COURSE OUTLINE. Introduction Signals and Noise Filtering: High-Pass Filters 2 HPF2 Sensors and associated electronics. Sensors, Signals and Noise 1

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1 esors, igals ad Noise COURE OULINE Iroducio igals ad Noise ilerig: High-ass ilers H esors ad associaed elecroics igal Recovery, 07/08 H

/ Noise ad High-ass ilers Measurig pulse sigals i presece o / oise wih cosa-parameer ilers Basic cosa-parameer High-ass iler (CR diereiaor) Cosa-arameer High-ass

2 / Noise ad High-ass ilers Measurig pulse sigals i presece o / oise wih cosa-parameer ilers Basic cosa-parameer High-ass iler (CR diereiaor) Cosa-arameer High-ass ilers i measuremes o pulses i sequece wiched-arameer High-ass iler: he Baselie Resorer Appedix: Bad-limi o he high-pass CR diereiaor or / oise igal Recovery, 07/08 H

3 3 Measurig pulse sigals i presece o / oise igal Recovery, 07/08 H

4 ulse sigals i presece o / oise 4 Case: ampliude measureme o pulse sigals wih / ad widebad oise. he classic approach o opimum ilerig (o id irs a oise-whieig iler ad he a mached iler) is arduous i his case because / oise ses a remarkably diicul mahemaical problem makes he whieig iler diicul o desig, o implemeable wih lumped circui compoes, bu wih disribued parameers (disribued RC delay lies, ec.) However, by oig ha a) or / oise he ilered power maily depeds o he spa o he bad-pass measured by he badlimi raio, hece i is markedly sesiive o he lower badlimi level weakly depeds o he shape o he iler weighig ucio b) or widebad oise he /N depeds o he spa o he bad-pass measured by he badlimi dierece, hece i is weakly sesiive o he lower badlimi level markedly depeds o he shape o he weighig ucio a aleraive approach leadig o quasi-opimum ilerig ca be devised igal Recovery, 07/08 H

5 ulses ad / oise: ilerig i wo-seps 5 IR E: Desig a mai iler or sigal ad widebad oise oly (ha is, cosiderig oexise he / oise) ad he ake he io accou he / compoe ad evaluae he addiioal oise power ha / oise brigs o he mai iler oupu. I he (lucky) cases where his / oise power is smaller ha he wide-bad oise (or a leas comparable), he mai iler may be cosidered suicie wihou urher ilerig. Oherwise, i he addiio due o / oise is excessive, proceed o he ECOND E : desig a addiioal iler or limiig he / oise power wihou worseig excessively he ilerig o he widebad oise. I is obviously a high-pass iler, which mus combie he goal o a) reducig eiciely he / oise power wih he urher requiremes o b) limiig o olerable level he icrease o he ilered wide-bad oise c) limiig o olerable level he reducio o he oupu sigal ampliude igal Recovery, 07/08 H

6 ilerig ulses ad / Noise: irs ep 6 he issue is beer clariied by cosiderig as IR E he opimum iler or sigal ad wide-bad oise (or is approximaio) composed by Noise-whieig iler, wih oupu whie oise B ad pulse sigal. Le be he upper bad-limi ad A he ceer-bad ampliude o he pulse rasorm. Mached iler, which has weighig ucio mached o he pulse sigal rom he whieig iler ad is hereore a low-pass iler wih upper badlimi. he oupu has a sigal wih ampliude roughly V A ad bad-limied whie oise wih bad-limi ad power B B or ocusig he ideas, le s cosider a well kow speciic case: ilerig o pulse-sigals rom a high impedace sesor wih a approximaely opimum iler, i.e. wih mached iler approximaed by a cosa-parameer RC iegraor. I his case, he oupu oise correspodig o he ipu wide-bad oise is a whie oise specrum wih bad-limi se by a pole wih ime cosa RC c igal Recovery, 07/08 H

7 ilerig ulses ad / Noise: ecod ep 7 Le s ow ake io accou also a / oise source, which brigs a he whieig iler oupu a sigiica / specral desiy B C /. A high rquecy, he / compoe is limied by he upper badlimi o he mached iler. A low requecy, he / compoe ca be limied by a lower bad-limi i se by a addiioal cosa-parameer iler. Wih i << he oupu power o he / oise ca be evaluaed as l i B C However, he cosa-parameer high-pass iler operaes also o he sigal: i aeuaes he low requecy compoes ad hus causes a loss i pulse ampliude, hece a loss i /N. he reduced ampliude is roughly evaluaed as or limiig he sigal loss, i / mus be limied; e.g. or keepig loss < 5% i mus be i ha is ( ) V A( ) A i i 0,05 ( ) l 3 i igal Recovery, 07/08 H

8 ilerig ulses ad / Noise: ecod ep 8 or reducig he / oise o he whie oise level or lower l ( ) B C i B We eed ha C l ( ) i ad sice or keepig he sigal loss <5% i mus be we eed o have C < 3 ( ) l 3 i his meas ha he goal ca be achieved oly i he / oise compoe is low or moderae. Noe ha C ad are daa o he problem, hey cao be chaged. I cases where C exceeds he above limi, a cosa-parameer high-pass iler is NO a suiable soluio or reducig he / oise power. CONCLUION: cosa-parameer high-pass ilers ca be useul as addiioal iler or limiig he / oise, bu jus i cases wih moderae / oise iesiy, because o heir derimeal eec o he sigal pulse ampliude. igal Recovery, 07/08 H

9 9 Basic cosa-parameer High-ass iler (CR diereiaor) igal Recovery, 07/08 H

10 Basic High-ass iler (CR diereiaor) 0 x() C R y() δ() δ-respose h () δ () () e p RC π # ep-respose u () () e raser ucio j π H( ) + j π ( ) π H( ) + π ( ) H() 3 db requecy (ole) % π # 0.5 igal Recovery, 07/08 H

11 A view o High-ass ilerig he iuiive view «High-ass iler All-ass - Low-ass iler» is coirmed by raser ucio j π H( ) + j π + j π h() δ-respose h () δ () () e Weighig ucio w( α) δ( α) ( α) e α α w(α) # igal Recovery, 07/08 H

12 A view o High-ass ilerig he circui mesh srucure isel coirms ha «High-ass iler All-ass - Low-ass iler» V C V i C R V R V i ipu volage V C low-pass ilered V i V R high-pass ilered V i Kircho s mesh volage law V i V C + V R V R V i V C hereore ha is High-pass ilered V i resisor volage ipu volage V i capacior volage ipu volage V i Low-pass ilered V i igal Recovery, 07/08 H

13 Bad-limi o CR diereiaor 3 High-pass bad-limi or Whie oise remise: wih oly a high-pass CR iler he whie oise power +, is diverge, hereore we cosider here also a low-pass iler wih bad-limi s >> /RC. he high-pass bad-limi i o he CR iler wih weighig ucio W() is deied by ( ) p B B 0 B 0 ( ) B i + p W ( ) d d ( ) he compuaio o he iegral ca be avoided by recallig ha CR high pass iler all-pass RC low-pass iler ad hereore high-pass bad-limi i o he CR iler low-pass bad-limi h o he RC iler icr hrc 4RC igal Recovery, 07/08 H

14 Bad-limi o CR diereiaor 4 High-pass bad-limi or / oise remise: wih oly a high-pass CR iler he / oise power #, is diverge, hereore we cosider here also a low-pass iler wih a high bad-limi s >> /RC. he high-pass bad-limi i o he CR iler is deied by ( ) p d d B c 0 ( ) B c B c i + p l( ) I his case he irs iegral is airly easily compued * ad shows ha i + p ( ) ha is, or s >> p i p p s π RC * ee he appedix. ice he high-pass limi o he CR diereiaor is also he low-pass limi o he RC iegraor, we ca avoid he compuaio or he laer (which is airly diicul). i igal Recovery, 07/08 H

15 Abou bad-limis ad oise power 5 he upper requecy limi : - is ecessary or limiig he whie oise power - is useul also or limiig he / oise power - he level o is dicaed by he pulse sigal o be measured he lower requecy limi i : - is ecessary or limiig he / oise power, - he seleced level o i is codiioed by he pulse sigal, i cao be arbirary - however, he reducio o / oise is sigiica eve wih airly low i, ha is, wih / i values ha are high, bu ayway iie. igal Recovery, 07/08 H

16 6 Cosa-arameer High-ass ilers i measuremes o pulses i sequece igal Recovery, 07/08 H

17 CR iler ad pulse sequece 7 Le s look i deail he eec o a high-pass iler (RC ) o a pulse sigal pulse area A V V View o HOR IME scale A INU View o LONG IME scale V RC A A OUU A NB: DC raser o CR is zero à Log ail A e e area o he oupu sigal is zero igal Recovery, 07/08 H

18 CR iler ad pulse sequece 8 V Δ V Δ A pulse ha ollows a previous oe wihi a airly shor ime ierval ( D < 5 ) seps o he slow ail o he irs pulse. hereore, i sars rom a dow-shied baselie, so ha he ampliude measured or i is smaller ha he rue oe. Δ D or periodic pulses wih airly shor repeiio period R <<, he superposiio o slow pulse-ails shis dow he baselie by a V ha makes zero he e area o he oupu sigal V INU V OUU R R V Repeiio-rae-depede baselie-shi V V A R R igal Recovery, 07/08 H

19 Drawbacks o he CR diereiaor iler 9 he high-pass ilerig (diereiaor acio) o he CR iler has MIXED eecs. he eec o oise is ADVANAGEOU: by cuig o he he low requecies i markedly decreases he / oise power (ad mildly reduces he whie oise power) he eec o he sigal is DIADVANAGEOU: Ø i decreases he sigal ampliude by cuig o he low requecies o he sigal, hece i mus be kep low ( i << o he pulse) i order o limi he sigal loss. However, his limis also he reducio o / oise Ø i geeraes slow ails aer he pulses, which shi dow he baselie ad hus cause a error i he measured ampliude o a ollowig pulse Ø Wih a periodic sequece o equal pulses, all pulses id he same baselie shi. he ampliude error is cosa, sisemaically depede o he repeiio rae. Ø Wih radom-repeiio pulses (e.g. pulses rom ioizig radiaio deecors) he pulses occur radomly i ime. Hece he radom superposiio o ails produces a radomly lucuaig baselie shi. he resulig ampliude error is radom: i his case he eec is equivale o ha o a addiioal oise source. CONCLUION: a diereiaor acio is desirable o oise, bu NO o he sigal. WANED: o a cosa-parameer diereiaor, bu a rue Base-Lie Resorer (BLR) igal Recovery, 07/08 H

20 0 wiched-arameer High-ass iler: he Baselie Resorer igal Recovery, 07/08 H

21 Baselie Resorer (BLR) priciple: swiched CR High-pass ilerig acio o he oise ad NO o he sigal: swiched-parameer CR iler wih CR à whe sigal is prese, iie CR whe o pulse is prese V C -dow -up -dow C α V i RC w( α ) ( α ) e R V R V i - V C α α x() V i ( α) δ( α) ( α ) w w B y() V R V i - V C As is ope a he pulse ose (a α ), chargig o C sops ad volage V C says cosa a he sored value pulse peakig ime δ(α) α 0 igal Recovery, 07/08 H

22 Comparig cosa CR iler ad BLR CONAN-ARAMEER ILER CR cosa a all imes WICHED-ARAMEER ILER wih -up R à ad CR à -dow -up -dow α x() x() α w (α) δ(α) α 0 α w (α- ) δ(α) α 0 igal loss y() y() igal Recovery, 07/08 H

23 BLR (All-ass) (Low-ass Boxcar Iegraor) 3 BLR V C -dow -up -dow -up -dow -up V i C R V R w B (α) BLR weighig m δ(α) V R V i - V C α α 0 BOXCAR V i R C α w (α- ) Boxcar weighig r, - r - - BLR weighig (All-ass Low-pass Boxcar) weighig ( α) δ( α) ( α ) w w B igal Recovery, 07/08 H

24 BLR weighig i requecy 4 α x() -dow -up -dow BLR priciple is alike ilered zero-seig, bu wih a basic advaage: much shorer much higher bad-limi i (high-pass) (he BLR swich is elecroically corolled, he ierval ca be very shor) α δ(α) α 0 BLR weighig i requecy: BLR weighig All ass Low-pass ( α) δ( α) ( α ) w w B Low-pass weighig i requecy: W ( ω) [ w ( α)] R ( ω) + i I ( ω) jω W ( ω) e W ( ω) [cosω jsi ω ] [ R + ji ] B [ R cosω I si ω ] j[ I cosω R si ω ] igal Recovery, 07/08 H

25 BLR weighig i requecy 5 BLR weighig or oise: B( ω) [ cosω si ω ] [ cosω si ω ] W R I + I R R I R cosω I siω + W R cosω I siω Le s cosider jus cases where he ierval bewee pulses is much loger ha α so ha w ( α) ( α) e ad W ( ω) + jω ad hereore W ( ω) + cosω + ω siω + ω + ω + ω B igal Recovery, 07/08 H

26 BLR cuo 6 I he low-requecy regio siω ω we ge wih he approximaios ω cosω W B ω ω ( ω) ω + ω + ω + ω ( ) ω ω ω ω ad i he lower regio ( ) B( ω) ω + W ha is, he BLR has a cuo equivale o a CR high-pass wih RC + igal Recovery, 07/08 H

27 BLR vs. CR High-ass iler: Cu-O 7 W W B BODE DIAGRAM highlighs he low-req cuo db W CR Example: BLR wih ad 0 CR iler wih RC + Log << / (i.e. << 0, i he example) ( ) B( ω) ω + W W ( ) CR ω ω R C / < << / (i.e. << i he example) W B W ( ω) CR ( ω) ( + ) ω + ω ω RC +ω RC igal Recovery, 07/08 H

28 BLR vs. CR High-ass iler: Whie Noise 8 W W B LIN LIN DIAGRAM highlighs whie oise power area o W W CR Example: BLR wih ad 0 CR iler wih RC + i π p 4RC 4 4 ( + ) i BLR high-pass bad-limi or whie oise. Noe ha: i is equal o ha o he equivale CR High-pass iler i is equal o badlimi o he low-pass secio i he BLR circui igal Recovery, 07/08 H

29 BLR vs. CR High-ass iler: / Noise 9 W W B LIN LOG DIAGRAM highlighs / oise power area o W W CR Example: BLR wih ad 0 CR iler wih RC + i p πrc π π ( + ) i BLR high-pass bad-limi or / oise. Noe ha: i is equal o ha o he equivale CR High-pass iler i is equal o badlimi o he low-pass secio i he BLR circui igal Recovery, 07/08 H

30 elecio o he BLR parameers 30 he BLR ilerig is ruled by:. ime delay rom swich opeig o pulse-ampliude measureme. here is o choice: is equal o he rise ime rom pulse ose o peak. I ac, ca be shorer ha he rise o he pulse sigal ad should be as shor as possible or ilerig eecively o he / oise.. RC diereiaio ime cosa:o be seleced or opimizig he overall ilerig o oise. he quesio is: how should be seleced or a) providig a good reducio o he / oise power ad b) avoidig o ehace sigiicaly he whie oise power ice he BLR cuo is se by /( + ), a very shor migh look advisable, bu i is o: a BLR wih << operaes like a CD, hece i doubles he whie oise ad remarkably ehaces also he / oise above he cuo requecy. I he ollowig discussio abou he selecio, or ocusig he ideas we will reer o a speciic case: sigals rom a high impedace sesor processed by a approximaely opimum iler, amely a CR-RC iler. he oupu correspodig o he ipu wide-bad oise is a whie specrum bad-limied by a simple pole. uch a siuaio is me i pracice also i may oher cases. A beer isigh i he issue is gaied wih a ime-domai aalysis o BLR ilerig igal Recovery, 07/08 H

31 BLR ilerig o Noise: ime-domai aalysis 3 α RC w w B δ(α) BLR weighig ucio α wb δ( α) w ( α) δ ( α) ( α ) exp δ(τ) k δδ auocorrelaio o δ(α) crosscorrel. δ wih w k δ k δ τ k auocorrelaio o w τ τ crosscorrel w wih δ k wwb auocorrelaio o w B k k + k + k + k wwb δδ δ δ τ igal Recovery, 07/08 H

32 BLR ilerig o Bad-Limied Whie Noise 3 ( ) ( ) ( ) ( ) ( ) k δ τ + k + τ w τ + τ + w τ + wwb τ w ( τ ) exp τ R xx Auocorrelaio o whie oise (bad-limied by a sigle pole) τ Rxx ( τ ) xe τ B Rxx( τ) kwwb( τ) dτ Rxx( 0) + R ( ) ( ) ( ) ( ) 0 xx τ w τ dτ Rxx τ w τ dτ Rxx ( 0) + Rxx ( τ) w ( τ) dτ R ( ) ( ) 0 xx β + w β dβ Rxx ( τ) Rxx ( τ) rxx ( τ ) R ( 0) Deoig We have xx x τ β B x + rxx( τ) e dτ r ( ) 0 xx β e dβ + igal Recovery, 07/08 H

33 BLR ilerig o Bad-Limied Whie Noise 33 τ β B x + rxx( τ) e dτ r ( ) 0 xx β e dβ + τ + β + x + e dτ e e dβ 0 x + e + + ad ially B x + e + Wih as diereiaio, i.e. wih <<, i is quaiaively coirmed ha he BLR acs like a CD wih B x e igal Recovery, 07/08 H

34 BLR ilerig wih as diereiaio 34 0 B x e Wih << he eec o BLR o bad-limied whie oise depeds o how log is he correlaio ime wih respec o he delay wih shor correlaio ime (wide bad) he oise is doubled: wih i is wih moderae correlaio ime (moderaely wide bad) he oise is ehaced: wih B x < 5 i is as diereiaio wih << B x /,73 B x oly wih log correlaio ime (low-requecy bad) he oise is aeuaed*: wih i is > 0 B x < 0,x * oe ha ayway he level is double o ha give by a simple CR iler wih equal cuo, ha is wih RC igal Recovery, 07/08 H

35 BLR ilerig wih as diereiaio 35 BLR WEIGHING wih << NOIE: 3 CAE R xx a) wih shor correlaio: R xx k wwb τ BLR weigh auocorrelaio k k + k + k + k wwb δδ δ δ Noise Auocorrelaio R b) wih moderae correlaio: xx ( τ ) e x τ B x, 73 B x > 0 R xx c) wih log correlaio: < 0, B x x igal Recovery, 07/08 H

36 BLR ilerig wih slow diereiaio 36 w ( τ ) exp τ ( ) ( ) ( ) ( ) ( ) k δ τ + k + τ w τ + τ + w τ + wwb τ Noise Auocorrelaio ( ) R xx τ e x τ R xx wih shor correlaio /0 or +, <,05 4, we eed > 0 τ wih moderae correlaio / R xx or +, <,05 4, we eed > 7 3,5 τ wih log correlaio > 0 R xx or +, <,05 4, o problem eve wih shor τ igal Recovery, 07/08 H

37 BLR ilerig wih slow diereiaio 37 Wih NO egligible wih respec o, he eec o whie oise depeds also o he size o compared o ad. A log ca limi he whie oise ehaceme Le s evaluae how log mus be i he various cases o oise correlaio wih shor correlaio ime /0 i is or keepig we eed > 0 wih moderae correlaio ime / i is or keepig <,05 <,05 B x + B x B x B x + e + + B x + 0, 73 x + + e + i his case we eed > 7 3,5 igal Recovery, 07/08 H

38 BLR ilerig wih slow diereiaio 38 wih log correlaio ime > 0 i is B x x + + No problem wih such a low-requecy oise: i is aeued by he BLR jus as by a CR cosa-parameer iler (wih equal ime cosa RC) he mos ieresig case or us is oise wih moderae. I ac, whe he BLR works o he oupu o a opimum (or approximae-opimum) iler or widebad oise, he correlaio ime ad delay p are comparable, sice hey are boh closely relaed o he bad-limi o he sigal pulse. We coclude ha or avoidig ehaceme o he whie oise i is ecessary o selec a airly slow BLR diereiaio, i.e. a airly log 5 his approach is saisacory also or ilerig he / oise, owihsadig ha makig loger ha shis dow he BLR cuo requecy, hece reduces he aeuaio o / oise. his is couerbalaced by he ac ha he ehaceme o / oise a requecies above he cuo is limied by he low-pass ilerig i he baselie subracio, whereas wih shor i is remarkable. igal Recovery, 07/08 H

39 BLR i summary 39 he BLR is a high-pass iler ha acs o oise ad disurbaces wihou aecig he pulse sigal he BLR is a swiched-parameer iler: he low-pass secio wihi he high-pass iler srucure is a boxcar iegraor ha acquires he baselie oly i he iervals ree rom pulses he BLR ca hus esablish a high-pass bad-limi a a high value (suiable or reducig eiciely he / oise oupu power) wihou causig he sigal loss suered wih a cosa-parameer high-pass iler havig he same bad-limi he high-pass bad-limi eorced by he BLR is give (wih good approximaio) by he low-pass badlimi o he low-pass secio i he BLR circui srucure he combiaio o: () opimum iler desiged or he case o pulse sigal i presece o widebad oise oly (i.e. wihou / oise) ad () BLR speciically desiged (or reducig he acual / oise wihou worseig he wide-bad oise) provides i mos cases a quasi-opimum ilerig soluio. igal Recovery, 07/08 H

40 Appedix: Bad-limi o he high-pass CR diereiaor or / oise 40 remise: wih oly a high-pass CR iler he / oise power #, is diverge, hereore we cosider here also a low-pass iler wih very high bad-limi s >> p /πrc. B c 0 + By chagig variable We have ad by seig p d ( ) ( ) p p equivale o d dx x x x dx x dx d x + x x + x + x x x x B c B c B c x y xdx dy x y d i B c B c p x l( ) ( ) i We obai dy l( + y) l( + y ) y B c B c y B c y igal Recovery, 07/08 H

41 Appedix: Bad-limi o he high-pass CR diereiaor or / oise 4 B c l( + y) B cl + y Bcl + p ice he high-pass bad-limi i is deied by d i B c B c l( ) i i is i + p ad hereore i + p ( ) p s ad akig io accou ha s >> p we have ially i p π RC igal Recovery, 07/08 H

SUMMATION OF INFINITE SERIES REVISITED

SUMMATION OF INFINITE SERIES REVISITED I several aricles over he las decade o his web page we have show how o sum cerai iiie series icludig he geomeric series. We wa here o eed his discussio o he geeral