (1.96)

Combining equations(1.95)and (1.96)for xt0and ˆϕ0yields:

x0sinϕetnt.(1.97)

Foramplitudesmuchlargerthanthe noise, we canapproximatesinϕeϕ

,resulting in:

ϕetnt

.(1.98)

Thevarianceofthe phase errorcan then be calculatedas:

σ2

eVarϕeVarn

0σ2

,(1.99)

Chapter1 TheoreticalBackground andMethods

where

σ2

is thevarianceofthe noise process

nt

.The phase deviationcan be expressed as:

σϕσ

,(1.100)

whichisthe same result obtainedin(1.86).

1.2.3Thermomechanical Noise

Oneofthefundamental noisesources in NEMS resonators is thermomechanical noise

[12,21,31]. This noisearises due to thecoupling (interaction)between theresonator anda

thermalreservoir ﬁlledwithrandomly distributed phonons.Itmanifests as a noisyforce

appliedtothe resonator, described by theﬂuctuation-dissipation theorem(FDT).The

NEMS resonatorcan be modeledasalumped-element system,asdiscussed in Section 1.1.1.

ThewhiteGaussian forcenoise with thepower spectraldensity (PSD)

SFωN

,acting

on theresonator, leadstoanamplitude noisePSD described by

SzωH

ω2SFω,(1.101)

where

HRω2

is thesquared magnitude of theresonator’stransfer function.For small

damping,this canbeapproximatedbyaLorentzian function

HRω2ω21

4m2

eﬀ ω2

0

ω0ω2ω

2



,(1.102)

where

ω

is theresonator’s susceptibility. Accordingtothe equipartition theorem[21],

thetotal energy of amechanical system with onedegreeoffreedom in thermalequilibrium is

kB

,where



is thetemperature and

is theBoltzmann constant.For alumped-element

modelwithelastic restorativeenergy, theconservationofenergyequationis:

2kB1

eﬀ z2

th1

eﬀ ω2

0z2

th,(1.103)

where

meﬀ

and

keﬀ

arethe eﬀectivemassand spring constant of theresonator,and

z2

th

themeansquaredisplacement due to thermalnoise.Solving this for z2

thgives

z2

th k



eﬀ ω2

.(1.104)

Sincethermomechanical noiseisazero-mean process,the totalnoise poweriscontained

within thevariance, implying that thevarianceofthe displacement ﬂuctuationis
σ
2
zz
2
th
.
Thevariancecan be determinedbyintegrating thePSD over allfrequencies. Usingthe
variance-PSDrelationfromTable 1.1 andtreating it in angularfrequency,weobtain
σ2
z1
2π∞
0
Szωdω1
2π∞
0
SFωω2dω0NFQ
4ω3m2
eﬀ
.(1.105)
Solving equations(1.104)and (1.105)yields thethermomechanicalforce noiselevel
NF4meﬀ ω0kB
Q.(1.106)
Using equation
(1.101)
,the peak of thethermomechanical displacementnoise at the
Lorentzian resonancecan be calculatedas:
Szω04k
B
Q
m
eﬀ ω3
0
.(1.107)
It is important to note that,althoughthe thermalnoise force
NF
is inverselyproportional
to Q,the peak displacementnoise scalesproportionally with Q[21].
1.2.4Detection (Transduction)Noise
Detection or transduction noise,inthiscontext,referstoany noiseintroduced duringthe
conversion of mechanical displacement(or velocity)intoanelectricalsignal[12,21,31]. This
noisetypically manifests as additiveamplitude noisethatissuperimposed on thesignal
afterthe resonator’smotionhas been convertedintoanelectricalform, as illustrated in
Figure 1.12a.Assuming thetransduction process does notlimit thesignalbandwidth,it
canbemodeled as asimple gainelement with aconversionfactordenoted by
G
.Sincethe
transduced signal is electrical, it canbeanalyzed within theframework of circuit theory.
Thenoise canbemodeled as aparasitic voltage source connected in series(Figure 1.12b)or
aparasitic currentsourceconnected in parallel(Figure 1.12c)tothe networkits coupling
to,e.g.equivalentresistance. In circuit theory,noise sources aretypically expressed in
termsoftheir root mean square (RMS)values, whichcorrespondtothe standard deviation
ofthe noisesignals (
VnσV
or
InσI
). Theprimary transduction noisesources relevant to
this work are
•
Johnson-Nyquistthermalnoise –Commonly encounteredwhere metaltraces
with ohmic resistance areused fortransduction.
33