Lecture 3 : Filter Banks

Transcription

1 Lectre : Filter Banks Part - : Filter Banks Preliminaries Filter bank introdction & applications Aliasing & perfect reconstrction PR Review of mlti-rate sstems st PR example : DFT/FFT filter bank Part - : Filter Banks Uniform filter banks, DFT-modlated filter banks Qadratre mirror filter banks Maximall decimated criticall downsampled PR filter banks Oversampled non-criticall downsampled PR filter banks VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p Filter Banks Introdction General `sbband processing set-p/overview: - Signals split into freqenc cannels/sbbands `analsis bank per-cannel/sbband processing reconstrction `sntesis bank - Mlti-rate strctre: down-sampling after analsis, p-sampling before sntesis - Aliasing vs ``Perfect reconstrction filter banks - Applications: coding, adaptive filtering, transmltiplexers I sbband processing sbband processing sbband processing G G G OUT sbband processing G VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p

2 Filter Banks Introdction Analsis filter bank: - collection of M filters `analsis filters, `decimation filters wit a common inpt signal - ideal bt non-practical freqenc responses ideal bandpass filters - tpical freqenc responses overlapping, marginall overlapping, non-overlapping M π I π π VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p Filter Banks Introdction Decimators downsamplers : I - sbband sampling rate redction b factor - maximall decimated filter banks, criticall downsampled filter banks: M nmber of filters/cannels # sbband samples # fllband samples sonds like maximm efficienc, bt aliasing see below! - non-criticall downsampled filter banks, oversampled filter banks: <M M # sbband samples> # fllband samples VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p

3 Filter Banks Introdction Sbband processing : - Example : coding compression transmission or storage decoding - Filter bank design mostl assmes sbband processing as `nit transfer fnction otpt signalsinpt signals, ie mostl ignores presence of sbband processing M sbband processing I sbband processing sbband processing sbband processing VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p 5 Filter Banks Introdction Expanders psamplers - restore original fllband sampling rate b -fold psampling insert - eros in between ever two samples M sbband processing I sbband processing sbband processing sbband processing VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p 6

4 Filter Banks Introdction 5 Sntesis filter bank: - collection of M filters `sntesis filters, `interpolation filters wit a `common smmed otpt signal - freqenc responses : preferabl `matced to freqenc responses of te analsis filters, eg, to provide perfect reconstrction see below M G G G OUT G G G G G G G G G G G G π π G π VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p 7 π Filter Banks Introdction Aliasing verss Perfect Reconstrction - Assme sbband processing does not modif sbband signals eg lossless coding/decoding - Te overall aim cold be to ave [k][k-d], ie tat te otpt signal is eqal to te inpt signal p to a certain dela - Bt: downsampling introdces ALIASIG, especiall in maximall decimated bt even so in non-maximall decimated filter banks - Qestion : Can [k][k-d] be acieved in te presence of aliasing? - AnswerYES, see below : PERFECT RECOSTRUCTIO banks wit sntesis bank designed to remove aliasing effects! [k] otptinpt otptinpt otptinpt otptinpt G G G G [k][k-d]? VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p 8

5 Filter Banks Applications Sbband coding: Coding Fllband signal split into sbbands & downsampled sbband signals separatel encoded sbband wit smaller energ content, encoded wit fewer bits Decoding reconstrction of sbband signals, ten fllband signal expanders sntesis filters Example : Image coding Example : Adio coding eg digital compact cassette DCC, MPEG, Filter bandwidts and bit allocations cosen to frter exploit perceptal properties of man earing perceptal coding, masking, etc VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p 9 Filter Banks Applications Sbband adaptive filtering: - Example : Acostic eco cancellation Adaptive filter models time-varing acostic eco pat and prodces a cop of te eco, wic is ten sbtracted from micropone signal VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p 5

6 Filter Banks Applications Sbband adaptive filtering contined: - Sbband filtering M simpler? sbband modeling problems instead of one more complicated? fllband modeling problem - Perfect reconstrction garantees distortion-free desired near-end speec signal adfilter adfilter adfilter adfilter G G G G OUT VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p Filter Banks Applications Transmltiplexers: - Freqenc Division Mltiplexing FDM in a digital telepone network - M different sorce signals mltiplexed into transmit signal b expanders & sntesis filters ps: ere interpolation factor > M - Received signal decomposed into M sorce signals b analsis filters & decimators signal- signal- signal- signal- 5 G 5 G 5 G 5 G transmission cannel - Again ideal filters ideal bandpass filters - on-ideal sntesis & analsis filters reslt in CROSS-TALK between cannels, ie eac reconstrcted signal contains nwanted contribtions from oter signals signal- signal- signal- signal- VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p 6

7 Filter Banks Applications Transmltiplexers: contined - Analsis & sntesis reversed, bt similar perfect reconstrction teor were analsis bank removes cross-talk introdced b sntesis bank, if transmission cannel distortion free - PS: special case is Time Division Mltiplexing TDM, if sntesis and analsis filters are replaced b dela operators and M [k],[k],[k],[k],[k],[k] [k],[k] [k],[k] [k],[k] [k],[k] transmission cannel [k-],[k] [k-],[k] [k-],[k] [k-],[k] [k],,,,[k],,,, VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p Review of Mlti-rate Sstems Decimation : decimator downsampler [],[],[] [], [], [] example : [k]:,,,,5,6,7,8,9, -fold downsampling:,,5,7,9, Interpolation : expander psampler [], [], [], [],,,[],,,,[] example : [k]:,,,,5,6,7,8,9, -fold psampling:,,,,,,,,5, VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p 7

8 Review of Mlti-rate Sstems Z-transform freqenc domain analsis of expander [], [], [], [],,,[],,,,[] U U `images π π π `expansion in time domain ~ compression in freqenc domain expander mostl followed b interpolation filter to remove all images π VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p 5 Review of Mlti-rate Sstems Z-transform freqenc domain analsis of decimator [],[],[] U [], [], [] i U e j π i i i i π π π decimation introdces ALIASIG if inpt signal occpies freqenc band larger tan π, for π ω π ence decimation mostl preceded b anti-aliasing decimation filter π π VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p 6 8

9 Review of Mlti-rate Sstems Z-transform analsis of decimator contined U jω - ote tat U e is periodic wit period π wile j / U e ω is periodic wit period π te smmation wit i - restores te periodicit wit period π! -Example: k [ k ] α, k U α Y α [ k ] α i k, k i U e j π i VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p 7 Review of Mlti-rate Sstems PS: sbband set-p revisited I sbband processing sbband processing sbband processing G G G OUT sbband processing G - analsis filters i are also decimation anti-aliasing filters, to avoid aliased contribtions in sbband signals - sntesis filters Gi are also interpolation filters, to remove images after expanders psampling VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p 8 9

10 Review of Mlti-rate Sstems Interconnection of mlti-rate bilding blocks : [k] [k] [k] x [k] x a a identities also old if all decimators are replaced b expanders x [k] [k] [k] [k] x VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p 9 Review of Mlti-rate Sstems Interconnection of mlti-rate bilding blocks : L L if and onl if L and are coprime!!!!! Example : [k],,,,5,6,7,8,9, L, a -fold p:,,,,,,,, a -fold down:,,7, b -fold down:,,,,7,, b -fold p:,,,,7, Example : [k],,,,5,6,7,8,9, L, a -fold p:,,,,,,,, a -fold down:,5,9, b -fold down:,,5,7,9, b -fold p:,,5,,9, VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p

11 VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p Review of Mlti-rate Sstems `oble identities I : onl for rational fnctions Example : [],[],,,, [k] [k] [k] [k] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] -fold downsampling -fold downsampling VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p Review of Mlti-rate Sstems `oble identities II : onl for rational fnctions Example : [],[],,,, [k] [k] [k] [k] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [5] [] [] [] [] [] -fold psampling -fold psampling

12 Review of Mlti-rate Sstems `oble identities III : Relevance of `noble identities : - efficient mlti-rate filter implementations trog `polpase representations - example : interpolation filters decimation filters filter strctres for sampling rate conversions - example : DFT-modlated filter banks VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p DFT/FFT Filter Bank First attempt to design a perfect reconstrction filter bank: -Starting point is tis : [k] [],,,,[],,,, [k-] as [k][k-d], tis can alread be viewed as a perfect reconstrction filter bank [-],[],,,[],[],,, [-],[-],[],,[],[],[],, [-],[-],[-],[],[],[],[],[], VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p

13 DFT/FFT Filter Bank - An eqivalent representation is tis [k] [k-] All analsis/sntesis filters are seen to be pre delas, ence are not freqenc selective ie far from ideal case wit ideal bandpass filters VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p 5 DFT/FFT Filter Bank - now insert DFT-matrix discrete Forier transform and its inverse I-DFT [k] F F [k-] as F F I tis clearl does not cange te inpt-otpt relation ence perfect reconstrction propert preserved VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p 6

14 VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p 7 DFT/FFT Filter Bank - and reverse order of decimators/expanders and DFTmatrices not done in an efficient implementation! : analsis filter bank sntesis filter bank Tis is te `DFT filter bank It is a first example of a maximall decimated perfect reconstrction filter bank! [k-] [k] F F VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p 8 DFT/FFT Filter Bank Analsis filters of te DFT filter bank: Tis is seen known to represent a collection of filters o,,, eac of wic is a freqenc sifted version of o : ie te i are obtained b niforml sifting te `prototpe o over te freqenc axis tis is referred to as a `niform DFT filter bank F [k] j F e / : : : : : : π / i j j i e e π ω ω

15 DFT/FFT Filter Bank o DFT analsis filter bank: Te prototpe filter o is a not-so-great lowpass filter wit first sidelobe onl db below te main lobe o and i s are ts far from ideal lowpass/ bandpass filters ence maximal decimation introdces significant ALIASIG in te decimated sbband signals Still, we know tis is a PERFECT RECOSTRUCTIO filter bank, wic means te sntesis filters can apparentl restore te aliasing distortion Tis is remarkable! VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p 9 Conclsions Seen te general sbband processing set-p applications Filter bank sstem is mlti-rate strctre, wit decimators and expanders, ence ALIASIG is a major concern Seen a first simple & not-so-great example of a PERFECT RECOSTRUCTIO filter bank niform DFT filter bank Part - : Oter better PR strctres, maximall decimated filter banks, oversampled filter banks Reference: `Mltirate Sstems & Filter Banks, PP Vaidanatan Prentice all 99 VLSI Signal Processing Lectre - K Filter Banks - Preliminaries p 5