EFFECT O QUANTIZATIO NOIS I DIGITA FILTER S Bernar Gol an Charle M Rade r Lincoln Laboratory  Massachusetts Institute of Technology Lexington Massachusetts GENERA EXPRESSION FO R QUANTIZATIO NOIS E a

EFFECT O QUANTIZATIO NOIS I DIGITA FILTER S Bernar Gol an Charle M Rade r Lincoln Laboratory Massachusetts Institute of Technology Lexington Massachusetts GENERA EXPRESSION FO R QUANTIZATIO NOIS E a - Description

Thes error ma b subdivide int thre classes namely th erro cause b discretizatio o th e syste parameters th erro cause b analo t o digita conversio o th inpu analo signal an d th erro cause b roundof o th result whic ar e neede i furthe computations ID: 27105 Download Pdf

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EFFECT O QUANTIZATIO NOIS I DIGITA FILTER S Bernar Gol an Charle M Rade r Lincoln Laboratory Massachusetts Institute of Technology Lexington Massachusetts GENERA EXPRESSION FO R QUANTIZATIO NOIS E a

Thes error ma b subdivide int thre classes namely th erro cause b discretizatio o th e syste parameters th erro cause b analo t o digita conversio o th inpu analo signal an d th erro cause b roundof o th result whic ar e neede i furthe computations

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EFFECT O QUANTIZATIO NOIS I DIGITA FILTER S Bernar Gol an Charle M Rade r Lincoln Laboratory Massachusetts Institute of Technology Lexington Massachusetts GENERA EXPRESSION FO R QUANTIZATIO NOIS E a




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EFFECT O QUANTIZATIO NOIS I DIGITA FILTER S Bernar Gol an Charle M Rade r Lincoln Laboratory * Massachusetts Institute of Technology Lexington, Massachusetts GENERA EXPRESSION FO R QUANTIZATIO NOIS E a discret tim linea system hereafte calle a digita filter, i programme o a digita compute r realize wit digita elements computationa l error du t finite wor lengt ar unavoidable . Thes error ma b subdivide int thre classes , namely th erro cause b discretizatio o th e syste parameters th erro cause b analo t o digita conversio o th inpu analo signal an d th erro cause b roundof o

th result whic ar e neede i furthe computations Th first typ o f erro result i a fixed deviatio i syste param eter an i aki t a slightl wron valu o (say ) inductanc i a analo filter. W shal no trea t thi proble here i ha bee treate i som detai l Kaiser. Th othe tw source o erro ar mor e complicate bu i reasonabl simplifyin assump tion ar mad the ca b treate b th technique s linea syste nois theory. I i ou ai t se u p mode o a digita filter whic include thes tw o latte source o erro and throug analysi o th e model t relat th desire syste performanc t o th require lengt o compute registers . Bot

analo t digita conversio an roundof f ma b considere a nois introducin processes , ver simila i nature I eac cas a quantit y know t grea precisio i expresse wit consider Operate wit suppor fro th U.S Ai Force . abl les precision I th digitize o rounde d quantit i allowe t occup th neares o a larg e numbe o level whos smalles separatio i E , then provide tha th origina quantit i larg e compare t E an i reasonabl wel behaved th e effec o th quantizatio o roundin ma b e treate a additiv rando noise Bennett ha s show tha suc additiv nois i nearl white wit h mea square valu o El/ 12. Furthermor th

e nois i reasonabl assume t b independen fro m sampl t sample an roundof noise occurrin g du t differen multiplication shoul b inde pendent I i possibl t sho pathologica ex ample whic disprov eac o thes assumptions , bu the ar reasonabl fo th grea majorit o f cases Ultimatel ou result mus res o experi menta verification o course . Sinc th nois o A- conversio i assume d independen o th nois create b roundoff w e ca comput th outpu o an filter du t eithe r excitatio alone o du t th signa alone an d combin the t ge th tru filter outpu (o cours e th nois term ar know onl statistically) there fore

w wil begi b finding a expressio fo th e mea square outpu o a arbitrar filter excite d a singl nois source Le th filter functio b e H(z); i i understoo tha H(z) i th transfe func tio betwee th outpu o th filte an th nod e wher nois i injected H(z) ma thu b differen t fro th transfe functio betwee th filter's norma inpu an output Le u thu conside th e 21 3
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21 4 PROCEEDINGS—SPRIN JOIN COMPUTE CONFERENCE 196 6 e(nT ) f(nT ) Figur 1 Rando nois applie t a filter. situatio o Fig 1 wher a give nois sequenc e e(nT) i applie t H(z), resultin i a outpu t nois sequence/(«7') . ca

convenientl examin thi mode usin g th convolutio sum Thus , f(nT) = £ h(mT)e(nT - mT) ) = 0 wher h(mT) i th invers z transfor o H(z). Th inpu nois e(nT) i presume t b zer fo r < 0 an th syste i initiall a rest Squarin g Eq (1 yield s f\nT)= £ £ h(mT)h(lT) = 0 / = 0 e(nT - mT)e(nT - IT) (2 ) Now i e(nT) i a rando variabl wit zer o mea an varianc a an recallin ou assumptio n tha e(nT) i independen fro sampl t sample , th statistica mea o Eq (2 reduce t o E[f\nT)] = a £ h (mT) (3 ) = 0 Fo a syste fo whic th righ sid o (3 con verges th stead stat mea square valu o f(nT) ca b obtaine b lettin n

approac infinity Fo r thi case a formul whic i usuall mor con venien ca b obtaine i term o th syste func tio H(z). Notin th definition . H(z) = £ h(mT)z (4 ) = 0 th z transform w ca for th produc H(z) H( —]z _1 and b performin a close contou inte gratio i th z plan withi th regio o conver \ genc o bot H(z) an Hi I, arriv a th identit y h (mT) = ±-&Hiz)Hl^\z- dz (5 ) Eithe th right o left-han sid o (5 ma b e use t evaluat th stead stat mea square valu e oif(nT). EXAMPLE—FIRS ORDE SYSTE M a example conside th first orde syste o f Fig 2 Le th analog-digita conversio nois e (nT) hav varianc a\ an

th roundof nois e (nT) hav varianc a Th syste functio H(z) Fig 2 i give b 1/( - Kz~ an h(mT) = Th outpu y(nT) ca b expresse a th su m a signa ter y (nT), cause b x(nT), an a nois term/(«r) whos mea square valu ca b e written fro (3) a s E[f(nT)] = ( a\) £ (K )' (6 ) fro whic th stead stat Valu ca b instantl y writte a s = li E(f(nT)) = (o\ + a\) - K 2 (7 ) Th implication o Eq (7 ar tricky Th mea n square valu o th nois clearl increase a K ap proache unity Th maximu gai o th filter als o increase (th gai o th syste o Fig 2 a d i s (1/( - K)). Fo thi filter wit lo frequenc y inpu th signa powe t

nois powe rati (S /N ) proportiona t ( + K)/(l - K) whic ap proache infinit a th pol o th filter approache s th uni circle Thi i a genera result However , wit a finite wor length th inpu signa mus b e kep smal enoug tha it doe no caus overflo w e.(nT ) Figur 2 Nois mod fo first orde system .
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EFFECT O QUANTIZATIO NOIS I DIGITA FILTER 21 5 th computation Thus th obtainabl signal to-nois rati decrease a K approache unity . Clearly eac cas deserve it ow considerations , th signal-to-nois rati i th filte depend ver y muc o th actua condition o th us o th e filter . Finally w commen

tha th case K = 0 K = i Eq (7 ar uniqu becaus become zer o sinc n multiplication ar performed . EFFEC O DIFFEREN REALIZATION S TH SAM FILTE R Ther ar a variet o way o programmin a secon orde digita filter (o i genera a filte wit h mor tha tw singularities) Suppos a particula r syste functio H(z) i desired I quantizatio i s ignored the onl th relativ spee an memor y requirement o th differen method ar o interes t decidin whic wa t use However Kaiser' s wor show tha th truncatio o syste constant s affect differen realization differently an ma i n fac lea t instabilit i som realizations Th e nois

effect describe her als yiel differen re sult fo differen programmin configurations Th e poin i illustrate throug th examinatio o th e tw system o Fig 3 Fig 3 represent a nois y programme realizatio o th differenc equation : y{nT) = 2r co bTyinT - T) - r y(nT - IT) x(nT) - r co bTx(nT - T) (8 ) an Fig 3 represent th pai o simultaneou dif ferenc equations : w(nT) = x(nT) + 2r co bTw(nT - T) r w(nT - IT) ) (9 ) y(nT) = w(nT) - r co bTw(nT - T) Bot system hav th transfe functio n - r co bTz~ l e,(nT ) co b T 1 - 2r cos bTz~ l rz' examinatio o th pole an zero o H(z) i n Fig 4 w se tha ou networ

behave a a reson ato tune t th radia cente frequenc b fo th e samplin interva T. Figs 3 an 3b X(nT) represent th noise les inpu t th filter e\(nT) represent th nois e du t A- conversio o th input an e (nT) represent th nois adde b rounding Th e roundof nois ca b cause eithe b a singl e roundof afte al product ar summed o b th e su o th roundof erro du t eac o th multi y(nT ) -r - x Figur 3a Nois mode fo secon orde system—direc t realization . (nT ) . e_(nT ) (+)—•y(nT ) r co b T Figur 3b Nois mode fo secon orde system—canonica l realization . plications I i simple t progra th latter bu t mor

nois i created Not that whil i th e realizatio o Fig 3 th nois term e (nT) an d ei(nT) ar injecte int th filte a th sam plac e th inpu X(nT) an thu se th sam transfe r functio H(z), i Fig 3 th nois ter e (nT) i s injecte i a differen par o th filter an see a differen transfe function : BM - ~, ~, L - . 2- (10 ) 1 2rco bTz~ + r z~ 2 Thu w ca expec tha th nois du t e (nT) wil b differen fo th filters o Figs 3 an 3b . Considerin first th realizatio o Fig 3a w e can afte som manipulation obtai th result , a\ = a]ui(r,bT) + o\u {r,bT) (H )
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21 6 PROCEEDINGS—SPRIN JOIN COMPUTE

CONFERENCE 196 6 z-PLAN E Figur 4 Pol zer representatio o Eqs (8 o (9) . wher a] an a\ ar th variance o e\(nT) an d (nT), an wit h 1 = + r 2 _ r x r + j _ 2r co 2bT an d i - r 2 - / sin &r( + r ) + 1 - 2r cos2bT. Mor insigh ca b obtaine int thes result s lettin r = 1 - e an allowin c t b quit e small o th orde o 0.0 o less The (11 re duce t th simpl for m 1 «- J . + «1 sin bT (12 ) Carryin throug a simila computatio fo th e realizatio o Fig 3 yield s = (aj + a Ul (r,bT) (13 ) whic ca als b reduced fo smal value o e t o ' ' - " (14 ) e Severa importan fact ca b deduce fro Eqs . (12 an (14)

First th so-calle "straightforward " realizatio o Fig 3 lead t increase nois fo r lo resonanc frequencie wherea th "canonic " realizatio o Fig 3 doe not Physically thi re sul ca b explaine b notin that i th straight forwar realization th nois "passe through onl y th pole o th filter s tha a lo frequencies , th comple conjugat pole interac t for a lo w pas filter. I th "canonic realizatio th nois e als filtered b a zer whic i clos t d an thu s th outpu nois i o a band-pas natur an les s tota energ i abl t pas throug th filter. Second w not tha Eqs (13 an (14 hav th e sam functiona dependenc o

pol positions , namely tha th mea square outpu nois i in versel proportiona t th distanc fro th pol e th uni circl an therefor directl proportiona l th gai o th filter . Fro thes result on can fo example estimat e th wor lengt neede fo a simulatio requirin g man filters. On suc syste i a vocode synthe size show i Fig 5 Typically a vocode syn SPECTRA L COEFFICIENT S PITC H PULSE S FRO M DEMULTIPLE X NOIS E -0 1 BP F 1 BP F 2 BP F 8 J J BP F BP F BP F 8 J t SYNTHESIZE D SPEEC H EXCITATIO PROCESSIN i CONVENTIONA SYNTHESI S VOCODE SYNTHESIZE R Figur 5 Vocode synthesizer . thesize wil contai abou

10 resonators Assum in tha th nois fro eac resonato i additiv t o th nois fro al othe resonator an pickin a n effectiv averag c o 0.01 w arriv a a tota nois e outpu o abou 7 o 8 bits I i clea tha wor d length o a leas 2 bit ar neede t avoi audi bl nois output superimpose o th vocode r generate syntheti speech . EXPERIMENTA VERIFICATIO FO R FIRS AN SECON ORDE FILTER S Th result o th precedin computation wer e experimentall verifie b programmin variou s realization o first an secon orde differenc e equation o th TX- digita computer T per for a measuremen o outpu nois fo a give n digita filter,

th computation wer performe wit h rounde arithmeti usin a 36-bi word an simul taneously usin rounde arithmeti wit a shorte r wor an exactl th sam input Th output o f
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EFFECT O QUANTIZATIO NOIS I DIGITA FILTER S 21 7 th tw filters wer subtracted squared integrate d an divide b th numbe o iteration o th e equation Th input t th filter wer rando m nois o sample sinusoids Th filter wer pro gramme Usin th PATSI compiler an th var iou waveform o interest includin th mea n square outpu noise wer displaye durin th e computation Th measuremen wa take whe n th mea square outpu nois seeme

t reac a stead value o i th cas o th ver hig gai n filters whe th patienc o th observe wa ex hausted A w shal se below th necessar ob servatio tim fo confidenc i suc a measuremen t highl dependen o th gai o th filter. Figur 6 show th predicte an measure out pu noise fo som one-pol filters, a Eq (7) wit h o\ = 0 Th horizonta axi i th pol positio an d th vertica axi i th mea square outpu nois e normalize t a\ Tabl 1 give th predicte versu s measure outpu noise fo severa two-pol filters (n rea zeros wit variou pol positions alon g wit th measuremen error Al o th result see m confir th theory . i

advisabl t determine o a statistica basis , th measuremen tim require befor th varianc e suc statistica observation i sufficientl small . Thus conside a rando variabl q define a s - - E f (nT) m = 0 (15 ) where/(«r i a outpu nois signa a indicate i n Fig 1 du t a se o mutuall independen inpu t nois sample e(nT). Assuming/(«r t hav zer mean w ca im mediatel perceiv tha th mea valu o th meas uremen q i give b y 3 Q O J 2 BITS RANDO M NOIS INPU T 2 BITS RANDO M NOIS INPU T = 2 BITS SIN E WAV IN PU T E[q] = aj (16 ) 1/ 3/ 1 POL POSITIO N Figur 6 Predicte v measure quantizatio nois fo first orde

system . Tabl 1 Two-Pol Filte Nois Measuremen t Mea Square Outpu Nois e Predicte d 20 4 28 9 50 8 101 1 282 4 555 3 555 3 1101 4 1101 4 330 6 330 6 Measure d 20 3 29 7 52 0 105 8 288 0 593 3 550 3 1145 0 1107 9 374 0 335 9 Erro r °/ o 0.49 % 2.7 7 2.3 6 4.6 5 1.9 8 6.4 0 0.9 0 3.9 6 0.5 9 13.1 2 1.6 0 Pol e Position s ± .5 j ± .707 j ± .778 j .7 ± .56 j .87 ± .332 j .9062 ± .235 j .9062 ± .235 j .92187 ± .169 j .92187 ± .169 j .7 ± .654 j .7 ± .654 j
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21 8 PROCEEDINGS—SPRIN JOIN COMPUTE CONFERENCE 196 6 wher oj i th varianc o th (stationary rando m variabl f(nT). No assumin

that f(nT) i a se o f stationar Gaussia variable wit correlatio co efficien p(rT), the it ca b show that 5 E[f (mT)f (lT)] = IT) (17 ) wher R i define a th covarianc betwee n f(mT) an f(lT). Fro Eqs (16 an (17) w ar riv a th expressio fo th varianc o q, o] = E[q - E [q] A £,.R {mT- IT) (18 ) Thi ca b evaluate fo firs orde syste o f Fig 2 Fo tha case/? (rn - IT) = K ^-^ and , fo larg n, Eq (18 reduce t o = 4o e 2\ 3 n{\ - K>) (19 ) wher a] i th varianc o th inpu e(nT) a i Fig . O majo interes i determinin th tim neede d perfor th measuremen i th rati o th e standar deviatio t th mea o q. Usin a

argu men simila t th on tha lead t Eq (7 w ca n fo th firs orde syste relat o] t a b th for mul a} = o /{\ K ), whic combine wit Eqs . (19 an (16 yield s E[q] ( - K )VE (20 ) Thus fo example i K = 0.99 w nee 10 8 term i th measuremen o Eq (15 i orde t o reduc th standar deviatio o th measuremen t 2 o th mea o th measurement Assumin g tha a iteratio coul b don i 10 /*sec 10 4 second woul b require fo suc accuracy . NOIS CONSIDERATION I N PROGRAMMIN ITERATIV SIN E WAV GENERATOR S On mus b especiall attentiv t nois con sideration i th programmin o iterativ sin e wav generators Variou efficien

routine exis t o comput th sin o cosin o a rando argumen t rapidly bu fo instance wher th argumen i s nT fo successiv integer n, th mos efficien wa y generat sinusoida function i b th us o f iterativ differenc equations Thes are o course , digita filters wit pole directl o th uni circle , input equa t zero an initia condition whic h specif th magnitud an phas o th output . Sinc th pole o th filter ar directl o th uni t circle th noise accordin t Eq (12 o (14 be come infinite Thi i indee th situation. Th e savin featur i th gradual increas o th nois e term s tha i on run th progra fo a limite d

time o periodicall reset th initia conditions , catastroph ca b avoided T stud thi proble m theoretically conside th simultaneou differenc e equation s y(nT + T) = cos bTy(nT) ^ sin bTx(nT) x(nT + T) = - sin bTy(nT) co bTx(nT) (2 D wit initia condition x(0) = 1 >>(0 = 0 . "circuit i show i Fig 7 . Th e sinb T ••y(n + T ) #'x(n + T ) Figur 7 Iterativ sin an cosin generator . Th z transfor X(z) o on outpu x(nT) ca b e writte n X(z) = -zcosbT+zE (z)- co bTE (z) - sinbTE (z) - 2z co bT + 1 (22 ) se tha th first tw term o th numerato r correspon t th signa an th remainin term t o th noise E (z) an

E (z) bein respectivel th z •Variou nonlinearitie ca b introduce t kee th nois e finite. Thi i adequat fo man application sinc th selectivit y th filter ca b relie o t kee th outpu spectrall pur e eve i th phas o th outpu i unpredictable .
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EFFECT O QUANTIZATIO NOIS I DIGITA FILTER S 21 9 transform o th adde noise e {nT) an Q (nT), bot introduce b roundof error . Defining : {nT) = Z (nT) = Z - co bT - 2 co bT + 1 sin bT - 2 co bT + 1 (23 ) wher Z ' i th invers z transform w ca fro m Eq (3 writ th tota nois a s E{f\nT)) = £ h]{nT) = 0 (24 ) o\ zZ hl(nT) = 0 Solvin Eq (23

explicitl an lettin a\ = o\ = 2 w arriv a th resul t 2 E(f (nT)) =^ cos (nbT - bT) sin (nbT - bT)\ = ^ n (25 ) Notic tha i wa impossibl t us Eq (5) sinc e th resul obtaine woul b infinit an thu n o time-dependen resul coul b formulated Equa tio (25 tell u tha th nois increase linearl y wit th numbe o iteration o th differenc equa tions Fo example afte 10 iterations th nois e abou 1 bits Assumin tha on iteratio i s performe i 10 /usee severa minute coul cer tainl pass eve i a 18-bi machine befor th e generate sin an cosin wave begi t loo noisy . Anothe progra fo generatin a cosin wav i s

expresse b th iteratio n y(nT + 2T) = 2 cos bTy(nT + T) - y(nT) (26 ) wit initia condition y(0) = 1 y(T) = co bT. Nois analysi o Eq (26 lead t a functiona de pendenc o th mea square noise o th for m n 2 ; thu appreciabl greate quantitie o nois e ar generate a lo frequencies an fewe itera tion ar availabl befor th progra become s unusable . Th compariso o Eqs (21 an (26 wa per forme qualitativel o TX- b programmin g identica sin wav generator usin bot methods . Fo al frequencies th metho o Eq (21 pro duce sinusoid o mor nearl constan amplitud e tha th metho o Eq (26) bu thi differenc i n

behavio wa negligibl fo frequencie greate tha n on fourt o th samplin frequency and usin g 36-bi arithmetic th distortion wer almos unob servabl fo thes frequencies Fo lo frequencie s (o th orde o on thousandt o th samplin g rate th metho o Eq (26 wa completel unus able wit th generate sin wav bein terribl y distorte i th firs period . REFERENCE S J F Kaiser "Som Practica Consideration s th Realizatio o Linea Digita Filter, 3r d Allerto Conferenc (Oct 20-22 1965) . J B Knowle an R Edwards "Effec o a Finite-Word-Lengt Compute i a Sampled-Dat a Feedbac System, Proc. IEEE, vol 112 no 6 , (Jun

1965) . W R Bennett "Spectr o Quantize Sig nals, Bell System Technical Journal, vol 27 pp . 446-47 (Jul 1948) , C M Rader "Speec Compressio Simula tio Compiler, J. Acoust, Soc. Am. (A) Jun 1965 . J L Lawso an G E Uhlenbeck Threshold Signals, MI Rad Lab Serie 24 McGraw-Hill , Ne York 1950 .