15 BPF 01 - home.deib.polimi.ithome.deib.polimi.it/rech/Download/15_BPF_01.pdf · SignalRecovery,...

Signal Recovery, 2017/2018 – BPF 1 Ivan Rech

Sensors, Signals and Noise 1

COURSEOUTLINE

• Introduction

• SignalsandNoise

• Filtering:Band-PassFilters1- BPF1

• Sensorsandassociatedelectronics


Band-Pass Filters 1 2

• Narrow-BandSignals

• RecoveringNarrow-BandSignalsfromNoise

• MovingSignals inFrequency(SignalModulation)


Narrow-Band Signals

3


Narrow-Band Signals 4

Powersignalswithanarrowpowerspectrum,thatis,apeakwith• center-frequencyfs• bandwidthΔfswhichissmall inabsolutevalue,typicallyΔfs <10Hz,

and/orwithrespecttothecenterfrequencyΔfs <<fsTheyapproximatewellasinusoidoverawidetimeintervalTs ≈1/Δfs

QUESTION:howcanwemeasuresuchnarrow-bandsignals inpresenceofintensewhitenoise?Andwhatifalso1/fnoise ispresent?


Recovering Narrow-Band Signalsfrom Noise

5


Recovering Narrow-Band Signals from noise 6

Let’sseesometypicalexamplesofsignalswith• narrowlinewidthΔfs=1Hz• smallamplitudeVs ≤100nV

forbringingthemtohigherlevel(suitableforprocessingcircuits:filters,meters,etc.)theyareamplifiedbyaDC-coupledwide-bandpreamplifierwith

§ upperband-limitfh =1MHz§ noisespectraldensity(referredtoinput) with

«white»componentand1/f componentwithcornerfrequencyfc=2kHz

Letusconsiderthreecaseswithdifferentcenter-frequencyfs :Ø Case1:high frequencyfs = 100kHzØ Case2:moderatelylow frequencyfs = 1kHzØ Case3:low frequencyfs = 10Hz

5 /bS nV Hz=


Recovering Narrow-Band Signals 7

CASE1:signalVs ≤100nV athighfrequencyfs = 100kHz

a) observingthevoltagewaveformintheTIMEDOMAIN,i.e.onoscilloscopedisplay

Thesignaltoberecoveredisatfrequencyfs = 100kHzmuchhigherthanthenoisecornerfrequencyfc=2kHz,sothatwecanuseasimplehigh-passfilterwithband-limitfi =10kHztocutoffthe1/fnoiseandobtainarmsnoise(referredtothepreampinput)

andtherefore

EventhehighestsignalVs =100nV ispracticallyinvisibleontheoscilloscopedisplay!Thenoisecoversaband ≈5xrmsvalue≈20μVandthesinusoidalsignal isburiedinit!

20,02 1S

n

VSN v= ≤ <<

( )2 5n b h i b hv S f f S f Vµ= ⋅ − ≈ ⋅ =

Verticalscale50μV/div

Horizontalscale5μs/div



CASE1:signalVs ≤100nV athighfrequencyfs = 100kHz

b)observingthepowerspectruminFREQUENCYDOMAIN,i.e.onspectrumanalyzerdisplay

SIGNAL:thepower iswithinabandwidthΔfS =1Hz

sothattheeffectivepowerdensityofthesignalis

NOISE:theeffectivepowerdensityatfs = 100kHzis

Onthespectrumanalyzerdisplaythesignalpeakisvery wellvisibleabovethenoise!

Conclusion:goodS/Ncanbeobtainedwithabandpassfilterhavingbandwidth∆𝑓#matchedtothesignalband∆𝑓# ≈ ∆𝑓%

216 250 10

2S

SVP V−= = ⋅

70SS

S

PS nV Hzf

≈ =Δ

5 /bS nV Hz=

14 1SS S S

b b b b b

SP S fSN S f S f S

Δ= = ≈ =

Δ Δ?

14 1S

b

SS

= ?



CASE2:signalVs ≤100nV atmoderatelylowfrequencyfs = 1kHza)observingthevoltagewaveformintheTIMEDOMAIN,i.e.onoscilloscopedisplayThesignal isnowatfs = 1kHzjustbelowthecornerfrequencyfc=2kHz.Forreducingthe1/fnoisewecanstilluseahigh-passfilter,butinordertopassthesignaltheband-limit fi mustbereduced:fi <<fs = 1kHz,typicallyfi =100Hz.Thermsnoisereferredtotheinput is

andtherefore

Thesituation ispracticallyequaltothatofCase1:thesignal ispracticallyinvisibleontheoscilloscopedisplay,it’sburiedinthenoise!

( )2 ln ln 5h hn b h i b c b h b c b h

i i

f fv S f f S f S f S f S f Vf f

µ⎛ ⎞ ⎛ ⎞

≈ − + ≈ + ≈ ≈⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

20,02 1S

n

VSN v= ≤ <<

ln hb c b h

i

fS f S ff

⎛ ⎞⎜ ⎟⎝ ⎠

=1/fnoiseisnegligible



CASE2:signalVs ≤100nV atmoderatelylowfrequencyfs = 1kHzb)observingthepowerspectruminFREQUENCYDOMAIN,i.e.onspectrumanalyzerdisplay



NOISE: duetothe1/fnoise,theeffectivepowerdensityatfs = 1kHzissomewhathigher

Anyway,onthespectrumanalyzerdisplaythesignalpeakisstillwellvisibleabovethenoise

Conclusion: abandpassfilterwithbandwidth∆𝑓# matchedtothesignal∆𝑓# ≈ ∆𝑓%stillgivesafairlygoodS/N

( ) 1 3 8,7c cn s b b b b

s s

f fS f S S S S nV Hzf f

= + = + = ⋅ ≈

216 250 10

2S

SVP V−= = ⋅

70SS

S

PS nV Hzf

≈ =Δ

( )8 1S

n S

SS f

= >

( ) ( ) ( )8 1SS S S

n S b n S b n S

SP S fSN S f f S f f S f

Δ= = ≈ = >

Δ Δ



CASE3:signalVs ≤100nV atlowfrequencyfs = 10Hza)observingthevoltagewaveformintheTIMEDOMAIN,i.e.onoscilloscopedisplayThesignal isnowatfs = 10Hzmuchbelowthecornerfrequencyfc=2kHz.Forreducingthethe1/fnoisewecanstilluseahigh-passfilter,butwithstronglyreducedband-limitfi <<fs = 10Hz,typicallyfi =1Hz.Thermsnoisereferredtoinputis

andtherefore

( )2 ln ln 5h hn b h i b c b h b c b h

i i

f fv S f f S f S f S f S f Vf f

µ⎛ ⎞ ⎛ ⎞

≈ − + ≈ + ≈ ≈⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

20,02 1S

n

VSN v= ≤ <<

Thesituation ispracticallyequaltothatofCase1:thesignal ispracticallyinvisibleontheoscilloscopedisplay,it’sburiedinthenoise!

ln hb c b h

i

fS f S ff

⎛ ⎞⎜ ⎟⎝ ⎠

=1/fnoiseisnegligible



CASE3:signalVs ≤100nV atlowfrequencyfs = 10Hzb)observingthepowerspectruminFREQUENCYDOMAIN,i.e.onspectrumanalyzerdisplay



NOISE:duetothe1/fnoise,theeffectivepowerdensityatfs = 10Hzisnowmuchhigher

Onthespectrumanalyzerdisplaythesignalpeakisbarely visible,it’sequaltothenoise

Conclusion:theS/Nisinsufficientevenwithabandpassfilterwithnarrowbandwidth∆𝑓#matchedtothesignal∆𝑓# ≈ ∆𝑓%

( ) 1 14,2 71c cn s b b b b

s s

f fS f S S S S nV Hzf f

= + = + = ⋅ ≈

216 250 10

2S

SVP V−= = ⋅

70SS

S

PS nV Hzf

≈ =Δ

( )1S

n S

SS f

≈

( ) ( )1SS S S

b b n S b n S

SP S fSN S f S f f S f

Δ= = ≈ ≤

Δ Δ



SUMMARY• Foranarrow-bandsignalplungedinwhitenoise(i.e.withfrequencyfS higher

thanthe1/fnoisecornerfrequencyfc)abandpassfiltermatchedtothesignal

bandisveryefficientandmakespossibletorecoversignals evensosmallthat

theyareburiedinthewide-bandnoise.

• Foranarrow-bandsignalplungedindominant1/fnoise(i.e.withfS lowerthan

the1/fnoisecornerfrequencyfc)abandpassfiltermatchedtothesignal isstill

quiteefficientandinmanycasesmakespossible torecoverthesignal.However,

ifweconsidersignalsatprogressivelylowerfrequencyfS,the1/fnoisedensity

at fSprogressivelyrises,sothattheavailableS/Nisprogressivelyreduced.



OPEN QUESTIONS• Weneedefficientband-passfilterswithverynarrowband-width.

Weneedtounderstandhowtodesignandimplement suchnarrow-bandfilters,

butweshalldealwiththisissueafterdealingwiththefollowingquestion.

• Iftheinformationiscarriedbytheamplitude ofalow-frequencysignal,ithasto

facealso1/fnoise.Itwouldbeadvantageoustoescapethisnoisebypreliminarly

transferringtheinformationtoasignalathigherfrequency.However:

a)howcanwetransferthesignaltohigherfrequency?

b)ifwetransfertothehigherfrequencyalsothe1/fnoisethatfacesthe

signal,thismakesthetransferuseless: howcanweavoidit?



OPEN QUESTIONS• Forescaping1/fnoise,alow-frequencysignalshouldbetransferredtohigherfrequency

beforeitmixeswith1/fnoiseofcomparablepowerdensity:thatis,frequencytransfer

shouldbedonebeforethestagewherethesignalmeets the1/fnoisesource.

• Thefrequency-transferstageshavetheirownnoise,withdifferentintensity indifferent

types.Unluckily,thetypeswithlowestnoisebearotherdrawbacks,typicallyalimited

capabilityoftransfer,restrictedtomoderatelyhighfrequency.

• Forachievingourgoal,thesignalmustbehigherthanthenoisereferredtotheinputof

thefrequency-transferstage.Ifwithagivenstagethesignalisnothighenough,

preamplifyingisnotadvisablebecauseapreampbringsits1/fnoise.Inmostcases itis

bettertotransferthesignal«asitis»bymeansofafrequencytransferstagewithlower

noiseandacceptthelimitationsofthisstage,typicallyamoderateoperatingfrequency.


Moving Signals in Frequency (Signal Modulation)

16


Amplitude Modulation with DC signal (ideal) 17

• Informationisbroughtbythe(VARIABLE)amplitudeA ofaDCsignalx(t)=A .(NB:arealDCsignal isasignalatverylowfrequencywithverynarrowbandwidth)

• Ananalogmultiplier circuitcombinesthesignalwithasinusoidalwaveformm(t)(calledreferenceorcarrier)withfrequencyfm andCONSTANT amplitudeB

• Theinformationistransferredtotheamplitudeofasinusoidalsignaly(t) atfrequencyfm

y(t)=A·B cos(2π fmt+φm)

ANALOGMULTIPLIER

x(t) y(t)=x(t)·m(t)A

t

t t

B

A·B

m(t)=Bcos(2π fmt+φm)m(t)


Amplitude Modulation with DC signal (ideal) 18

y(t)=x(t)·m(t)=

=A·Bcos(2π fmt+φm)

A

t

t

B

t

A·B

TIMEDOMAIN FREQUENCYDOMAIN

Aδ(f)

f

f

f

|X|x

m

y

fm- fm

- fm fm

|M|

|Y|

( )2 mB f fδ −( )

2 mB f fδ +

Y(f)=X(f)*M(f)

( )2 mBA f fδ⋅ + ( )

2 mBA f fδ⋅ −

Thesignal isshiftedinfrequencyby+fm and–fmandinphaseby+φm and– φm respectively


Convolution in the Frequency Domain 19

Inthetimedomain(TD) the amplitudemodulation isthemultiplicationofthesignalx(t)(withvariableamplitudeA)

bythereferencewaveformm(t) (withstandardamplitudeB)

Inthefrequencydomain(FD) itistheconvolutionofthetransformedsignalX(f) bythetransformedreferenceM(f)

( ) ( ) ( )y t x t m t= ⋅

( ) ( ) ( ) ( ) ( )Y f X f M f X M f dα α α∞

−∞= ∗ = −∫

Convolutionismorecomplicated inFDthaninTDbecause:1. thefunctionstobeconvolvedaretwofold,thatis,theyruninthepositive

andnegativesenseofthefrequencyaxis2. Complex valuesmustbesummedateveryfrequencyforobtainingY(f).IngeneraltheresultofFDconvolutionisnotasintuitiveasthatofTDconvolutionandthemodule|Y(f)|isNOTgivenbytheconvolutionof|X(f)|and|M(f)|

wemustfirstcomputetherealandimaginarypartsofY(f)andthenobtain|Y(f)|

( ) ( ) ( )Y f X f M f≠ ∗


Amplitude Modulation with Narrow-Band Signal 20

Inthecaseshereconsidered, however,the issue isremarkablysimplified becausea) X(f) isconfinedinanarrow bandwidthΔfsb) M(f)hasalinespectrumwith(fundamental)frequencyfm thatismuchgreater

thanthesignalbandwidthfm >>ΔfsIntheconvolutionX(f)*M(f) eachlineofM(f)actsonX(f) asfollows§ ShiftsinfrequencyeverycomponentofX(f) by+ fm and -fm

(i.e.addstoeachfrequency+ fm and -fm )§ ShiftsinphaseeverycomponentofX(f) by+φm and - φm

(i.e.addstoeveryphase+φm and - φm )IncaseswithΔfs << fm,thereisnosumofcomplexnumbers tobecomputedbecauseatanyfrequencyfthereisatmostonetermtobeconsidered,allothertermsarenegligible.Theresultoftheconvolutioniseasilyvisualized:everylineofM(f) shifts X(f) infrequencyandaddstoX(f) itsphase.Therefore,|Y(f)|iswellapproximatedbytheconvolutionof|X(f)|and|M(f)|and|Y(f)|2 bytheconvolutionof|X(f)|2 and|M(f)|2

( ) ( ) ( )Y f X f M f≅ ∗ ( ) ( ) ( )2 2 2

Y f X f M f≅ ∗


Amplitude Modulation with Narrow-Band Signal 21

y(t)=x(t)·m(t)

f

f

f

|X|

fm- fm

- fm fm

|M|

|Y|

( )2 mB f fδ −( )

2 mB f fδ +

Y(f)=X(f)*M(f)

( )2 mB X f f+ ( )

2 mB X f f−

m(t)=Bcos(2π fmt+φm)

( ) ( )1 S

tT

S

Ax t t eT

−

= ⋅

( ) 11 2 S

X f Aj f Tπ

=+1

S

fTπ

Δ ≈

FREQUENCYDOMAINExampleofquasi-DCNBsignal(withverylongTS)

with fm >>1/TS


Amplitude Modulation with Sinusoidal Signal 22

f

f

f

|X|

fm- fm

- fm fm

|M|

|Y|

( )2 mB f fδ −( )

2 mB f fδ +

( ) cos (2 )Sx t A f tπ= ( )2 SA f fδ + ( )

2 SA f fδ −

( )4 m SAB f f fδ − −( )

4 m SAB f f fδ − +( )

4 m SAB f f fδ + + ( )

4 m SAB f f fδ + −

fS-fS

( ) cos (2 )m mm t B f tπ ϕ= +

with fm >>fS


Amplitude Modulation with sinusoidal signal 23

Byexploitingtheawellknowntrigonometric equation

incaseswithsinusoidalsignalandsinusoidal reference

theresult isdirectlyobtained

( ) cos(2 )Sx t A f tπ=

( ) cos(2 )m mm t B f tπ ϕ= +

( ) ( )( ) ( ) ( ) cos 2 cos 22 2S m m S m mAB ABy t x t m t f f t f f tπ ϕ π ϕ= ⋅ = − − + + +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

( ) ( )1 1cos cos cos cos2 2

α β α β α β= − + +


Squarewave Amplitude Modulation 24

Modulationwithasquarewavereferencem(t) canbeimplementedwithcircuitsbasedsimplyonswitchesandamplifiers,withoutanalogmultipliers

A

t

A·Bx(t)

t

t

y(t)=x(t)·m(t)

B

B

• Insuchcases,thecircuitnoisereferredtotheinputisduemostlytotheswitch-devices andismuchlowerthanthatofanalogmultiplier circuits.

• Metal-contactswitcheshavethelowestnoise,buttheycanoperateatlimitedswitchingfrequency,typicallyuptoafew100Hz

• Electronicswitches(MOSFET,diodes,etc.)operateuptoveryhighfrequenciesbuthavefairlyhighernoise(anywayMOSFETsoperatingasswitch-devicehavelowernoisethanMOSFETsoperatingasamplifierdevices).

+



y(t)=x(t)·m(t)

+Bx(t)

Switchingexample:differentialamplifierwithalternated inputpolarity

t

A·Bt

A

y(t)=x(t)·m(t)+Bx(t)

Switchingexample:chopper(ON-OFFmodulation)

t

A·Bt

A


Squarewaves and F-transforms 26

Bmsq(t) =symmetricalsquarewave(from+1to-1)atfrequencyfm

ffm- fm 3fm 5fm

B·|Msq|

-3fm-5fm

2 Bπ

23B

π⋅ 2

5B

π⋅

2 Bπ

23B

π25B

π

[ ]2 12 1 2 1

0( ) ( ) ( )

2k

sq k kk

bB M f B f f f fδ δ∞

++ +

=

⋅ = ⋅ − + +∑ with ( )( )

2 1

2 1

(2 1)

1 42 1

k mk

k

f k f

bk π

+

+

= +⎧⎪

−⎨= ⋅⎪ +⎩

t

B·msq

B0

Intheamplitudemodulation:• eachlineofthereferenceMacts likeasimplesinusoidalreference,i.e.shiftsbyits

frequencyanditsphasethesignalXandmultiplies itbytheamplitudeB·b2k+1• ifthesquarewaveisnotperfectlysymmetrical(e.g.ithasasymmetricalamplitudeand/or

durationofpositiveandnegativeparts)thereisalsoafiniteDCcomponentwithamplitudeB0 (possiblyverysmall)

• theDCcomponentdoesNOTtransferthesignalXinfrequency,just«amplifies» itbyB0


Squarewaves and F-transforms 27

t

B mch(t) =choppersquarewave(from+1tozero)atfrequencyfm

B·mch

ffm- fm 3fm 5fm

B·|Mch|

-3fm-5fm

Bπ 1

3Bπ⋅ 1

5Bπ⋅

2B

( )( (2

))2ch sqB M f B fB M f δ⋅ = + ⋅⋅

Bπ

13

Bπ⋅1

5Bπ⋅

2B

2B

ChoppersquarewavewithamplitudeB==Symmetricalsquarewavewithamplitude B/2+DCcomponentwithamplitudeB/2

Intheamplitudemodulationbyachopper:• a replicaofthesignalX«amplified»byB/2istransferredinfrequencybythesquarewave• anotherreplicaofX«amplified»byB/2isNOT transferred,itstayswhereitis



ffm- fm 5fm

B·|Msq|

-3fm-5fm

B0

|X|

ffm- fm 5fm

|Y|=B·|Msq|*|X|

-3fm-5fm

2 B Aπ

⋅

25B A

π⋅B0·A

A

3fm

3fm

2 Bπ 2

3B

π25B

π

23B A

π⋅

Narrow-bandDCinputsignal

Squarewavereference(slightlyasymmetricalcase)

Squarewavemodulatedoutputsignal


Summary and Prospect 29

• Asintuitive,narrow-bandfilteringisveryeffectiveforrecoveringnarrow-bandsignals immersed inwide-bandnoise

• Besideswide-bandnoise,however,othercomponentswithpowerdensityincreasingasthe inversefrequency(1/fnoise)areubiquitous inelectroniccircuitry(amplifiersetc.).Inthelow-frequencyrangetheyareindeeddominant.

• Atlowfrequenciesthe1/fnoiseaddedbythecircuitryisoverwhelming,sothatthesolutionofnarrow-bandfilteringbecomesprogressivelylesseffectiveandfinallyinsufficientforrecoveringsignalswithprogressivelylowerfrequency

• Aneffectiveapproachtorecoveralow-frequencysignal istomoveittohigherfrequencybeforetheadditionof1/fnoise.Thatis,tomodulatethesignalbeforethecircuitrythatcontainsthe1/fnoise sources

• Narrow-bandfilteringcanthenbeemployedtorecoverthemodulatedsignal;wewillnowproceedtoanalyzemethodsandcircuitsfornarrowbandfiltering

15 BPF 01 - home.deib.polimi.ithome.deib.polimi.it/rech/Download/15_BPF_01.pdf · SignalRecovery,...

Documents

Transcript of 15 BPF 01 - home.deib.polimi.ithome.deib.polimi.it/rech/Download/15_BPF_01.pdf · SignalRecovery,...