Fixed Point Effects in Digital Filters

Transcription

1 Fixd Point Effcts in Digital Filtrs Cimarron Mittlstadt David Hwang Finit-prcision Problms Quantizrs ar nonlinar dvics Charactristics may b significantly diffrnt from th idal linar filtr Ovrflow Cofficint quantization Limit-Cycl Oscillations

2 Quantizrs Nonlinar ffcts mak it xtrmly difficult to prcisly analyz th filtr s prformanc. How do w modl a fixd-point filtr thn? Adopt a statistical modl of th quantization ffcts Rsults in a linar modl for th filtr Statistical Charactrization v(n) = Q[av(n-)]+x(n) x(n) v(n) Q[av(n-)] = av(n-)+(n) v(n) = av(n-)+x(n)+(n) a W can now viw th rspons of th filtr as coming from two inputs. (n)

3 Basic Assumptions Th nois sourc is stationary whit nois. Th squnc (n) is uncorrlatd with th squnc (m) for n m. Squnc is man rgodic and corrlation rgodic. Th rror squnc (n) is uncorrlatd with th input squnc x(n). Man Th man of th output gnratd by th filtr with impuls rspons h(n) whn xcitd by (n) is m = m h( n) q n= quivalntly put m q = m H () 3

4 Autocorrlation Th autocorrlation is computd to b qq γ ( n ) = h( k) h( l) γ ( k l + n) k= l= This rducs to = q k= h ( k) By Parsval s thorm q = π π π H ( ω) dω Typs of Quantization b fractional bits Rounding m = = b Truncation m = b = b Magnitud Truncation m = = b 3 4

5 Sction Ordring is Important Exampl H ( z) = H( z) H ( z) Whr H( z) = H ( z) = z 4 z and thir corrsponding impuls rsponss ar givn by h(n), h (n) and h (n) Ralization x(n) v(n) ½ ¼ (n) (n) With rounding q = h ( n) + h ( n). 9 n= n= 5

6 Ralization x(n) v(n) ¼ ½ (n) (n) With rounding q = h ( n) + h ( n) 3. 6 n= n= Sction Ordring Comparison Th ovrall noiss wr found to b 9 q q q q.9 Thus, th scond ralization lads to 9% mor nois powr than th first. 6

7 Ovrflow Wrap Around Ex. 7 () + () = -8 () Saturation Ex. 5 () + 4 () = 7 () Scaling to Prvnt Ovrflow Pssimistic Scaling Narrowband Scaling A x < k m= h ( m) Ax < max H ω π k ( ω) 7

8 Practical Round-Off Effcts on Digital Filtrs Cofficint Quantization Frquncy Rspons Charactristics Intrnal Wordlngth Quantization Dynamic Rang Signal-to-Nois Ratio Limit Cycls Cofficint Quantization Cofficint quantization altrs th valus of th cofficints => changs your frquncy rspons A filtr dsignd in floating point arithmtic to mt crtain spcs may not mt thos spcifications aftr cofficint quantization 8

9 Cofficint Quantization Exampl W usd th Parks-McClllan optimal FIR filtr algorithm to dsign a -tap filtr Dsignd th filtr in floating point format Took ach of th cofficints and roundd to th narst 8-bit two s complmnt numbr Ex:.99 =>.35 ( two s complmnt) Lsson: Ovr-dsign th filtr and/or us an optimization algorithm to mt th spc Frquncy Rspons Floating Point -Passband Rippl:.3 db -Stopband Attnuation: 48 db. Floating Point Passband Fixd Point -Passband Rippl:.4 db -Stopband Attnuation: 36 db. Fixd Point Passband Magnitud Rspons (db).. Magnitud Rspons (db) Normalizd frquncy (Nyquist == ) Normalizd frquncy (Nyquist == ) Magnitud Rspons (db) Floating Point Stopband Normalizd frquncy (Nyquist == ) Magnitud Rspons (db) Fixd Point Stopband Normalizd frquncy (Nyquist == ) Cofficint Quantization => Frquncy Rspons Dgradation 9

10 Intrnal Wordlngth Quantization Quantization of intrnal wordlngth lads to finit-wordlngth ffcts x(n) Z - Z - Z- ) Dynamic Rang ) Signal-to-Nois Ratio Q Q Q Q 3) Limit Cycls y(n) intrnal wordlngth b Dynamic Rang Constraints Dynamic Rang is dfind as: log rang of rprsntabl numbrs smallst non-zro rprsntabl numbr Th largr th dynamic rang spcification, th largr intrnal wordlngth b rquird Ex. b-bit numbr (x. x x x 3 x 4.x b- ) DR: log -(b-) / -(b-) = log ( -b ) ~ 6 db / bit

11 Dynamic Rang (cont d) b = 8 => DR ~ 48 db b = => DR ~ 7 db b = 6 => DR ~ 96 db Nowadays, most hi-fi audio systms hav a DR in th rang of 8- db Signal-to-Nois Ratio Th signal-to-nois ratio is dfind as: SNR = log signal powr / nois powr = log x = /= = For our systm (wordlngth b): SNR = log x = /=( -(b-) /) = log x = + 6.b Ex. x(n) =.75 sin (ωn) => SNR = 6.b b = 8 => SNR = 47 db

12 Signal-to-Nois Ratio (cont d) Thus, SNR incrass by ~ 6 db / bit To maximiz SNR, you want to scal th signal x(n) as larg as possibl (to incras x ) Howvr, thr is a tradoff you nd to kp all intrnal signals small nough to prvnt ovrflow / saturation (us normalization and scaling tchniqus) Limit Cycls Limit cycls occur whn th output of a digital filtr dos not dcay to zro whn th input gos to zro Limit cycls occur only in IIR filtrs and nvr occur in FIR filtrs Thy ar causd by quantizing th data aftr a fdback multiplir in a rcursiv loop

13 Limit-Cycl Exampl x(n) b y(n) lt M = -.96, y(-) = 4, x(n) = and rounding to th narst intgr Z - n -.96 * y(n-) y(n) -3 b Q b M => Limit cycl puts nrgy at Fs/ which is dtrimntal.5 Limit Cycls in Matlab Output of th systm Input to th systm.5 Limit cycl of Input FFT Output FFT INPUT causs a spik at DC OUTPUT 3

14 Eliminating Limit Cycls Us magnitud truncation (which always dcrass th nrgy of th signal) Us a filtr for which a Lyapunov function xists Us controlld rounding Us novl filtr structurs dsignd to liminat limit cycls Rfrncs J. Proakis and D. Manolakis. Digital Signal Procssing: Principls, Algorithms, and Applications. 3 rd Edition. Prntic Hall, Uppr Saddl Rivr, Nw Jrsy, 996. M. Wrtr. EE A Lctur Nots. Los Angls, California, 998. R. Schafr. MEAD DSP IC Dsign Cours, Lctur #3, Quantization Effcts in Digital Filtrs. Atlanta, Gorgia. 4