Analysis of Finite Word-Length Effects

Transcription

1 T-6.46 Digital Signal Pocessing and Filteing Intoduction Analysis of Finite Wod-Length Effects Finite wodlength effects ae caused by: Quantization of the filte coefficients ounding / tuncation of multilication esults Quantization of the inut signal Dynamic ange constaints of the imlementation 4 Olli Simula T-6.46 / Mita: Chate 9 Analysis of Finite Wodlength Effects Ideally, the system aametes along with the signal vaiables have infinite ecision taking any value between and In actice, they can take only discete values within a secified ange since the egistes of the digital machine whee they ae stoed ae of finite length The discetization ocess esults in nonlinea diffeence equations chaacteizing the discete-time systems 4 Olli Simula T-6.46 / Mita: Chate 9 3 Coyight, S. K. Mita Analysis of Finite Wodlength Effects These nonlinea equations, in incile, ae almost imossible to analyze and deal with exactly Howeve, if the quantization amounts ae small comaed to the values of signal vaiables and filte aametes, a simle aoximate theoy based on a statistical model can be alied 4 Olli Simula T-6.46 / Mita: Chate 9 4 Coyight, S. K. Mita Analysis of Finite Wodlength Effects Using the statistical model, it is ossible to deive the effects of discetization and develo esults that can be veified exeimentally Souces of eos - ( Filte coefficient quantization ( A/D convesion (3 Quantization of aithmetic oeations (4 Limit cycles 4 Olli Simula T-6.46 / Mita: Chate 9 5 Coyight, S. K. Mita e [ n ] e j [ Analysis of Noise Poeties and Dynamic ange Constaints Σ Σ * G G * j z ( x [ y[ * F H ( F * i v * [ n ] v * i n [ ] 4 Olli Simula T-6.46 / Mita: Chate 9 6 Σ Mita: Chate 9 / Coyight Olli Simula

2 T-6.46 Digital Signal Pocessing and Filteing Examle: Fist Ode II Filte y [ αy[ n ] + x[ z αz z α H Quantization of coefficients α: H '( α' z Quantization of inut x[: x '[ x[ + e[ ounding/tuncation of v[: v' [ v[ + e [ Outut y[ with finite wodlength: y' [ y[ + η[ α The Quantization Pocess and Eos Factional numbes (sign bit + factional at The quantization ocess model Eo : ounding / Tuncation: ε Q( x x 4 Olli Simula T-6.46 / Mita: Chate Olli Simula T-6.46 / Mita: Chate 9 8 The Quantization Eos Quantization Eo ε ounding b β ( b β ( Two s comlement tuncation b β ( ε t Sign-magnitude and one s comlement tuncation b β ( ε t fo x > b β εt ( fo x < 4 Olli Simula T-6.46 / Mita: Chate Olli Simula T-6.46 / Mita: Chate 9 Quantization of Floating-Point Numbes Only mantissa is quantized; the elative eo is elevant! E E x M, Q( x Q( M Eo : Q( x x Q( M x e x x Analysis of Coefficient Quantization Effects The tansfe function Hˆ of the digital filte imlemented with quantized coefficients is diffeent fom the desied tansfe function H( Main effect of coefficient quantization is to move the oles and zeos to diffeent locations fom the oiginal desied locations 4 Olli Simula T-6.46 / Mita: Chate 9 4 Olli Simula T-6.46 / Mita: Chate 9 Coyight, S. K. Mita Mita: Chate 9 / Coyight Olli Simula

3 T-6.46 Digital Signal Pocessing and Filteing Analysis of Coefficient Quantization Effects The actual fequency esonse ˆ H ( e is thus diffeent fom the desied fequency esonse H ( e In some cases, the oles may move outside the unit cicle causing the imlemented digital filte to become unstable even though the oiginal tansfe function H( is stable Analysis of Coefficient Quantization Effects Diect fom ealizations ae moe sensitive to coefficient quantization than cascade o aallel foms The sensitivity inceases with inceasing filte ode Usually second ode blocks in cascade o aallel ae used 4 Olli Simula T-6.46 / Mita: Chate 9 3 Coyight, S. K. Mita 4 Olli Simula T-6.46 / Mita: Chate 9 4 Coyight, S. K. Mita Coefficient Quantization Effects On a Diect Fom II Filte Gain esonses of a 5-th ode ellitic lowass filte with unquantized and quantized coefficients Fullband Gain esonse Passband Details oiginal - solid line, quantized - dashed line oiginal - solid line, quantized - dashed line ω/ ω/ 4 Olli Simula T-6.46 / Mita: Chate 9 5 Gain, db Gain, db Coyight, S. K. Mita Coefficient Quantization Effects On a Diect Fom II Filte Pole and zeo locations of the filte with quantized coefficients (denoted by x and o and those of the filte with unquantized coefficients (denoted by + and * eal Pat 4 Olli Simula T-6.46 / Mita: Chate 9 6 Coyight, S. K. Mita Imaginay Pat Coefficient Quantization Effects On a Cascade Fom II Filte Gain esonses of a 5-th ode ellitic lowass filte imlemented in a cascade fom Gain, db with unquantized and quantized coefficients Fullband Gain esonse Passband Details - -4 oiginal - solid line, quantized - dashed line oiginal - solid line, quantized - dashed line ω/ ω/ 4 Olli Simula T-6.46 / Mita: Chate 9 7 Gain, db Coyight, S. K. Mita Coefficient Quantization Effects On A Diect Fom FI Filte Gain esonses of a 39-th ode equiile lowass FI filte with unquantized and quantized coefficients Gain, db - -4 Fullband Gain esonse oiginal - solid line, quantized - dashed line oiginal - solid line, quantized - dashed line ω/ ω/ 4 Olli Simula T-6.46 / Mita: Chate 9 8 Gain, db - - Passband details Coyight, S. K. Mita Mita: Chate 9 / Coyight Olli Simula 3

4 T-6.46 Digital Signal Pocessing and Filteing Examle of Coefficient Quantization in 6 th Ode Diect Fom ealization Examle of Coefficient Quantization in 6 th Ode Cascade Fom ealization Amlitude esonses Pole-zeo locations Amlitude esonses Pole-zeo locations 4 Olli Simula T-6.46 / Mita: Chate Olli Simula T-6.46 / Mita: Chate 9 Examle: 6 th ode bandsto filte with unquantized coefficients Cascade fom with coefficients quantized to 6 bits Paallel fom with coefficients quantized to 6 bits 4 Olli Simula T-6.46 / Mita: Chate 9 Coefficient Quantization in FI Filtes Conside an (M-th ode FI tansfe function H M n h[ z Quantization of the filte coefficients esults in a new tansfe function M M n n H ' h'[ z ( h[ + e[ z H( n n 4 Olli Simula T-6.46 / Mita: Chate 9 n H '( H + E( E( Linea hase: h[ + h[n-- Symmety of the imulse esonse not affected by quantization + A/D Convesion Noise Analysis Quantization Noise Model Two s comlement eesentation Analog inut Inut analog samle x[ Quantized inut samle Binay equivalent of quantized inut Quantization of the inut signal intoduces eo at the inut of the filte This eo is oagated though the filte togethe with the inut signal Affects the signal-to-noise atio of the system 4 Olli Simula T-6.46 / Mita: Chate 9 3 ˆ [ < Inut signal is assumed to be scaled to be in the ange of + by dividing its amlitude by FS / 4 Olli Simula T-6.46 / Mita: Chate 9 4 x eq xˆ[ xˆ eq[ FS δ + b FS Mita: Chate 9 / Coyight Olli Simula 4

5 T-6.46 Digital Signal Pocessing and Filteing Quantization Eo The quantization eo e[: e[ Q( x[ x[ xˆ[ x[ Fo two s comlement ounding: δ δ < e[ Outside FS the eo inceases linealy; e[ is called the satuation eo o the oveload noise The outut value is clied to the maximum value 4 Olli Simula T-6.46 / Mita: Chate 9 5 e[ is called ganula noise Model of the Quantization Eo x[ + x ˆ[ x[ + e[ e[ Assumtions: The eo sequence {e[} is a samle sequence of a widesense stationay (WSS white noise ocess, with each samle e[ being unifomly distibuted ove the quantization eo The eo sequence is uncoelated with its coesonding inut sequence {x[} 3 The inut sequence is a samle sequence of a stationay andom ocess The assumtions hold in most actical situations with aidly changing inut signals 4 Olli Simula T-6.46 / Mita: Chate 9 6 Quantization Eo Distibutions (a ounding (b Two s comlement tuncation m e ( δ / ( δ / (( δ / ( δ / δ e m e δ δ ( δ δ e Signal-to-Noise atio Additive quantization noise e[ on the signal x[ Signal-to-quantization noise atio in db is defined as log db x SN e whee x is the signal vaiance (owe and e is the noise vaiance (owe The vaiance eesents the noise owe 4 Olli Simula T-6.46 / Mita: Chate Olli Simula T-6.46 / Mita: Chate 9 8 Signal-to-Noise atio A/D convesion: (b+l bits: δ -(b+ FS, whee FS is the full-scale ange ( b+ b FS FS e δ / log 6.b 6.8- log db x FS SN + A D b FS x Thus, SN inceases 6 db fo each added bit in the wodlength 4 Olli Simula T-6.46 / Mita: Chate 9 9 Effect of Inut Scaling on SN Let the inut scaling facto be A with A> The vaiance of the scaled inut Ax[ is A x The SN changes to SN A/ D FS 6.b log A x 6.b log ( K + log ( A whee FS K x ( x is the MS value of the signal Scaling down the inut signal (A< deceases the SN Scaling u the inut signal (A> inceases the ossibility to exceed the full-scale ange FS esulting in cliing SN 4 Olli Simula T-6.46 / Mita: Chate 9 3 Mita: Chate 9 / Coyight Olli Simula 5

6 T-6.46 Digital Signal Pocessing and Filteing Poagation of Inut Quantization Noise to Digital Filte Outut Due to lineaity of H( and the assumtion that x[ and e[ ae uncoelated the outut can be exessed as a linea combination (sum of two sequences: yˆ [ h[ xˆ[ The outut noise is: [ x[ + e[ ] h[ x[ + h[ e[ ] h[ n m v [ e[ m] h[ n m] 4 Olli Simula T-6.46 / Mita: Chate 9 3 Poagation of Inut Quantization Noise to Digital Filte Outut The mean and vaiance of v[ chaacteize the outut noise j The mean m v is: m v meh ( e The noise vaiance v is: e v The outut noise owe sectum is: vv 4 Olli Simula T-6.46 / Mita: Chate 9 3 e H ( e P ( ω H ( e j ω dω Poagation of Inut Quantization Noise to Digital Filte Outut The nomalized outut noise vaiance is given by ω H e j, ( v v n e which can be witten as: dω Analysis of Aithmetic ound-off Eos j C v, n H H ( z z An equivalent exession is: v, n h[ n 4 Olli Simula T-6.46 / Mita: Chate 9 33 dz Quantization of Multilication esults Assumtions: The eo sequence {e α [} is a samle sequence of a stationay white noise ocess, with each samle e α [ being unifomly distibuted The quantization eo sequence {e α [} is uncoelated with the signal {v[}, the inut sequence {x[} to the filte, and all othe quantization eos The assumtion of {e α [} being uncoelated with {v[} holds fo ounding and two s comlement tuncation 4 Olli Simula T-6.46 / Mita: Chate 9 35 Quantization of Multilication esults The quantization model can be used to analyze the quantization effects at the filte outut Quantization befoe summation The numbe of multilications k l at adde inuts The th banch node with signal value u [ needs to be scaled to event oveflow 4 Olli Simula T-6.46 / Mita: Chate 9 36 Mita: Chate 9 / Coyight Olli Simula 6

7 T-6.46 Digital Signal Pocessing and Filteing Quantization of Multilication esults Statistical model of the filte: f [ Imulse esonse fom filte inut to banch node g l [ Imulse esonse fom inut of lth adde to filte outut 4 Olli Simula T-6.46 / Mita: Chate 9 37 Quantization of Multilication esults Banch nodes to be scaled u [ α lead to multilies and ae v l [ + oututs of summations: Scaling tansfe function: F ( Noise tansfe function: G l ( Let be the vaiance of each individual noise souce; then k l is the noise vaiance of e l [ The outut noise vaiance is: [ ( ] kl G l ( z z dz k G e dω j l ( C 4 Olli Simula T-6.46 / Mita: Chate 9 38 Quantization of Multilication esults The total outut noise vaiance: γ L kl l ( ( z z d j whee L is the numbe of summation nodes to which noise souces ae connected The noise vaiance can also be witten as γ C L l n kl g [ 4 Olli Simula T-6.46 / Mita: Chate 9 39 l The Outut Quantization Noise The amount of noise deends on the imlementation Quantization of multilication esults afte summation educes the numbe of noise souces to one The vaiance of the noise souce e l [ is now DSP ocesso cay out multily-accumulate oeation using double ecision aithmetic 4 Olli Simula T-6.46 / Mita: Chate 9 4 Dynamic ange Scaling Digital filte The th node value u [ has to be scaled Assume that the inut sequence is bounded by unity, i.e., x[ < fo all values of n The objective of scaling is toensue that u [ < fo all and all values of n Dynamic ange Scaling Thee diffeent conditions to ensue that u [ satisfies the conditions: An absolute bound L infinity -bound 3 L -bound Diffeent bounds ae alicable unde cetain inut signal conditions 4 Olli Simula T-6.46 / Mita: Chate Olli Simula T-6.46 / Mita: Chate 9 4 Mita: Chate 9 / Coyight Olli Simula 7

8 T-6.46 Digital Signal Pocessing and Filteing An Absolute Bound f [ Digital filte F ( is the scaling tansfe function The node value u [ is detemined by the convolution f k u [ [ k] x[ n k] 4 Olli Simula T-6.46 / Mita: Chate 9 43 An Absolute Bound Assuming that x[ satisfies the dynamic ange constaint x[ < k 4 Olli Simula T-6.46 / Mita: Chate 9 44 u [ f [ k] x[ n k] f [ k] k The node value u [ now satisfies the dynamic ange constaint, i.e., u [ < if k f [ k] fo all This is both necessay and sufficient condition to guaantee that thee will be no oveflow Scaling with the Absolute Bound If the dynamic ange constaint is not satisfied the filte inut has to be scaled with the multilie K K max f[ k k] The scaling ule based on the absolute bound is too essimistic and educes the SN significantly Moe actical and easy to use scaling ules can be deived in the fequency domain if some infomation about the inut signal is known a ioi 4 Olli Simula T-6.46 / Mita: Chate 9 45 Scaling Noms Definethe L -nom of a Fouie tansfom F(e as j ω F F( e d ω L -nom, F, is the oot-mean-squae (MS value of F(e, and L -nom, F, is the mean absolute value of F(e ove ω Moeove, lim -> F exists fo a continuous F(e and is given by its eak F max F( e ω 4 Olli Simula T-6.46 / Mita: Chate 9 46 Scaling Noms: L -Bound U ( e F ( e An invese Fouie tansfom X ( e n [ F ( e X ( e e u u [ F ( e F ( e F ( e X ( e X ( e X ( e 4 Olli Simula T-6.46 / Mita: Chate 9 47 dω dω dω Scaling Noms: L -Bound If X <, then the dynamic ange constaints satisfied if F If the mean absolute value of the inut sectum is bounded by unity, then thee will be no adde oveflow if the eak gains fom the filte inut to all adde outut nodes ae scaled satisfying the above bound The scaling ule is aely used since with most inut signals encounteed in actice X < does not hold 4 Olli Simula T-6.46 / Mita: Chate 9 48 Mita: Chate 9 / Coyight Olli Simula 8

9 T-6.46 Digital Signal Pocessing and Filteing Scaling Noms: L -Bound n u [ F ( e X ( e e dω Alying Schwaz inequality u [ F ( e dω o equivalently u [ F ( e If the filte inut has finite enegy bounded by unity, i.e., X <, then the adde oveflow can be evented by scaling the filte such that the MS value of the scaling tansfe functions ae bounded by unity: F,,,..., 4 Olli Simula T-6.46 / Mita: Chate 9 49 X ( e X ( e dω A Geneal Scaling ule A moe geneal scaling ule is obtained using Holde s inequality u [ F ( e X ( e q ( ( + q fo all,q >, with Afte the scaling the tansfe functions become F and the scaling constants should be chosen such that F',,,..., In many stuctues the scaling multilies can be absobed to the existing feedfowad multilies 4 Olli Simula T-6.46 / Mita: Chate 9 5 Scaling of a Cascade Fom II Filte The nodes ( * need to be scaled Bi + b iz + bi z H K H i, whee Hi i Ai + a iz + aiz K' Scaling tansfe functions: F H ' l,,,..., A l F ( can be exessed by oles and zeos of the oiginal H( 4 Olli Simula T-6.46 / Mita: Chate 9 5 Scaling - Back-Scaling α FILTE The effect of inut scaling is comensated by back-scaling at the outut of the filte Scaling block-by-block in cascade ealization foms H ( H ( H ( α α α α α Each second ode block is scaled individually The scaling coefficients between the blocks contain the backscaling of the evious block and the scaling of the the next block 4 Olli Simula T-6.46 / Mita: Chate 9 5 α α Scaled Cascade Fom II Filte Stuctue H K l H l, whee + b z Hl + a Scaling tansfe functions: K F Hl, A l,,..., i l z + b z + a l l z The scaled stuctue has new values of the coefficients in the feed-fowad banches Only one citical banch node in each second ode block has to be checked fo oveflow 4 Olli Simula T-6.46 / Mita: Chate 9 53 Otimum Section Odeing and Pole-Zeo Paiing of a Cascade Fom II Digital Filte Odeing of second-ode sections as well as aiing of oles and zeos affects the outut noise owe of the filte Mita: Chate 9 / Coyight Olli Simula 9

10 T-6.46 Digital Signal Pocessing and Filteing Noise Tansfe Functions The noise tansfe functions can be exessed using the tansfe functions of the cascaded second-ode blocks The scaled noise tansfe functions ae given by K Hi βi, l,,..., ; and G+ i l i l 4 Olli Simula T-6.46 / Mita: Chate 9 55 Noise Tansfe Functions The outut noise owe sectum due to oduct ound-off is given by + Pyy ( ω kl ( e l and outut noise vaiance is + + y kl ( e dω kl l l whee the integal in the aenthesis is the squae of the L -nom of the noise tansfe function 4 Olli Simula T-6.46 / Mita: Chate 9 56 Noise Model of Second-Ode Blocks The noise model intoduces noise souces to the inut/outut summation of each block The numbe of elementay noise souces, k l, has diffeent values deending on the location of ounding (befoe o afte the summation and deending on the block (fist, intemediate, last Let k l be the total numbe multilies connected to the l th adde ounding befoe summation: k k + 3, k l 5, fo l, 3,..., ounding afte summation: k l, fo l, Noise Tansfe Functions The scaling coefficients ae The outut noise owe sectum of the scaled filte is + + Pyy ( ω k + H k l Fl H l and outut noise vaiance is ( e + + y k + H kl Fl G l H l,...,+ 4 Olli Simula T-6.46 / Mita: Chate Olli Simula T-6.46 / Mita: Chate 9 58 i l α βi α l + F l H Minimizing the Outut ound-off Noise The scaling tansfe function F l ( contains sections H i (, i,,..., l- The noise tansfe function G l ( contains sections H i (, i l, l+,..., Evey tem in the sum fo the noise owe o the noise vaiance includes the tansfe function of all sections in the cascade ealization To minimize the outut noise owe the noms of H i ( should be minimized fo all values of i by aoiately aiing the oles and zeos 4 Olli Simula T-6.46 / Mita: Chate 9 59 Paiing the Poles and Zeos Poles close to unit cicle intoduce gain and zeos (on the unit cicle intoduce attenuation Fist, the oles closest to the unit cicle should be aied with the neaest zeos Next, the oles closest to the evious set of oles should be aied with the next closest zeos 3 This ocess is continued until all oles and zeos ae aied 4 Olli Simula T-6.46 / Mita: Chate 9 6 Mita: Chate 9 / Coyight Olli Simula

11 T-6.46 Digital Signal Pocessing and Filteing Section Odeing A section in the font at of the cascade has its tansfe function H i ( aeaing moe fequently in the scaling tansfe functions A section nea the outut end of the cascade has its tansfe function H i ( aeaing moe fequently in the noise tansfe function exessions > The best location fo H i ( deends on the tye of noms being alied to the scaling and noise tansfe functions 4 Olli Simula T-6.46 / Mita: Chate 9 6 Section Odeing L scaling: The odeing of aied sections does not influence too much the outut noise owe since all noms in the exessions ae L -noms L scaling: The sections with oles closest to the unit cicle exhibit a eaking magnitude esonse and should be laced close to the outut end > The odeing should be fom least-eaked to most-eaked On the othe hand, the odeing scheme is exactly oosite if the objective is to minimize the eak noise P yy (ω and L - scaling is used The odeing has no effect on the eak noise with L -scaling 4 Olli Simula T-6.46 / Mita: Chate 9 6 Eo Sectum Shaing Quantization eo can be comensated using the so called eofeedback (o eo sectum shaing The filteed eo signal is added to the signal banch befoe quantization (Q[.]. 4 Olli Simula T-6.46 / Mita: Chate 9 63 Eo Sectum Shaing Without eo-feedback the eo signal e[ is the ue quantization eo, i.e., e[ y[ - x[ In the comensated stuctue the eo signal is the diffeence between the outut y[ and the comensated inut signal 4 Olli Simula T-6.46 / Mita: Chate 9 64 Eo Sectum Shaing w[ x[ + ae[ n ] + be[ n ] e[ y[ w[ Substitutingw[: e[ y[ x[ ae[ n ] be[ n ] Total eo between outut and inut is still: e[ y[ x[ Eo Sectum Shaing Solving y[ - x[: y[ x[ e[ + ae[ n ] + be[ n ] Taking the z-tansfom: Y X E( + az E( + bz E( ( + az + bz E( G( E( whee G( is the eo shaing tansfe function 4 Olli Simula T-6.46 / Mita: Chate Olli Simula T-6.46 / Mita: Chate 9 66 Mita: Chate 9 / Coyight Olli Simula

12 T-6.46 Digital Signal Pocessing and Filteing Eo Sectum Shaing Examle: a- and b G( ( + az + bz ( z + z ( z Double zeo is at z Noise sectum is modified by attenuating noise at low fequencies 4 Olli Simula T-6.46 / Mita: Chate 9 67 Mita: Chate 9 / Coyight Olli Simula