B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

124 ANALYSIS AND TRANSMISSION OF SIGNALS
I
Because the signal has a jump discontinuity at t = 0, the first sample (at t = 0) is 0.5,
the averages of the values on the two sides of the discontinuity. The MATLAB program,
which implements the DFT by using the FFT algorithm is as follows:
Ts=l/64; T0=4; NO=TO/Ts;
t=O:Ts:Ts* (NO-l) ;t=t';
g=Ts* exp(-2*t) ;
g(l)=Ts* 0.5;
G=fft (g) ;
$[Gp ,Gm) $=cart2pol ( $real (G) ,imag (G) $) ;
k=O :N0-1; k=k' ;
w=2 *pi*k/TO;
subplot (211) ,stem ( w(l:32) ,Gm(l:32) );
subplot (212 ) ,stem ( w(l:32) ,Gp(l:32) )
Figure 3.40
Discrete Fourier
transform of an
exponential
signal e- 21 u(t).
Notice that the
horizontal axis in
this case is w (in
radians per
second).
Because G q
is No-periodic, G q
= G (q+ 256)
so that G2s6 = Go. Hence, we need
to plot G q
over the range q = 0 to 255 (not 256). Moreover, because of this periodicity,
G -q
= G (-q+256) , andthe Gq over the range of q = - 127 to -1 are identical to the G q
over
the range of q = 129 to 255. Thus, G - 121 = G129, G - 126 = G130, ... , G - 1 = G2ss­
In addition, because of the property of conjugate symmetry of the Fourier transform,
G_ q
= c;, it follows that G129 = Gi 27
, G130 = Giw ... , G255 = Gi- Thus, the plots
beyond q = No/2 (128 in this case) are not necessary for real signals (because they are
conjugates of G q
for q = 0 to 128).
The plot of the Fourier spectra in Fig. 3.40 shows the samples of magnitude and phase
of G(f) at the intervals of I/To = 1/4 Hz or wo = 1.5708 rad/s. In Fig. 3.40, we have
shown only the first 28 points (rather than all 128 points) to avoid too much crowding of
the data.
0.5 t
IG(f)I
0.4
0.3
0.2
0.1
Exact
FFT values
0
10
20 30
I
I
40 w--
I
-0.5
-1 - \
"
-1t/2 ---·-··
Exact
I
/-.
FFTvalues
...... .__.
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