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Unformatted text preview: n t he v alue j ust o btained by using 1% a mplitude criterion.
T he second issue is to determine To. Because t he signal is n ot t imelimited, we have
t o t runcate i t a t To such t hat f (To) « 1. A r easonable choice would be To = 4 because
f (4) = e  8 = 0.000335 « 1. T he r esult is No = T o/T = 254.6, which is n ot a power of 2.
Hence, we choose To = 4, a nd T = 0.015625 = 1 /64, yielding No = 256, which is a power
of 2.
N ote t hat t here is a g reat deal of flexibility in determining T a nd To, d epending on
t he a ccuracy desired a nd t he c omputational capacity available. We could j ust as well have
chosen T = 0.03125, yielding No = 128, although this choice would have given a slightly
higher aliasing error.
Because t he s ignal has a jump d iscontinuity a t t = 0, t he first sample ( at t = 0) is 0.5,
t he averages of t he values o n t he two sides of t he discontinuity. We compute Fr ( the D FT)
from t he s amples of e  2t u(t) according t o Eq. (5.18a). Note t hat Fr is t he r th s ample of
F (w), a nd t hese samples are spaced a t:Fo = l /To = 0.25 Hz. (wo = t r/2 r ad/s.)
Because Fr is Noperiodic, Fr = F(r+256) so t hat F256 = Fo. Hence, we need to
p lot Fr over t he range r = 0 t o 255 ( not 256). Moreover, because of this periodicity,
F  r = F( r+256), a nd t he values of Fr over t he r ange r =  127 t o  1 a re identical to o
LF(oo) 0.5
I 1tI2 10 20 I 30 40 00 I ~
~
t \ ',
Exact F ig. 5 .15 ...., j
FFT values D iscrete Fourier transform of an exponential signal e  2t u(t). o C omputer E xample C 5.1
Use D FT ( implemented by t he F FT, t he fast Fourier transform algorithm) t o c ompute
t he Fourier transform of e  2t u(t).
T =O.015625jTO=4jNO=TO/Tj
t =O:T:T* ( NOI) j t=t ' j
f =T*exp(2*t)j 5 Sampling 346
f (1)=T*O.5 ;
F =fft(f);
[ Fp,Fm}=cart2pol(real(F) , imag(F»;
k =O:NOl; k =k';
w =2*pi*k/ TO;
s ubplot(2l1.) , plott w (l: 1 28) , Fm(l :128»
s ubplot(2l2) , plot ( w(1:128) , Fp(1:128» 5.2 Numerical Computation of the Fourier Transform: T he D FT 347 (a)
t_ o • E xample 5 .7
Use O FT t o c ompute t he F ourier transform of 8 rect ( t).
T his g ate f unction a nd its Fourier transform are illustrated in Fig. 5.16a a nd b. To
determine t he va.lue of t he s ampling interval T , we must first decide o n t he essential
b andwidth B . I n Fig. 5.16b, we see t hat F(w) decays r ather slowly with w. Hence,
t he essential b andwidth B is r ather large. For instance, a t B = 15.5 Hz (97.39 r ad/s),
F{w) =  0.1643, which is a bout 2% of t he p eak a t F(O). Hence, t he essential b andwidth
is well above 16 Hz if we use t he 1% criterion for computing t he essential bandwidth.
However, we s hall d eliberately take B = 4 for two reasons: ( I) t o show t he effect of
aliasing a nd (2) t he use of B > 4 will give enormous number of samples, which cannot
b e conveniently displayed on t he b ook size page without losing sight of t he essentials.
Thus, we s hall intentionally accept approximation in order t o clarify t he c oncepts...
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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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