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94
Signals and Spectra Chap. 2
w s (t) = w(t) a
q
= a
q
n=-q
n=-q
(2–171)
where T as illustrated in Fig. 2–18. † s = 1/f s ,
In the figure, the weight (area) of each impulse,
w(nTs), is indicated by the height of the impulse.
The spectrum for the impulse-sampled waveform w s (t) can be evaluated by substituting
the Fourier series of the (periodic) impulse train into Eq. (2–171), giving
q
1
w s (t) = w(t) a e jnv st
(2–172)
n=-qT s
Taking the Fourier transform of both sides of this equation, we get
= 1 q
T
W(f)* a d(f - nf s )
s n=-q
d(t - nT s )
w(nT s ) d(t - nT s )
W s (f) = 1 q
W(f) * c
T a e jnvst d = 1 q
s n=-q T
W(f)* a
s
n=-q
[e jnv st ]
w(t)
W(f)
T s
–f s –f s f s 2 f s
t
–B B
f
(a) Waveform and Its Spectrum
Low-pass filter
T s W s (f)
w s (t)
T s t
f
––– ––– s
2 2
B
(b) Impulse Samples Waveform and Its Spectrum (f s > 2B)
Figure 2–18 Impulse sampling.
f
† For illustrative purposes, we assume that W(f) is real.