7.2 Uniform Sampling 421
and its Fourier transform is
XPAM()=
∑
k
PkX(−k 0 )
showing that PAM is a modulation of the train of pulsesp(t)by the signalx(t). The spectrum of
xPAM(t)is the spectrum ofx(t)shifted in frequency by{k 0 }, weighted byPk, and superposed.
7.2.2 Ideal Impulse Sampling
Given that the pulse width 1 is much smaller thanTs,p(t)can be replaced by a periodic sequence of
impulses of periodTs(see Figure 7.1) orδTs(t). This simplifies considerably the analysis and makes
the results easier to grasp. Later in the chapter we consider the effects of having pulses instead of
impulses, a more realistic assumption.
Thesampling functionδTs(t), or a periodic sequence of impulses of periodTs, is
δTs(t)=
∑
n
δ(t−nTs) (7.3)
whereδ(t−nTs)is an approximation of the normalized pulse [u(t−nTs)−u(t−nTs−1)]/1when
1 <<Ts. The sampled signal is then given by
xs(t)=x(t)δTs(t)
=
∑
n
x(nTs)δ(t−nTs) (7.4)
as illustrated in Figure 7.1.
There are two equivalent ways to view the sampled signalxs(t)in the frequency domain:
n Modulation: SinceδTs(t)is periodic, of fundamental frequencys= 2 π/Ts, its Fourier series is
δTs(t)=
∑∞
k=−∞
Dkejkst
FIGURE 7.1
Ideal impulse sampling.
t
t
t
×
0
0
0
x(t)
xs(t)
Ts 2 Ts
−Ts Ts 2 Ts
··· ···
δTs(t)
