EE123 Digital Signal Processing

Transcription

1 Aoucemets HW solutios posted -- self gradig due HW2 due Friday EE2 Digital Sigal Processig ham radio licesig lectures Tue 6:-8pm Cory 2 Lecture 6 based o slides by J.M. Kah SDR give after GSI Wedesday Fiish readig Ch. 8, start Ch. 9 Cool thigs DSP Last Time Cosmic Microwave Backgroud radiatio Discrete Fourier Trasform Similar to DFS Samplig of the DTFT (subtitles.more later) Properties of the DFT Today Liear covolutio with DFT Fast Fourier Trasform Circular shift Iherited from DFS (EE2/2) so o eed to be proved ~ x! " ~ x! # m" Liearity Circular Time Shift m))n ] $ X[k]e j(2 /N )km m N # x!$$ # m %%N " x! " x [] + 2 x2 [] $ X [k] + 2 X2 [k] x[(( N # N # m N # = X[k]WNkm

2 Circular frequecy shift x[]e j(2 /N )l = x[]wn $ X[((k l)) N ] Complex Cojugatio x [] $ X [(( k)) N ] Cojugate Symmetry for Real Sigals x[] =x [] $ X[k] =X [(( k)) N ] l Show. Parseval s Idetity = x[] 2 = N Proof (i matrix otatio) k= X[k] 2 x x = N W N X N W N X = N 2 X W N WN X = {z } N X X N I Circular Covolutio Sum Circular Covolutio: x [] N x 2 [] = m= for two sigals of legth N x [m]x 2 [(( m)) N ] Compute Circular Covolutio Sum x [] x 2 [] Note: Circular covolutio is commutative 2 y[] =x [] 7 x 2 [] =? x 2 [] N x [] =x [] N x 2 [] Compute Circular Covolutio Sum Compute Circular Covolutio Sum x [] x [] 2 4 x 2 [] x 2 [] Circular flip multiply ad add Here: y[] y[] =x [] 7 x 2 [] =? Equivalet periodic covolutio over a period y[] =x [] 7 x 2 [] =?

3 Result y[] =x [] 7 x 2 [] =? Circular Covolutio: Let x[], x2[] be legth N 4 x [] N x 2 [] $ X [k] X 2 [k] 2 Very useful!!! ( for liear covolutios with DFT) Multiplicatio: Let x[], x2[] be legth N x [] x 2 [] $ N X [k] N X 2 [k] Liear Covolutio Next. Usig DFT, circular covolutio is easy But, liear covolutio is useful, ot circular So, show how to perform liear covolutio with circular covolutio Used DFT to do liear covolutio Liear Covolutio We start with two o-periodic sequeces: x[] apple apple L h[] apple apple P for example x[] is a sigal ad h[] a impulse respose of a filter We wat to compute the liear covolutio: y[] =x[] h[] = LX m= Requires L P multiplicatios x[m]h[ m] y[] is ozero for L+P-2 with legth M=L+P- Liear Covolutio via Circular Covolutio Zero-pad x[] by P- zeros x[] apple apple L x zp [] = L apple apple L + P 2 Zero-pad h[] by L- zeros h[] apple apple P h zp [] = P apple apple L + P 2 Now, both sequeces are of legth M=L+P- Liear Covolutio via Circular Covolutio Now, both sequeces are of legth M=L+P- We ca ow compute the liear covolutio usig a circular oe with legth M = L+P- Liear covolutio via circular ( x zp [] M h zp [] apple apple M y[] =x[] y[] = otherwise

4 Example Example x [] L= x [] 2 4 x 2 [] P=4 2 4 x 2 [] M = L + P - = 8 M = L + P - = 8 Example Liear Covolutio usig DFT x [] I practice we ca implemet a circulat covolutio usig the DFT property: 2 4 x 2 [] 2 Circular flip 6 7 M = L + P - = 8 y[] =x [] 8 x 2 [] =x [] x 2 [] x[] h[] = x zp [] M h zp [] = DFT {DFT {x zp []} DFT {h zp []}} for M-, M=L+P- Advatage: DFT ca be computed with Nlog2N complexity (FFT algorithm later!) Drawback: Must wait for all the samples -- huge delay -- icompatible with real-time Block Covolutio Problem: A iput sigal x[], has very log legth (could be cosidered ifiite) A impulse respose h[] has legth P We wat to take advatage of DFT/FFT ad compute covolutios i blocks that are shorter tha the sigal Approach: Break the sigal ito small blocks Compute covolutios Combie the results Block Covolutio Example: h[] Impulse respose, Legth P=6 2 x[] Iput Sigal, Legth P= y[] Output Sigal, Legth P=8 2 2

5 Overlap-Add Method We 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 x r [] The output sigal is the sum of the output segmets x r [] h[]: r= y[] =x[] h[] = X x r [] h[] () r= Each of the output segmets x r [] h[] is of legth N = L + P. Miki Lustig UCB. Based o Course Notes by J.M Kah SP Fall 2 22, EE2 Digital Sigal Processig Overlap-Add Method We ca compute each output segmet x r [] h[] withliear covolutio. DFT-based circular covolutio is usually more e ciet: Zero-pad iput segmet x r [] to obtai x r,zp [], of legth N. Zero-pad the impulse respose h[] to obtai h zp [], of legth N (this eeds to be doe oly oce). Compute each output segmet usig: x r [] h[] =DFT {DFT {x r,zp []} DFT {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. Miki Lustig UCB. Based o Course Notes by J.M Kah SP Fall 2 22, EE2 Digital Sigal Processig Example of overlap ad add: Overlap-Save Method x[]. x[] y [] x[] 2 x2[] y 2 [] y [] x[] = x[]+x[]+x2[] y[] = y[]+y[]+y2[]. x[] x2[] Basic Idea We 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 DFT. 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. 2 2 Miki Lustig UCB. Based o Course Notes by J.M Kah SP Fall 2 22, EE2 Digital Sigal Processig Recall: x [] Overlap-Save Method Example of overlap ad save: Overlap-Save, Iput Segmets, Legth L = 6. Overlap-Save, Output Segmets, Usable Legth L - P + Usable (y []). Uusable x [] y p [] 2 4 x 2 [] 6 x [] y p [] -. 2 Usable (y []) Uusable Usable (y 2 []) Uusable. 4 2 Valid liear covolutio! x 2 [) -. 2 y 2p [] -. 2 Overlap-Save, Cocateatio of Usable Output Segmets. y[] M. Lustig, Miki EECS Lustig UC Berkeley UCB. Based o Course Notes by J.M Kah Fall SP 2 22, EE2 Digital Sigal Processig