Digital Signal Processing Reference
In-Depth Information
F
(
)
A
B
B
(a)
F
s
(
)
A
T
s
• • •
• • •
2
s
s
B
B
2
s
B
s
s
B
s
(b)
Figure 5.23
The frequency spectrum of a sampled-data signal.
Frequency domain characteristics of the sampling operation are now derived from
this result.
We first let the frequency spectrum of the signal be limited such that
for [This is illustrated in Figure 5.23(a).] We assume that the
highest frequency in is less than one-half the sampling frequency; that is,
From (5.43), we see that the effect of sampling
f(t)
ƒ
v
ƒ
7 v
B
.
F(v) = 0
F(v)
v
B
6 v
S
/2.
f(t)
is to replicate the
frequency spectrum of about the frequencies
This result is illustrated in Figure 5.23(b) for the signal of Figure 5.23(a). For this
case, we can, theoretically, recover the signal exactly from its samples by using
an ideal low-pass filter. We call the recovery of a signal from its samples
data recon-
struction
. Data reconstruction is discussed, as an application of the Fourier trans-
form, in Chapter 6.
The frequency is called the
Nyquist frequency
. One of the requirements
for sampling is that the sampling frequency must be chosen such that
where is the highest frequency in the frequency spectrum of the signal to be
sampled. This is stated in Shannon's sampling theorem [1]:
F(v)
kv
S
, k =;1, ; 2, ; 3, Á .
f(t)
v
S
/2
v
S
7 2v
M
,
v
M
A function of time that contains no frequency components greater than
hertz is determined uniquely by the values of
f(t),
f
M
f(t)
at any set of points spaced
