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(
58
Signals and Spectra Chap. 2
Time Domain
(
t
Frequency Domain
1.0T
T Sa(∏Tf)
1.0
T
– –– 2
T ––2
3
– –– T
2
– –– T
1
– –– T
1 ––T
2 ––T
(a) Rectangular Pulse and Its Spectrum
2WSa(2∏Wt)
–– T
T
0.5T
t
f
2W
(
–––
f
2W
W
1.0
3
– ––––
2W
1
– –––
W
1
– ––––
2W
1
–––
2W
1
t
–W
W
f
(b) Sa(x) Pulse and Its Spectrum
(
–– t
T
(
1.0T
0.5T
T Sa 2 (∏Tf)
– T
–– W
3 ––T
1.0
t
–
2
– –– T
1
– –– T
1 ––T
2 ––T
f
(c) Triangular Pulse and Its Spectrum
Figure 2–6 Spectra of rectangular, ( sin x)/x, and triangular pulses.
where W is the absolute bandwidth in hertz. This Fourier transform pair is also shown in Fig. 2–6b.
The spectra shown in Fig. 2–6 are real because the time domain pulses are real and even. If
the pulses are offset in time to destroy the even symmetry, the spectra will be complex. For
example, let
v(t) = e 1, 0 6 t 6 T t = T/2
f =ßa b
0, t elsewhere T
Then, using the time delay theorem and Eq. (2–55), we get the spectrum
V(f) = T e -jpfT Sa (pTf)
(2–57)