Digital Signal Processing Reference
In-Depth Information
and, if the ROCs of
X
(
z
)
,
Y
(
z
)
,and
H
(
z
)
include the unit circle, expression
(2.24) reduces to
Y
exp
)
=
H
exp
)
X
exp
)
(
j
2π
f
(
j
2π
f
(
j
2π
f
(2.25)
2.2 Digital Signal Processing
Discrete-time signals can be characterized and stored very easily. This relies
on a very relevant feature of discrete-time signals: given a finite time interval,
there is a finite set of values that fully characterize a sequence, whereas the
same does not hold for a continuous-time signal. This essential difference is
a reflex of the profound structural divergences between the domains of these
classes of information-bearing functions.
The world of digital computers excels in storage capacity and potential
of information processing, and is essentially a “discrete-time world.” There-
fore, it is not surprising that digital signal processing is a widespread tool
nowadays. Nevertheless, it is also clear that many of our physical models are
inherently based on continuous-time signals. The bridge between this “real
world” and the existing digital tools is established by the sampling theorem.
2.2.1 The Sampling Theorem
The idea of sampling is very intuitive, as it is closely related to the notion
of measure. When we measure our height or weight, we are, in a certain
sense, sampling the continuous-time signal that expresses the time-evolution
of these variables. In the context of communications, the sampling process
produces, from a continuous-time signal, a representative discrete-time sig-
nal that lends itself to proper digital processing and storage. Conditions for
equivalent representation and perfect reconstruction of the original signal
from its samples were achieved through the sampling theorem, proposed by
Harry Nyquist (1926), D. Gabor (1946), and Claude Shannon (1949), and are
related to two requirements:
1. The continuous-time signal must be band-limited, i.e., its Fourier
spectrum must be null for
f
f
M
.
2. The sampling rate, i.e., the inverse of the time-spacing
T
S
of the
samples must be higher than or equal to 2
f
M
.
>
Given these conditions, we are ready to enunciate the sampling
theorem [219]
