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Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Sensors, Signals and Noise 1
COURSEOUTLINE
• Introduction
• SignalsandNoise
• Filtering:Band-PassFilters3– BPF3
• Sensorsandassociatedelectronics
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Band-Pass Filtering 3 2
• AsynchronousMeasurementofSinusoidalSignals
• PrincipleofSynchronousMeasurements ofSinusoidalSignals
• NoiseFilteringinSynchronousMeasurements
• Lock-inAmplifierPrincipleandWeighting Function
• APPENDIX1:Stage-by-stageviewofSignalsintheLock-inAmplifier
• APPENDIX2:Stage-by-stageviewofNoise intheLock-inAmplifier
• APPENDIX3:Bandwidth,Response TimeandS/NoftheLock-inAmplifier
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Asynchronous measurement of sinusoidal signals 4
• Asynchronous (orphase-insensitive) techniquesweredevisedformeasuringasinusoidal signalwithoutneedinganauxiliaryreference thatpointsoutthepeakingtime(i.e.thephaseofthesignal).
• TheyarecurrentlyemployedinACvoltmetersandamperometers.
• Thebasiccircuitsofsuchmetersarethemean-squaredetectorthehalf-waverectifierthefull-waverectifier
• Foracorrectmeasurementoftheamplitudeofthesinusoidal signal,itisnecessarytoavoidfeedingaDCcomponenttotheinputofanasynchronousmetercircuit.Therefore,themetermustbeprecededbyafilterthatcutsoffthelow-frequencies,thatis,aband-passorahigh-passfilter.
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Asynchronous measurement of sinusoidal signalswith Mean-Square Detector
5
Low-PassFilter
( ) ( )cosx t A tω ϑ= + ( ) ( ) ( )2 2
2 2cos cos 2 22 2A Ay t A t tω ϑ ω ϑ= + = + +
( )2
2Az t =
t
• Itisapower-meter:theoutputisameasureofthetotal inputmeanpower,sumofsignalpower(proportionaltothesquareofamplitudeA2)plusnoisepower.
• Thelow-passfilterhasNOEFFECTOFNOISEREDUCTION.Infact,itdoesnotaveragetheinput,itaveragesthesquareoftheinput.
• ForimprovingtheS/Nitisnecessarytoinsertafilterbefore theMean-SquareDetector
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Low-PassFilter
( ) Az tπ
=
Asynchronous measurement of sinusoidal signals with Rectifier
6
( ) ( )cosx t A tω=
Half-WaveRectifier(HWR)
Full-WaveRectifier(FWR)( ) ( )cosx t A tω=
t
Low-PassFilter
( ) 2Az tπ
=
t
( )4 2
4 2
1cos cos2
T
T
A Az t dt A dT
π
π
ω ϕ ϕπ π
− −
= = =∫ ∫
( )4 2
4 2
2 1 2cos 2 cos2
T
T
A Az t dt A dT
π
π
ω ϕ ϕπ π
− −
= = =∫ ∫
2T πω
=
2T πω
=
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Asynchronous measurement of sinusoidal signals with Rectifier
7
• Themeasurementwitharectifier isnotreallyasynchronous,itisself-synchronized.Thesinusoidal signalitselfdecideswhenithastobepassedwithpositivepolarityandwhenpassedwithnegativepolarity(inthefull-waverectifier)ornotpassedatall(inthehalf-waverectifier).
• In such operation, the LPF reduces the contribution of the wide-band noise,thus improving the output S/N. However, this is true only if the input signalis remarkably higher than the noise, i.e. if the input S/N is high.
• Astheinputsignalisreducedthenoisegainsincreasinginfluenceontheswitchingtimeoftherectifier,whichprogressivelylosessynchronismwiththesignalandtendstobesynchronizedwiththezero-crossingsofthenoise.
• ThelossofsynchronizationprogressivelydegradesthenoisereductionbytheLPF.WithmoderateS/NtheimprovementduetoLPFismodest;withlowS/Nitisveryweak.WithS/N< 1 thereisnoimprovement,there isnotevenameasureofthesignal:theoutputisameasureofthenoisemeanabsolutevalue.
• Inconclusion,metersbasedonrectifierscanjustimproveanalreadygoodS/N.Theycan’thelptoimproveamodestS/NanditisoutofthequestiontousethemwhenS/N<1.ForimprovingS/Nitisnecessarytoemployfiltersbeforethemeter.
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Synchronous (or Phase-Sensitive) Measurements of Sinusoidal Signals
8
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Out+
_
R2R4
R3 RT
VA
Signal and Reference for Synchronization 9
( ) ( )cos mx t A tω ϕ= +
( ) ( )cos mm t B tω=
ACvoltagesupply
REFERENCE
A tobemeasured
Showsthefrequencyandphaseofthesignali.e.pointsoutthepeakinstantsofthesignal
*inthisexample ϕ=0 sincethepreamppassbandlimitismuchhigherthanthesignalfrequencyfm
SIGNALOUT
ϕconstantandknown*
KEYEXAMPLEforthestudyofsynchronousmeasurements andnarrow-bandfiltering
VA
RT =RT(θ)resistancetracksavariableθ,(e.g.a temperatureorastrain);resistancesR2 ,R3 ,R4 areconstant
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Out+
_
R2R4
R3 RT
VA
Signal and Reference for Synchronization 10
( ) ( )cos mx t A tω ϕ= +
( ) ( )cos mm t B tω=
A tobemeasuredϕconstant
RT e.g.strainsensor,theresistancevariesfollowingamechanicalstrainθ
a)incaseswithconstant strainθconstant A à x(t)isapuresinusoidalsignal
b)incaseswithslowlyvariable strainθ =θ(t)variable A=A(t)à x(t) isamodulatedsinusoidalsignal
SLOWvariations=theFouriercomponentsofA(f)=F[A(t)] havefrequencies f<<fm
VA
KEYEXAMPLEforthestudyofsynchronousmeasurements andnarrow-bandfiltering
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Sample&Hold
outV A=inV A=
DCoutputsignal
Signal
δ(t)
Elementary synchronous measurement: peak sampling 11
fm-fm f
|X|
|W|
f
t
x(t)
t
w(t)=δ(t)
A
( )2 mA f fδ −( )
2 mA f fδ +
Timedomain Frequencydomain
Signal
Weighting
NOFILTERINGACTION:outputnoisepower=fullinputnoisepower
1
Reference
Samplingdriver
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Synchronous measurement with averaging over many samples N >>1 of the peak
13
t
t
tT-T
rT(t)
t
w(t)
( ) ( ) ( )Tw t m t r t= ⋅
2 1mN f T= ?
( ) ( )cos 2 mx t A f tπ=
T T
m(t)
TotakeNsamplesisequivalenttogate
afree-runningsampler
x(t)
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Synchronous measurement with averaging over many samples N >>1 of the peak
14
fm-fm f
|X|
f
|RT|
f
t
t
tT-T
rT(t)
t
fm-fm f
fm-fm
-2fm 2fm
........m(t) |M|
2fm-2fm
FILTERING:narrowbandsatfrequencies0,fm,2fm,3fm ....
........
( ) ( ) ( )Tw t m t r t= ⋅ ( ) * TW f M R≈
12mf T
?
( ) ( )cos 2 mx t A f tπ=
12nf T
Δ =
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Poor Noise Filtering by Sample-Averaging 15
fm-fm
|X|
f
|W|2
fm-fm 2fm-2fm
........
3fm-3fm
f
f
Sn1/fnoise whitenoise
• Atfm useful band:itcollects thesignal andsomewhitenoisearoundit• atf=0VERYHARMFULband:itcollects1/fnoiseandnosignal• at2fm ,3fm ,.... harmful bands:theycollect justwhitenoisewithoutanysignal
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Synchronous measurement with DC suppression by summing positive peak and subtracting negative peak samples
16
t
x(t)
t
tT-T
rT(t)
t
m(t)
w(t)
T T( ) ( ) ( )Tw t m t r t= ⋅
Totake2Nsamplesisequivalenttogate
afree-runningsampler
NB:equalnumberofpositiveandnegativesamples(zeronetarea)
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Synchronous measurement with DC suppression by summing positive peak and subtracting negative peak samples
17
fm-fm f
|X|
f
|RT|
f
t
x(t)
t
tT-T
rT(t)
t
fm-fm f
fm-fm
-2fm 2fm
........m(t) |M|
2fm-2fm
FILTERING:narrowbandsatfm andatodd multiples3fm,5fm ....
........
( ) ( ) ( )Tw t m t r t= ⋅ ( ) * TW f M R≈
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Improved Filtering by Sample-Averaging with DC Suppression
18
fm-fm
|X|
ffm-fm 2fm-2fm
........
3fm-3fm
f
f
Sn 1/fnoisewhitenoise
• atfm useful bandthatcollectsthesignal andsomewhitenoisearoundit• Nomorebandatf=0,nomorecollectionof1/fnoise• at3fm ,5fm....residualharmful bandsthatcollect justwhitenoisewithoutanysignal:
howcanwegetridalsoofthem?
|W|2
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
-T T
Continuous sinusoidal weighting instead of peak sampling 19
fm-fm f
|X|
f
|RT|
f
t
t
tT-T
rT(t)
t
fm-fm f
fm-fm
|M|
TRULYEFFICIENTFILTERING:justonenarrowbandatfm
( ) ( ) ( )Tw t m t r t= ⋅( ) * TW f M R≈
( ) ( )cos mm t B tω=
( ) ( )cos mx t A tω=
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Optimized noise filtering in synchronous measurement 20
fm-fm
|X|
ffm-fm 2fm-2fm 3fm-3fm
f
f
Sn 1/fnoisewhitenoise
• atfm useful bandthatcollectsthesignal andsomewhitenoisearoundit• Nobandatf=0,nocollection of1/fnoise• Noresidualbandsat3fm ,5fm ...nomorecollectionofwhitenoisewithoutanysignal
Howtoimplementthisoptimizedsynchronousmeasurement?
|W|2
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Basic set-up for Synchronous Measurement with optimized noise filtering
21
Sy A∝A DCOutput
InputSignal
Reference
GatedIntegrator
2T
-T T ft fm-fm
( ) ( ) ( )Tw t m t r t= ⋅ ( ) * TW f M R≈Weighting intimedomain Weighting infrequencydomain
z(t)=x(t)·m(t)( ) ( )cos mx t A tω=
( ) ( )cos mm t B tω=
AnalogMultiplier
NB:Low-PassFilter(withswitchedparameters)
• NB thereferenceinputtothemultiplier isaSTANDARDwaveform,whichabsolutelydoes NOTdependonthesignal: itisthesameforanysignal!!
• Thereforetheset-up isaLINEAR filter(withtime-variantparameters)
12nf T
Δ =GIcutGIcut
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Main Advantages ofSynchronous Measurements with optimized noise filtering
22
Thislineartime-variantfiltercomposedbyAnalogMultiplier(Demodulator)andGatedIntegrator(Low-passfilterwithswitched-parameter)hasaweightingfunctionsimilartothatofatunedfilterwithconstant-parameter,buthasbasicadvantagesoverit:• Centerfrequencyfm andwidthΔfn areindependentlyset
• Thecenter frequencyissetbythereferencem(t) andlockedatthefrequencyfm
• Incaseswherefm hasnotaverystablevaluethefilterband-centertracksit:thesignal isthuskeptintheadmissionbandevenifthewidthΔfn isverynarrow.
• The width Δfn = 1/2T issetbytheGI,itisthe(bilateral)passbandoftheGI
• NarrowΔfn andhighqualityfactorQcanthusbeeasilyobtainedatanyfm
1m
n
fQf
=Δ
?n mf fΔ =
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Intuitive Interpretation ofSynchronous Measurement with optimized noise filtering
23
Itis instructivetofollowtheactionofthefilterthroughthevariousstages,thatis• Multiplierstage,wherethereferencemodulatestheinputsignal,
thusshiftingthesignal infrequencydownby-fm andupby+fm• GatedIntegratorstage,whichintegratesover2Ttheoutputofthemultiplier
Bynotingthattheinputsignal isamodulatedsignalwithA(t)slowwithrespecttofm (i.e.withFouriercomponentsA(f) onlyatlowfrequencies f<<fm)
weunderstandthat
• themultiplier performsaDEMODULATION thata)shiftsx(t)infrequencydowntof=0,thatis,bringsA(t)backtoits«BASEBAND»b)shiftsx(t)alsouptothehigherfrequency2fm
• theGatedIntegrator1)performsameasurement ofA(t)averagedover2T2)cutsoffthecomponentswithfrequencyhigherthantheGIlow-passband-limit
( ) ( ) ( )cos 2 mx t A t f tπ=
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
From Discrete to Continuous Synchronous Measurements:principle of the Lock-in Amplifier (LIA)
25
Sy A∝
Aquasi-DC
Outputsignal
Inputsignal
Reference
z(t)=x(t)·m(t)( ) ( )cos mx t A tω=
( ) ( )cos mm t B tω=
AnalogMultiplier
R
C
wF(α)LPFweightingf.
Theconstant parameterLPFperformsarunningaverageoftheoutputz(t) ofthedemodulator.TheoutputiscontinouslyupdatedandtrackstheslowlyvaryingamplitudeA(t).Thisbasicset-up isdenotedPhase-Sensitive Detector(PSD)andisthecoreofthe instrumentcurrentlycalled Lock-inAmplifier.
FT RC=
Constant-param.Low-PassFilter
Withaveragingperformedbyagatedintegrator,theamplitudeAcanbemeasuredonlyatdiscretetimes (spacedbyatleasttheaveragingtime2T).However,byemployingaconstant-parameterlow-passfilterinsteadoftheGI,continuousmonitoringoftheslowlyvaryingamplitudeA(t)isobtained.
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Principle of the Lock-in Amplifier (LIA) 26
Sy A∝A
quasi-DCOutputsignal
Inputsignal
Reference
z(t)=x(t)·m(t)( ) ( )cos mx t A tω=
( ) ( )cos mm t B tω=
AnalogMultiplier
R
C
wF(α)LPFweightingf.
Theconstant parameterLPFperformsarunning averageofz(t) overafewTFthatcontinouslyupdatestheoutput
FT RC=
Constant-param.Low-PassFilter
( ) ( ) ( ) ( ) ( ) ( )0 0F Fy t z w d x m w dα α α α α α α∞ ∞
= =∫ ∫
( ) ( ) ( )0 Ly t x w dα α α∞
= ∫
BycomparisonwiththedefinitionoftheLIAweightingfunctionwL(α)
( ) ( ) ( )L Fw m wα α α= ⋅
weseehowthedemodulation andLPFarecombined intheLIA
( ) *L FW f M W≅
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Weighting Function wL of the Lock-in Amplifier 27
fm-fm f
|X|
f
f
t
t
t
t
fm-fm f
fm-fm
|M|
( ) ( ) ( )L Fw m wα α α= ⋅( )LW f
( ) ( )cos mm t B tω=
( ) ( )cos mx t A tω=
2A
2A
2B
2B
( )21
1 2F
F
WfTπ
=+
12 FT
LPFnoisebandwidth(bilateral)
α
α2B
LPFweightingfunctionwF 12n
F
fT
Δ =
( ) *L FW f M W≅
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
S/N of the Lock-in Amplifier 28
fm-fm
|X|
ffm-fm
f
f
Snb 1/fnoise
whitenoise
|W|2
2A
2A
2
2B⎛ ⎞
⎜ ⎟⎝ ⎠
12n
F
fT
Δ =
SBb
InputSignal(inphase)
Weighting
Noisedensity(bilateral)
22 2 2SA B By A= ⋅ =
2 22 2
2 2yL Bb n Bb nB Bn S f S f⎛ ⎞= ⋅ ⋅Δ = ⋅ ⋅Δ⎜ ⎟
⎝ ⎠
Outputsignal
OutputNoise2 2S
L Bb nyL
yS AN S fn
⎛ ⎞ = =⎜ ⎟Δ⎝ ⎠
( ) ( )cos 2 mx t A f tπ=
Bb Cnb Bb
S fS Sf
= +
BilateralnoisebandwidthoftheLPF
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
S/N of the Lock-in Amplifier 29
2L Bb n
S AN S f
⎛ ⎞ =⎜ ⎟Δ⎝ ⎠
S/Nequation intermsoftheunilateralparameters
Byintroducing
• fFn theLPFunilateralbandwidth(upperband-limit fornoise),i.e.
• Sbu theunilateralnoisedensity,i.e.
wecanwrite
2L Bu Fn
S AN S f
⎛ ⎞ =⎜ ⎟⎝ ⎠
2n Fnf fΔ =
2 Bb BuS S=
orinpowerterms2 2 2
L Bb n
S AN S f
⎛ ⎞ =⎜ ⎟ Δ⎝ ⎠ n
in phase signal powerhalf power of white noise in the band f
−=
Δ
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
OutputNoise(𝑆"#$ meandensity intheLPFband)
Case of DC signal with LPF compared to AC signal with LIA 30
Out+
_
R2R4
R3 RT
VA
( )x t A=DCvoltagesupplyVA
RC
FT RC=
LPF
Letusconsidertheset-upofthekeyexample(measurementwithresistivesensor)nowwithDCsupplyvoltageVA equaltotheamplitudeofthepreviousACsupply.ThesignalnowisaDCvoltageequaltotheamplitudeA ofthepreviousACsignal.WithaLPFequaltothatemployedinthepreviousLIAweobtain:
Cy A=∂2
yC nu Fnn S f= ⋅
Outputsignal
∂2
C
C nu FnyC
yS AN S fn
⎛ ⎞ = =⎜ ⎟⎝ ⎠
ThisS/Nmaylookbetterbythefactor 𝟐 thantheS/NobtainedwiththeLIA,butisthisconclusiontrue?NO,suchaconclusionisgrosslywrongbecause𝑺𝒏𝒖$ ≫ 𝑺𝑩𝒖 !!
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
DC signal and LPF compared to AC signal and LIA 31
fm-fm
ffm-fm
f
f
1/fnoise
whitenoise
|WF|2
2n F nf fΔ =
SBb
DCInputSignal
WeightingoftheLPF
Noisedensity(bilateral)
x A=
Bb Cnb Bb
S fS Sf
= +
nbS
1
( ) ( )X f A fδ= ⋅
𝑆"+$ meanspectraldensityintheLPFbandatf=0
𝑆,+ SpectraldensityintheLIAbandatf=fm
Apassbandatf=0 isarisk:1/fnoisegives𝑺𝒏𝒃$ ≫ 𝑺𝑩𝒃 !!
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Fake LIA passbands arise from imperfect modulation 32
• Ideally,thereferencewaveformshouldbeaperfectsinusoidatfrequencyfmwithamplitudeB1
• Inreality,deviationsfromtheidealcangeneratespuriousharmonicsatmultiples kfm(k=0,1,2...)withamplitudesBk (smallBk <<B1 incaseofsmalldeviations)
• Moreover,effectsequivalenttoanimperfectreferencewaveformcanbecausedbynon-idealoperation(non-linearity)ofthemultiplier
• Sinceitis
eachspuriousharmoniccomponentofM(f) addstotheLIAweightingfunctionWLaspuriouspassbandatfrequencykfm withamplitudeBk andshapegivenbytheLPF
• Afakepassbandatf=0is particularlydetrimentalevenwithsmallB0 <<B1because itcoversthehighspectraldensityof1/f noise ....
• ....andunluckilyanydeviationfromperfectbalanceofpositiveandnegativeareasofthereferenceproducesaDCcomponentwithassociatedpassbandatf=0 !!
( ) ( ) ( )*L FW f M f W f≅
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Theratiocanmatchorexceedtheamplituderatio
sothatthenoise inthefakepassband
canequalorexceedthatinthecorrectpassband
Fake LIA passband at f = 0 33
fm-fm
ffm-fm
f
f
|WL|2
SBb
ImperfectreferencewithDCcomponent
LIAweightingfunction
Noisedensity(bilateral)
Bb Cnb Bb
S fS Sf
= +
nbS
( )M f
( )0B fδ⋅ ( )1
2 mB f fδ⋅ −
2 22 1 11 2 2yL Bb n Bu FnB Bn S f S f= ⋅ ⋅Δ = ⋅ ⋅
20B
21
2B⎛ ⎞
⎜ ⎟⎝ ⎠
Cnb Bb Bb
fS S Sf
= +
2n F nf fΔ =
2 21 02B B
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Appendix:About the «Gain» of the Lock-in Amplifier 35
Gisdifferentinprinciplefromthegainofanordinaryamplifier: itisaTransfer GainthatcharacterizestheINFORMATIONTRANSFERINFREQUENCYfromfm toDC.
Lock-inAmplifier
Reference
outV GA=inV A=
Output:DCvoltage
Input:ACsignal
2
2in
inVP =
2out outP V=out
Pin
Poutput powerGinput power P
= =
out
in
Voutput voltageGinput voltage V
= =
22
2 2
2
outP
in
VG GV
= =
Vout isaDC voltageVin istheamplitudeofanAC voltageLIAGain however
LIAPowerGain howeverPout isDCpower
Pin isACpower
Therefore anditisprecisely2PG G≠
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Appendix1:Stage-by-Stage view of the Signal
in the Lock-in Amplifier
36
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Processing Signals with f =fm in the Lock-in Amplifier (LIA) 37
A
quasi-DCOutput
Signal
Reference
z(t)=x(t)·m(t)( ) ( )cos 2 mx t A f tπ ϕ= +
( ) ( )cos 2 mm t B f tπ=
AnalogMultiplier
LPFBandlimit fFn
y(t)
Fn mf f=
• TheMultiplier translatesthesignalintworeplicasshifted infrequencydowntoω=0 anduptoω=2ωm
• TheLPF(thathascutofffFnmuchlowerthanthesignal frequencyfm)passes totheoutputpracticallyonlytheconstantcomponent
Wecangainabetterinsightbyconsideringthespecialcasesofsignalthatwithrespecttothereferenceareinphase [ϕ=0ork·π] andinquadrature [ϕ=π/2or(2k+1)· π/2]
( ) ( ) ( ) ( )cos cos cos cos 22 2m m mAB ABz t A t B t tω ϕ ω ϕ ω ϕ= + = + +
( ) cos2ABy t ϕ≈
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Processing Signals with f =fm in the Lock-in Amplifier (LIA) 38
( ) ( )cos mx t A tω=
( ) ( )cos mm t B tω=
( ) ( )cos sin2m mx t A t A tπ
ω ω⎛ ⎞= + =⎜ ⎟⎝ ⎠
Signalinphaseϕ=0 Signalinquadrature ϕ =π/2
( ) ( ) ( )z t m t x t= ⋅( ) ( ) ( )z t m t x t= ⋅
( ) ( )cos mm t B tω=
𝑥 𝑡 𝑥 𝑡
( )2ABy t ≈ ( ) 0y t ≈
2AB
t
t
t
t t
t
t
t
• Anysinusoidcanbeanalyzedassumofin-phaseandinquadraturecomponents
• PhaseselectionbytheLIA:thequadrature components donotcontribute totheoutput( ) ( ) ( ) ( )cos cos cos sin sinm m mx t A t A t A tω ϕ ϕ ω ϕ ω= + = ⋅ − ⋅
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Processing Signals with fs ≠ fm in the Lock-in Amplifier (LIA) 39
• Themultiplier convertsthesinusoidalsignaloffrequencyfs ≠fm in:ahigh-frequencycomponentatfh =fs +fmplusalow-frequencycomponentatfl=│fs – fm │
• Thehigh-frequencycomponent iscutoffbytheLPFwithbandlimit fFn<<fm< fs +fm
• Thelow-frequencycomponentisfirstsubjecttoselection infrequencyandisadmittedonlyiffl iswithintheLPFpassband(i.e.if 𝑓1 − 𝑓3 < 𝑓5"),otherwise itis(approximately)cut-off.
• Asignalwithfrequencyfs admittedbythefrequencyselection issubjectedtofurtherselectioninphase: onlyitsin-phasecomponent(ϕ=0)contributestotheoutput.
• Insummary:asinusoidalsignalatfrequencyfs ≠fm- iffs isintheadmissionbandwidthΔfn =2fFn centeredatfm (i.e. 𝑓1 − 𝑓3 < 𝑓5")thesignalisprocessedbytheLIAjustlikeasignalatfs= fm- iffs isoutoftheadmissionbandwidth,itiscutoff
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Appendix2:Stage-by-Stage view of Noise
in the Lock-in Amplifier
40
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
• Selectioninfrequency:noisecomponentsareadmittedonlyfrom(fm – fFn)to (fm +fFn),thatis,withinabandwidthΔfn =2fFn setbytheLPFandcenteredatfm .ThetotalmeanpoweroftheinputnoiseSnu(f) inthisadmissionbandis
• Selectioninphase:onlycomponents inphasewiththereferencecontributetotheoutput.Thenoisecomponentshaverandomphaseϕ ,withuniformprobabilityforallphasesfromϕ=0toϕ=2π.Intheensemblenophaseisfavouredandthemeanpowerisequallysplitbetween in-phaseandquadraturecomponents:onlythein-phase noisemeanpoweristransmittedtotheoutputandishalfofthetotal
Noise in the Lock-in Amplifier (LIA) 41
Lock-inAmplifier
Reference
Noise(unilateraldensity)Snu(f)
( ) ( )cos mm t B tω=
( ) ( )2 2nu m Bu m Fnn S f f S f f= Δ = ⋅
( )2
2 212 2o P Bu m Fn
Bn G n S f f= ⋅ ⋅ = ⋅
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
S/N of the Lock-in Amplifier (LIA) 42
Lock-inAmplifier
Reference
Noise(unilateraldensity)Snu(f)
( ) ( )cos mm t B tω=
( ) ( )cos mx t A tω ϕ= +Signal ( )cos2By A ϕ=
Outputsignalpower( )
( )2
22cos 1 cos
2 2 2y P
A BP G Aϕ
ϕ⎡ ⎤⎣ ⎦= = ⋅ ⎡ ⎤⎣ ⎦
( )( )
22 cos2 Bu m Fn
ASN S f f
ϕ⎛ ⎞ =⎜ ⎟⎝ ⎠
( ) ( )2
2
2o P nu m Fn Bu m FnBn G S f f S f f= ⋅ = ⋅Outputnoisepower
( )cos
2 Bu m Fn
S AN S f f
ϕ⎛ ⎞ =⎜ ⎟⎝ ⎠
Signal-to-NoiseRatio: itisconfirmedthat
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Notice about the noise transfer in frequency 43
AnalogMultiplier
LPFBandlimit fFn
Reference ( ) ( )cos 2 mm t B f tπ=
StationaryNoise nx(t)withspectrumSxb (bilateral)
nz(t) ny(t)
( ) ( )cos mm t B tω=
( )xn t
( ) ( ) ( )z xn t m t n t=
t
t
t
( ) ( )2 2 2 2cosz x mn t n B tω=
Referencewaveform
inputnoisewaveform
multiplieroutnoisewaveform
multiplieroutnoisepowert
We might think that the noise spectrum at the multiplier output is simply thesuperposition of two replicas of the input spectrum shifted up and down by fm ,but this is not exactly the case. The input noise nx(t) is stationary, whereas themultiplier output noise nz(t) is not stationary, it has cyclically varying intensity sothat it is denoted «cyclostationary» noise
However,wewillshowthattheLIAoutputnoisepower canbecomputedwithanequivalentstationary noise,namelywiththeaverageintime ofthecyclostationarynoise
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Notice about the noise transfer in frequency 44
Atthemultiplierout,thenoiseautocorrelationfunctionisnotstationary,butcyclic
Sincetheinputisstationary𝑅77(𝑡, 𝑡 + 𝛾) = 𝑅77(𝛾) ;with𝑚 𝑡 = 𝐵cos 𝜔3𝑡 weget
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), ,zz xxR t t z t z t m t m t x t x t m t m t R t tγ γ γ γ γ γ+ = + = + ⋅ + = + ⋅ +
TheLPFwith𝑓5" ≪ 𝑓3 performsthenanaverageoveratimemuchlongerthantheperiodoffm.Therefore,thenoisepowerattheoutputoftheLPFcanbecomputedemployingthetime-averageofthecyclostationarynoisenz .Wecanthusemployasequivalent stationaryautocorrelationRzz,eq(ϒ) thetime-averageofRzz(t,t+ϒ)
Inthefrequencydomain,theFouriertransformofRzz,eq(ϒ) canbeemployedasequivalentstationarynoisespectrum
(NB:thespectraldensitieshereareBILATERAL )
( ) ( ) ( )2
, cos cos 22zz m xxBR t t t Rγ γ ω γ γ+ = + + ⋅⎡ ⎤⎣ ⎦
( ) ( ) ( ) ( ) ( ) ( ) ( ), ,zz eq zz xx mm xxR R t t m t m t R K Rγ γ γ γ γ γ= + = + ⋅ = ⋅
( ) ( ) ( ),z eq m xbS f S f S f= ∗
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Notice about the noise transfer in frequency 45
Withsinusoidalreference
wehave
ThankstotheaveragingactionoftheLPF,theoutputpoweractuallycanbecomputedasifthemultiplieroutputwereastationaryspectrum,obtainedbyshiftinginfrequencyby-fmandby+fm theinput spectrumandbysuperposingthetworeplicas.Thisoperationisusuallydenoted«spectrumfolding».Notethatthespectrumfolding doublesthespectraldensityat f=0
sothat
or,withunilateralparameters
( ) ( )cos mm t B tω=
( ) ( )2
cos2mm mBK γ ω γ= ( ) ( ) ( )
2
4m m mBS f f f f fδ δ= − + +⎡ ⎤⎣ ⎦
( ) ( ) ( )2
, cos2zz eq m xxBR Rγ ω γ γ= ⋅ ( ) ( ) ( )
2
, 4z eq x m x mBS f S f f S f f= − + +⎡ ⎤⎣ ⎦
( )2
2, 0 2
4o Bz eq BbBn S f S f= Δ = ⋅ Δ
( ) ( ) ( )2 2
, 0 24 4z eq x m x m BbB BS S f S f S= − + + = ⋅⎡ ⎤⎣ ⎦
22
2o Bu FnBn S f= ⋅
Ä
Ä
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Appendix3:Bandwidth, Response Time and S/N
of the Lock-in Amplifier
46
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
ByprocessingasinusoidalsignalofamplitudeA inpresenceofnoiseSBu aLIAgives
• TheequationshowsthatS/NisimprovedbyreducingtheLPFbandlimitfFn .• Thisconclusion,however,isnotindefinitely validinrealcases:thereisalimit
bandwidth,belowwhichtheS/Nisdegraded.• Thereason isthatrealsignalsaremodulatedsinusoids, theyhaveamplitude thatvaries
withtimeA(t) (thoughveryslowlywithrespecttothesignalperiod1/fm )
• Let’sseehowandwhythisreasonlimits theS/Nimprovementobtainedbybandwidthreduction.Thestage-by-stageviewofsignalprocessingreadilyandintuitivelyclarifies it.
Limit to narrow bandwidth convenience 47
( )2 Bu m Fn
S AN S f f
⎛ ⎞ =⎜ ⎟⎝ ⎠
( ) ( ) ( )cos 2 mx t A t f tπ=
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Real Signals are Modulated Sinusoids 48
t
t2T
A(t)
( ) ( ) ( )cos mx t A t tω=t2T0
ε
vA
Out+
_
R2R4
R3 RT
VA
RT strainsensorsubjectedtovariablestrainε(t)
( ) ( )cosA A mv t V tω=
strainεà variationΔRT∝ ε à
à Bridgeunbalanceà signalA∝ ε
( ) ( ) ( )cos mx t A t tω=
0
Bridgesupplyvoltage
Bridgeoutputsignal
Amplitudemodulation=BaseBandsignal
TypicalexampleofBaseBandsignal(signaltoberecovered):
Strainε withfiniteduration2T
t
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
t
Real Signals are Modulated Sinusoids 49
t
ε
Bridgesupply
Strain(Base-Bandsignal)
Bridgeoutputsignal(modulatedsignal)
fm-fm f
f
|ε|
f
fm-fm
|X|
TIMEDOMAIN FREQUENCYDOMAIN
t
x
vA
2T
1. TheBase-bandsignalisnotconstant,itisslowlyvariable,hencethemodulatedsignalhasafinitebandwidthΔfS aroundfm
2. intheLIA,themodulatedsignal isde-modulatedbythemultiplier3. thedemodulatedsignal isthenfilteredbytheLPFintheLIA
ΔfS
ΔfS
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Signal de-modulation by the multiplier 50
LIAinput
ffm-fm
fm-fm f
|X|
ffm-fm
t
TIMEDOMAIN FREQUENCYDOMAIN
LIAReference
LIAmultiplieroutput
t
t
|M|
|Z|
x
m
z
-2fm 2fm
t
t
ffm-fm
|ZBB|ffm-fm
|ZUB|
-2fm 2fmzBB
zUBissumof
Upper-bandsignal
Base-bandsignal
plus
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Base-Band signal filtered by the LPF in LIA 51
• amplitudeAofBase-BandsignalzBB(t) ∝ areaof|ZBB(f)| ∝ itsbandwidthΔfS
• amplitudeAu ofLIAoutputsignaly(t) ∝ areaof|Y(f)| ∝ bandwidthof|Y(f)|
• LIAoutputsignaly(t) =Base-BandsignalzBB(t)filteredbyLPFintheLIA
• a)forLPFwithbandΔfn >>ΔfS :- bandwidthof|Y(f)|= bandwidthΔfS ofBBsignal- fulloutputsignalamplitudeAu =A- S/NisimprovedbyreducingΔfn becausesignalstaysconstantandrmsnoise isreduced
• b)forLPFwithbandΔfn <<ΔfS :- bandwidthof|Y(f)|= bandwidthΔfn ofLPF- reducedoutputsignalamplitudeAu ≈A·Δfn /ΔfS <<A- S/NisdegradedbyreducingΔfn :signaldecreasesasthebandwidthΔfn whereasrmsnoisedecreases onlyasthesquarerootofthebandwidth 𝛥𝑓"
TheLPF 1)cutsofftheUpper-Bandsignaland2)filterstheBase-Bandsignal.Wemaynotethat:
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Base-Band signal filtered by the LPF in LIA 52
TIMEDOMAIN FREQUENCYDOMAIN
tf
|ZBB|zBBBase-Bandsignal
Casea)LPFbandΔfn >>ΔfS BBsignalband
δ-response Transferfunction
A
A LIAoutput
|HF|
y
t
|Y|
hLPFfilter
f
f
-fm fm
Δfn
ΔfS
Au=A
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Base-Band signal filtered by the LPF in LIA 53
TIMEDOMAIN FREQUENCYDOMAIN
tf
|ZBB|zBB Base-Bandsignal
δ-response
Caseb)LPFbandΔfn <<ΔfS BBsignalband
Transferfunction
A
ALIAoutput
|HF|
t
t
y
LPFfilter
|Y|
Au<<A
f
f
-fm fm
ΔfS
Δfnh
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Time-domain study of the LIA response time 54
ThewaveformoftheoutputsignalofaLIAcanalsobecomputeddirectlyintime.With LIAinputsignal 𝑥 𝛼 = 𝑎 𝛼 ⋅ 𝑚 𝛼
LIAreference𝐵 ⋅𝑚 𝛼LPFinLIAweightingfunction𝑤5 𝛼
wehave
Inthesquareofreference𝑚K 𝛼 wecanseparatetheconstantaverageintime 𝑚K
andtheoscillatingcomponent 𝑚K 𝛼 − 𝑚K
andsincetheweighting𝑤5 𝛼 variesveryslowlywithrespecttotheoscillation(inaperioditisalmostconstant)itaveragestheoscillation,sothatwehave
Inotherwords,theoscillatingcomponentiscut-offbytheLPFasillustratedinslide49
( ) ( ) ( ) ( ) ( ) ( ) ( )2F Fy t x Bm w d B a m w dα α α α α α α α
∞ ∞
−∞ −∞= ⋅ = ⋅∫ ∫
( ) ( ) ( ) ( ) ( ) ( )2 2 2F Fy t B m a w d B a m m w dα α α α α α α
∞ ∞
−∞ −∞⎡ ⎤= ⋅ + ⋅ −⎣ ⎦∫ ∫
( ) ( ) ( )2 2 0Fa m m w dα α α α∞
−∞⎡ ⎤⋅ − ≅⎣ ⎦∫
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Therefore
theoutputissimplytheBase-BandsignalweightedbytheLPF.SincetheLPFisaconstant-parameterfilter,wecanuseitsδ-responseh(t)
andconfirmthattheoutputwaveformisthatoftheBase-BandsignalfilteredbytheLPF
Inthecaseofsinusoidalmodulationandsinusoidal de-modulationbyLIA
andwehave
Time-domain study of the LIA response time 55
( ) ( )cos mm t tω= 2 12
m = ( ) ( )2 2 1 cos 22 mm m tα ω⎡ ⎤− =⎣ ⎦
( ) ( ) ( )2Fy t B m a w dα α α
∞
−∞≅ ⋅∫
( ) ( ) ( ) ( )2 2 *y t B m a h t d B m a hα α α∞
−∞≅ ⋅ − = ⋅∫
( ) ( )Fw h tα α= −
( ) ( ) ( ) *2 2B By t a h t d a hα α α
∞
−∞≅ ⋅ − =∫
Signal Recovery, 2017/2018 – BPF 3 Ivan Rech
Time-domain study of the LIA response time 56
t
t
x
y
t
a(t)
t
( ) ( ) ( )1 exp Fh t t t T= ⋅ −
h
Base-Bandsignala(t)=1(t)
Modulatedsignalx(t)(LIAinput)
δ-responseofLPF
LIAoutput
( ) ( ) ( )1 1 exp2 FBy t t t T≅ − −⎡ ⎤⎣ ⎦
Example:stepstrainisappliedtothesensoratt=0andthenmaintainedLPFisasinglepole integratorwithRC=TF