Signal Processing and Linear Systems-B.P.Lathi copy

Signal Processing and Linear Systems-B.P.Lathi copy

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: e error in F2 is j ust a bout 1.3%. However, t he aliasing error increases rapidly with r . For instance, t he e rror in F6 is a bout 12%, a nd t he error in FlO is 33%. The error in H 4 is a whopping 72%. T he percent error increases rapidly near the folding frequency ( r = 16) because J (t) has a j ump discontinuity, which makes F(w) decays slowly as l /w. Hence, near t he folding frequency, t he i nverted tail (due to aliasing) is very nearly equal to F(w) itself. Moreover, the final values a re t he difference between t he e xact and the folded values (which are very close to t he exact values). Hence, t he percent error near the folding frequency ( r = 16 in this case) is very h lgh, although t he absolute error is very small. Clearly, for signals with jump discontinuities, t he aliasing error near t he folding frequency will always b e high (in percentage terms), regardless of t he choice of No. To ensure a negligible aliasing error a t any value r , we m ust make sure t hat No » r . T his observation is valid for all signals with jump discontinuities . • (5.28) T he p roof is trivial. 2. C onjugate S ymmetry £ff 0 ) C omputer E xample C 5.2 Use O FT (implemented by the F FT algorithm) to compute the Fourier transform of 8 r ect (t). P lot t he resulting Fourier spectra. T he MATLAB program, which implements this O FT equation using t he F FT algorithm, is given next. First we write a MATLAB program to generate 32 samples o f /k, a nd then we c ompute t he O FT. (5.29) T his follows from t he c onjugate s ymmetry p roperty o f t he F ourier t ransform ( F; = F -r) a nd t he p eriodic p roperty o f D FT ( F-r = F No-r). B ecause o f t his p roperty, we n eed c ompute o nly h alf t he O FTs for r eal / k. T he o ther h alf a re t he c onjugates. 3. Time Shifting (Circular Shifting) (5.30) Proof: U sing Eq. (5.18b), we find t he inverse O FT o f F re-jrflon as 1 - N " o N o-1 N o-l ~ F e -jrflonejrflok r = .N " !.... ~ 0 r=O r =O F e j rflo(k-n) r = I k -n 4. Frequency Shifting (5.31) P roof: T his p roof is i dentical t o t hat o f t he t ime s hifting p roperty e xcept t hat we s tart w ith E q. (5.18a). 5. Circular (also called periodic) Convolution (5.32a) a nd % ( c52.m) N O=32;k=O:NO.l; f =[ones(1,4) 0 .5 z eros(1,23) 0 .5 o nes(1,3)]; F r=fft(f); s ubplot(2,1,1),stem(k,f) x label('k');ylabel('fk'); s ubplot(2,l,2),stem(k,Fr) x label('r');ylabel('Fr'); 0 ) 5.2-1 Some Properties o f O FT T he d iscrete F ourier t ransform is basically t he F ourier t ransform o f a s ampled s ignal r epeated p eriodically. Hence, t he p roperties d erived earlier for t he F ourier t ransform a pply t o t he O FT a s well. (5.32b) For two No-periodic sequences / k a nd g k, t he c ircular convolution is defined b y N o-l / k ® 9k = L No-l I ngk-n = n=O L g n/k-n n=O To prove (5.32a), we find t he D FT o f c ircular convolution i k ® g k a s 1:11:1 ( k=O n=O f ngk-n) e -jrflok = 1:1 (1:1 In n=O N o-1 L n=O g k_ne-jrflOk) k=O I n(Gre-jrflon) = F rG r (5.33) 5 S ampling...
View Full Document

This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

Ask a homework question - tutors are online