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

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

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

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