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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 ( Fr = F Nor). 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 rejrflon as 1
 N " o N o1 N ol
~ F e jrflonejrflok
r = .N "
!.... ~
0 r=O r =O F e j rflo(kn)
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 ) 5.21 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 Noperiodic sequences / k a nd g k, t he c ircular convolution is defined b y
N ol / k ® 9k = L Nol I ngkn = n=O L g n/kn 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 ngkn) e jrflok = 1:1 (1:1
In n=O
N o1 L n=O g k_nejrflOk) k=O
I n(Grejrflon) = 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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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