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Sec. 3–2 Pulse Amplitude Modulation 139
where t … T s = 1f s and f s Ú 2B.
THEOREM.
The spectrum for a flat-top PAM signal is
where
W s (f) = 1 q
H(f)
T a W(f - kf s )
s k =-q
H(f) = [h(t)] = t a
sin ptf
ptf
b
(3–10)
(3–11)
This type of PAM signal is said to consist of instantaneous samples, since w(t) is
sampled at t kT s and the sample values w(kT s ) determine the amplitude of the flat-top
rectangular pulses, as demonstrated in Fig. 3–5c. The flat-top PAM signal could be generated
by using a sample-and-hold type of electronic circuit.
Another pulse shape, rather than the rectangular shape, could be used in Eq. (3–8), but
in this case the resulting PAM waveform would not be flat topped. Note that if the h(t) is of
the (sin x)x type with overlapping pulses, then Eq. (3–8) becomes identical to the sampling
theorem of Eq. (2–158), and this sampled signal becomes identical to the original unsampled
analog waveform, w(t).
PROOF. The spectrum for flat-top PAM can be obtained by taking the Fourier transform
of Eq. (3–8). First we rewrite that equation, using a more convenient form involving the
convolution operation:
w s (t) = ak
w(kT s )h(t) * d(t - kT s )
Hence,
= h(t) * ak
w(kT s )d(t - kT s )
The spectrum is
w s (t) = h(t) * cw(t) a
k
d(t - kT s )d
W s (f) = H(f)cW(f) * ak
e -j2pfkT s
d
(3–12)
But the sum of the exponential functions is equivalent to a Fourier series expansion (in the
frequency domain), where the periodic function is an impulse train. That is,
1
T s
a k
d(f - kf s ) = 1 T s
q
a
n =-q
c n e j(2pnT s)f ,
(3–13a)