EE123 Digital Signal Processing

Transcription

1 Last Time EE Digital Sigal Processig Lecture 7 Block Covolutio, Overlap ad Add, FFT Discrete Fourier Trasform Properties of the Liear covolutio through circular Today Liear covolutio with Overlap ad add Overlap ad save Fast Fourier Trasform (start) based o slides by JM Kah M Lustig, EECS UC Berkeley M Lustig, EECS UC Berkeley Block Covolutio Problem: A iput sigal x[], has very log legth (could be cosidered ifiite) A impulse respose h[] has legth P e wat to take advatage of /FFT ad compute covolutios i blocks that are shorter tha the sigal Approach: Break the sigal ito small blocks Compute covolutios Combie the results M Lustig, EECS UC Berkeley Block Covolutio Example: h[] Impulse respose, Legth P x[] Iput Sigal, Legth P y[] Output Sigal, Legth P8 M Lustig, EECS UC Berkeley Overlap-Add Method e decompose the iput sigal x[] ito o-overlappig segmets x r [] of legth L: ( x[] rl apple apple (r + )L x r [] otherwise The iput sigal is the sum of these iput segmets: x[] x r [] The output sigal is the sum of the output segmets x r [] h[]: r y[] x[] h[] x r [] h[] () r Each of the output segmets x r [] h[] is of legth M L + P SP Fall, EE Digital Sigal Processig Overlap-Add Method e ca compute each output segmet x r [] h[] withliear covolutio -based circular covolutio is usually more e ciet: Zero-pad iput segmet x r [] to obtai x r,zp [], of legth M Zero-pad the impulse respose h[] to obtai h zp [], of legth (this eeds to be doe oly oce) Compute each output segmet usig: x r [] h[] { {x r,zp []} {h zp []}} Sice output segmet x r [] h[] startso set from its eighbor x r [] h[] byl, eighborig output segmets overlap at P poits Fially, we just add up the output segmets usig () to obtai the output SP Fall, EE Digital Sigal Processig

2 Example of overlap ad add: Overlap-Add, Iput Segmets, Legth L x[] Overlap-Add, Output Segmets, Legth L+P x[] Overlap-Save Method y [] - x[] x[] y [] y [] - - x[] x[]+x[]+x[] y[] y[]+y[]+y[] x[] x[] Basic Idea e split the iput sigal x[] ito overlappig segmets x r [] of legth L + P Perform a circular covolutio of each iput segmet x r [] with the impulse respose h[], which is of legth P usig the Idetify the L-sample portio of each circular covolutio that correspods to a liear covolutio, ad save it This is illustrated below where we have a block of L samples circularly covolved with a P sample filter SP Fall, EE Digital Sigal Processig Recall: x [] Overlap-Save Method Example of overlap ad save: Overlap-Save, Iput Segmets, Legth L Overlap-Save, Output Segmets, Usable Legth L - P + Usable (y []) Uusable x [] y p [] x [] x [] - y p [] - Usable (y []) Uusable - - Usable (y []) Uusable Valid liear covolutio! x [) - y p [] - Overlap-Save, Cocateatio of Usable Output Segmets y[] - M Lustig, Miki EECS Lustig UC Berkeley UCB Based o Course otes by JM Kah Fall SP, EE Digital Sigal Processig vs DTFT (revisit) Back to movig average example: (e j! ) e j! e j! si(!) si(! ) ad Samplig the DTFT (e j! )e j! si (!/) si (!/) x[] (e jω ) ω recostructed x[] (e jω ) w ω M Lustig, EECS UC Berkeley Fall, EE Digital Sigal Processig

3 Circular Covolutio as Matrix Operatio Circular covolutio: h[] x[] H c x h[] h[ ] h[] h[] h[] h[] h[ ] h[ ] h[] x[] x[] 7 7 x[] [] H c is a circulat matrix The colums of the matrix are Eige vectors of circulat matrices Eige vectors are coe ciets How ca you show? Proof i H Circular Covolutio as Matrix Operatio Diagoalize: H c H[] H[] H[ ] Right-multiply by H[] H c H[] 7 H[ ] Multiply both sides by x H[] H c x H[] 7 x H[ ] 7 Fast Fourier Trasform Algorithms e are iterested i e ad iverse : where [k] x[] ciet computig methods for the x[] k, k,, [k] k,,, k e j( ) Recall that we ca use the to compute the iverse : Hece, we ca just focus o e { [k]} ( { [k]}) ciet computatio of the Straightforward computatio of a -poit (or iverse ) requires complex multiplicatios Fast Fourier trasform algorithms eable computatio of a -poit (or iverse ) with the order of just log complex multiplicatios This ca represet a huge reductio i computatioal load, especially for large log log 8,8 89 8,,8,7, 8,9 7,8,8,9 7 * Mp image size Most FFT algorithms exploit the followig properties of k : Cojugate Symmetry Periodicity i ad k: Power: k( ) k ( k ) k k(+) (k+) /

4 Decimatio-i-Time Fast Fourier Trasform Most FFT algorithms decompose the computatio of a ito successively smaller computatios Decimatio-i-time algorithms decompose x[] ito successively smaller subsequeces Decimatio-i-frequecy algorithms decompose [k] ito successively smaller subsequeces e mostly discuss decimatio-i-time algorithms here Assume legth of x[] is power of ( ) If smaller zero-pad to closest power e start with the [k] x[] k, k,, Separate the sum ito eve ad odd terms: [k] x[] k + x[] k eve odd These are two s, each with half of the samples Decimatio-i-Time Fast Fourier Trasform Decimatio-i-Time Fast Fourier Trasform Let r ( eve) ad r +( odd): Hece: [k] (/) r (/) r x[r] rk x[r] rk (/) + r + k x[r + ] (r+)k (/) r x[r + ] rk [k] (/) r (/) x[r]/ rk + k r G[k]+ k H[k], k,, where we have defied: x[r + ] rk / ote that: rk e j( )(rk) e j rk / rk / Remember this trick, it will tur up ofte G[k] H[k] (/) r (/) r x[r] rk / ) of eve idx x[r + ] rk / ) of odd idx Decimatio-i-Time Fast Fourier Trasform A 8 sample ca the be diagrammed as Eve Samples Odd Samples x[] x[] x[] x[] x[] / - Poit / - Poit G[] G[] G[] G[] H[] H[] H[] H[] 7 [] [] [] [] [] [] [] [7] Decimatio-i-Time Fast Fourier Trasform Both G[k] ad H[k] are periodic, with period / For example so G[k + /] (/) r (/) r (/) r G[k] x[r] r(k+/) / x[r] rk / r(/) / x[r] rk / G[k +(/)] G[k] H[k +(/)] H[k]

5 Decimatio-i-Time Fast Fourier Trasform Decimatio-i-Time Fast Fourier Trasform The periodicity of G[k] adh[k] allowsustofurthersimplify For the first / poits we calculate G[k] ad k H[k], ad the compute the sum [k] G[k]+ k H[k] 8{k :applek < } How does periodicity help for apple k <? [k] G[k]+ k H[k] 8{k :apple k < } for apple k < : k+(/)? [k +(/)]? Decimatio-i-Time Fast Fourier Trasform [k +(/)] G[k] e previously calculated G[k] ad k H[k] k H[k] ow we oly have to compute their di erece to obtai the secod half of the spectrum o additioal multiplies are required Decimatio-i-Time Fast Fourier Trasform The -poit has bee reduced two /-poit s, plus / complex multiplicatios The 8 sample is the: Eve Samples Odd Samples x[] x[] x[] x[] x[] / - Poit / - Poit G[k] H[k] k [] [] [] [] [] [] [] [7] Decimatio-i-Time Fast Fourier Trasform Decimatio-i-Time Fast Fourier Trasform ote that the iputs have bee reordered so that the outputs come out i their proper sequece e ca defie a butterfly operatio, eg, the computatio of [] ad [] from G[] ad H[]: G[] H[] This is a importat operatio i DSP [] G[] + H[] [] G[] - H[] Still O( ) operatios hat shall we do? Eve Samples Odd Samples x[] x[] x[] x[] x[] / - Poit / - Poit G[k] H[k] k [] [] [] [] [] [] [] [7]

6 Decimatio-i-Time Fast Fourier Trasform Decimatio-i-Time Fast Fourier Trasform e ca use the same approach for each of the / poit s For the 8case,the/ s look like At this poit for the 8 sample, we ca replace the / sample s with a sigle butterfly The coe ciet is x[] x[] / - Poit / - Poit / / *ote that the iputs have bee reordered agai G[] G[] G[] G[] / 8/ e j The diagram of this stage is the x[] x[] + x[] - Decimatio-i-Time Fast Fourier Trasform Decimatio-i-Time Fast Fourier Trasform Combiig all these stages, the diagram for the 8 sample is: x[] [] I geeral, there are log stages of decimatio-i-time / x[] / x[] x[] / x[] / This the decimatio-i-time FFT algorithm [] [] [] [] [] [] [7] Each stage requires / complex multiplicatios, some of which are trivial The total umber of complex multiplicatios is (/) log The order of the iput to the decimatio-i-time FFT algorithm must be permuted First stage: split ito odd ad eve Zero low-order bit first ext stage repeats with ext zero-lower bit first et e ect is reversig the bit order of idexes Decimatio-i-Time Fast Fourier Trasform Decimatio-i-Frequecy Fast Fourier Trasform This is illustrated i the followig table for 8 Decimal Biary Bit-Reversed Biary Bit-Reversed Decimal 7 7 The is [k] x[] k If we oly look at the eve samples of [k], we ca write k r, [r] x[] (r) e split this ito two sums, oe over the first / samples, ad the secod of the last / samples (/) [r] x[] r (/) + x[ + /] r(+/)

7 Decimatio-i-Frequecy Fast Fourier Trasform Decimatio-i-Frequecy Fast Fourier Trasform But r(+/) r r e ca the write r / [r] (/) (/) (/) x[] r x[] r (/) + (/) + (x[]+x[ + /]) r / x[ + /] r(+/) x[ + /] r This is the /-legth of first ad secod half of x[] summed [r] {(x[]+x[ + /])} [r + ] {(x[] x[ + /]) } (By a similar argumet that gives the odd samples) Cotiue the same approach is applied for the / s, ad the / s util we reach simple butterflies Decimatio-i-Frequecy Fast Fourier Trasform The diagram for ad 8-poit decimatio-i-frequecy is as follows x[] x[] x[] x[] x[] / / / / This is just the decimatio-i-time algorithm reversed! The iputs are i ormal order, ad the outputs are bit reversed [] [] [] [] [] [] [] [7] o-power-of- FFT s A similar argumet applies for ay legth, where the legth is a composite umber For example, if, a decimatio-i-time FFT could compute three -poit s followed by two -poit s x[] x[] x[] x[] x[] -Poit -Poit -Poit -Poit -Poit [] [] [] [] [] [] o-power-of- FFT s Good compoet s are available for legths up to or so May of these exploit the structure for that specific legth For example, a factor of / e j (/) e j j hy? just swaps the real ad imagiary compoets of a complex umber, ad does t actually require ay multiplies Hece a of legth does t require ay complex multiplies Half of the multiplies of a 8-poit also do t require multiplicatio Composite legth FFT s ca be very e ciet for ay legth that factors ito terms of this order For example 9 factors ito (7)(9)() each of which ca be implemeted e 9 s of legth 7 7 s of legth 9, ad 7 9 s of legth cietly e would perform

8 Historically, the power-of-two FFTs were much faster (better writte ad implemeted) For o-power-of-two legth, it was faster to zero pad to power of two Recetly this has chaged The free FFT package implemets very e ciet algorithms for almost ay filter legth Matlab has used FFT sice versio FFT as Matrix Operatio FFT as Matrix Operatio [] [k] [ ] C A ( ) k k k( ) ( ( ( ) ) )( ) B A x[] x[] x[ ] C A [] [k] [ ] C A ( ) k k k( ) ( ( ( ) ) )( ) B A x[] x[] x[ ] C A is fully populated ) etries is fully populated ) etries FFT is a decompositio of ito a more sparse form: apple I/ D F / I / D / apple / / apple Eve-Odd Perm Matrix I / is a idetity matrix D / is a diagoal with etries,,, / FFT as Matrix Operatio Example: F Beyod log hat if the sigal x[] has a k sparse frequecy A Gilbert et al, ear-optimal sparse Fourier represetatios via samplig H Hassaieh et al, early Optimal Sparse Fourier Trasform Others O(K Log ) istead of O( Log ) From: M Lustig, EECS UC Berkeley