Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 71

Fig. 2.6 Block diagram of a
practical signal processing
system

Digital
Signal

Digital
Signal

r(n) p(n)

x ( t ),
Analog
Input
Signal

y ( t ),
Analog
Output
Signal

LPF &
A / D

D / A

DSP

& LPF

2.3 Time-Domain / Frequency-Domain Representations

Like analog signals and systems, digital signals and systems can be analyzed either
in the time-domain or in the frequency domain. The two domains are equivalent,
provided that suitable transformations are used. The most common time-to-fre-
quency transformations for continuous-time signals are the Fourier transform and
the Laplace transform. The counterparts of these transforms for sampled signals
are the discrete-time Fourier transform (DTFT) and the z-transform. The DTFT is
useful for analyzing the frequency content of signals, while the z-transform
is useful for both analyzing the frequency content of signals and analyzing the
stability of systems. The DTFT and z-transforms will be defined and studied in
more detail in Sects. 2.3.2.2 and 2.3.3 .

2.3.1 Time-Domain Representation of Digital Signals and Systems

This section considers the representation and analysis of digital signals and sys-
tems. Fundamental to time domain analysis of discrete-time signals is discrete-
time convolution, which is defined in what follows.

2.3.1.1 Discrete Linear Convolution

If x(n) and y(n) are two discrete signals, their discrete linear convolution w(n)is
given by:

w ð n Þ¼ x ð n Þ y ð n Þ¼ X

1

x ð k Þ y ð n k Þ :

k ¼1

Note that k is a dummy variable of summation. The above formula is similar to
that of continuous-time linear convolution, except that summation is used instead
of integration.
If both x(n) and y(n) are finite signals of lengths N 1 and N 2 samples, respectively,
the length of w(n) is finite and Is given by L = N 1 ? N 2 - 1. Hence, if x(n) and
y(n) are of equal length N, then the result of the convolution is L = 2N - 1
samples long.