Digital Signal Processing Reference
In-Depth Information
T
(
t
)
(
t
nT
)
n
f
(
t
)
f
s
(
t
)
Figure 5.21
Impulse sampling.
The
ideal impulse sampling
operation is modeled by Figure 5.21 and is seen to be a
modulation process (modulation is discussed in Chapter 6) in which the carrier sig-
nal
d
T
(t)
is defined as the train of impulse functions:
q
d
T
(t) =
a
d(t - nT
S
).
(5.39)
n=-
q
An illustration of appears as Figure 5.19(a) if in the figure is replaced by
(We justify this model later.) The output of the modulator, denoted by
d
T
(t)
T
0
T
s
.
f
s
(t)
, is given by
q
q
f
s
(t) = f(t)d
T
(t) = f(t)
a
d(t - nT
S
) =
f(nT
S
)d(t - nT
S
).
(5.40)
a
n=-
q
n=-
q
Ideal sampling is illustrated in Figure 5.22. Figures 5.22(a) and (b) show a con-
tinuous-time signal and the ideal sampling function respectively. The
sampled signal is illustrated in Figure 5.22(c), where the heights of the impuls-
es are varied to imply graphically their variation in weight. Actually, all impulses
have unbounded height, but each impulse in the sampled signal has its weight de-
termined by the value of at the instant that the impulse occurs.
We make two observations relative to First, because impulse functions
appear in this signal, it is not the exact model of a physical signal. The second ob-
servation is that the mathematical sampling operation does correctly result in the
desired sampled sequence as weights of a train of impulses. It is shown in
Chapter 6 that the modeling of the sampling operation by impulse functions is
mathematically valid, even though cannot appear in a physical system.
To investigate the characteristics of the sampling operation in Figure 5.21 and
(5.40), we begin by taking the Fourier transform of
f(t)
d
T
(t),
f
s
(t)
f(t)
f
s
(t).
f(nT
s
)
f
s
(t)
f
s
(t).
From Example 5.14,
q
q
d(t - nT
S
)
Î
f
"
v
S
a
d
T
(t) =
a
d(v - kv
S
),
(5.41)
n=-
q
k=-
q
where is the sampling frequency in radians/second. The sampling fre-
quency in hertz is given by therefore,
Recall that the Fourier transform of an impulse function in time is not an im-
pulse function in frequency; however, the Fourier transform of a train of periodic
v
s
= 2p/T
s
f
s
= 1/T
s
;
v
s
= 2pf
s
.
