Digital Signal Processing Reference
In-Depth Information
P1p=dt*abs(P1);
P2p=dt*abs(P2);
%
% Plot the magnitude frequency spectra of the two pulses.
subplot(2,2,3), plot(w, P1p), xlabel('Frequency (rad/s)')
subplot(2,2,4), plot(w, P2p, 'g), xlabel('Frequency (rad/s)')
■
Example 5.3 and the waveforms shown in Figure 5.6 give insight into an im-
portant physical relationship between the time domain and the frequency domain,
which is implied by the time-scaling property. Notice how the frequency spectrum
of the signal spreads as the time-domain waveform is compressed. This implies that
a pulse with a short time duration contains frequency components with significant
magnitudes over a wider range of frequencies than a pulse with longer time dura-
tion does. In the study of communication systems, this reciprocal relationship be-
tween time-domain waveforms and their frequency spectra is an important
consideration. This is known as the
duration-bandwidth
relationship and is dis-
cussed in greater detail in Chapter 6.
The property of time shifting previously appeared in the Fourier transform of the
impulse function
(5.7) derived in Section 5.1, although it was not recognized at that
time. This property is stated mathematically as
f(t - t
0
)
Î
f
"
F(v)e
-jvt
0
,
(5.13)
where the symbol
t
0
represents the amount of shift in time.
The time-shifting property of the Fourier transform
EXAMPLE 5.4
We now find the Fourier transform of the impulse function, which occurs at time zero.
From (5.8),
q
d(t)e
-jvt
dt = e
-jvt
f
5
d(t)
6
=
L
t = 0
= 1.
-
q
If the impulse function is shifted in time so that it occurs at time
t
0
instead of at
t = 0,
we see
from the time-shifting property (5.13) that
= (1)e
-jvt
0
= e
-jvt
0
,
f
5
d(t - t
0
)
6
which is recognized as the same result obtained in (5.7).
■
