Signals and Systems - Electrical Engineering

440 CHAPTER 7: Sampling Theory

FIGURE 7.10

Sampling using a sample-and-hold system(δ=Ts).

Δ t

1

×

t t

0

x(t)

δTs(t)

xs(t)

h(t)

ys(t)

h(t)

x(t)

(^0) Ts
xs(t)
Ts
ys(t)
The system shown in Figure 7.10 generates the desired signal. Basically, we are modulating the ideal
sampling signalδTs(t)with the analog inputx(t), giving an ideally sampled signalxs(t). This signal
is then passed through azero-order hold filter, an LTI system having as impulse responseh(t)a pulse
of the desired width 1 ≤Ts. The output of the sample-and-hold system is a weighted sequence of
shifted versions of the impulse response. In fact, the output of the ideal sampler isxs(t)=x(t)δTs(t),
and using the linearity and time invariance of the zero-order hold system its output is
ys(t)=(xs∗h)(t) (7.20)
with a Fourier transform of
Ys()=Xs()H(j)

=

[

1

Ts

∑

k

X(−ks)

]

H(j) (7.21)

where the term in the brackets is the spectrum of the ideally sampled signal and

H(j)=

e−^1 s/^2

s

(e^1 s/^2 −e−^1 s/^2 )|s=j

=

sin(1/ 2 )

/ 2

e−j1/^2 (7.22)

is the frequency response of the LTI system.

Remarks

n Equation (7.20) can be written as

ys(t)=

∑

n

x(nTs)h(t−nTs)

That is, ys(t)is a train of pulses h(t)=u(t)−u(t−1)shifted and weighted by the sample values x(nTs),

a more realistic representation of the sampled signal.