Digital Signal Processing Reference
In-Depth Information
CHAPTER
2
Introduction to systems
In Chapter 1, we introduced the mathematical and graphical notations for repre-
senting signals, which enabled us to illustrate the effect of linear time operations
on transforming the signals. A second important component of signal process-
ing is a system that usually abstracts a physical process. Broadly speaking, a
system is characterized by its ability to accept a set of input signals
x
i
, for
i
∈
{
1, 2, . . . ,
m
}
, and to produce a set of output signals
y
j
, for
j
∈{
1, 2, . . . ,
n
}
,in
response to the input signals. In other words, a system establishes a relationship
between a set of inputs and the corresponding set of outputs.
Most physical processes are modeled by multiple-input and multiple-output
(MIMO) systems of the form illustrated in Fig. 2.1(a), where the
x
i
(
t
)'s repre-
sent the CT inputs while the
y
j
(
t
)'s represent the CT outputs. Such systems,
which operate on CT input signals transforming them to CT output signals,
are referred to as CT systems. Using the principle of superimposition, a linear
MIMO CT system is often approximated by a combination of several single-
input CT systems. The block diagram representing a single-input, single-output
CT system is illustrated in Fig. 2.1(b). Throughout this topic, we will restrict
our discussion to the analysis and design of single-input, single-output sys-
tems, knowing that the principles derived for such systems can be generalized
to MIMO systems.
In comparison to CT systems, DT systems transform DT input signals, often
referred to as sequences, into DT output signals. Two DT systems are shown in
Fig. 2.2. In Fig. 2.2(a), the schematic of a MIMO DT system is illustrated with
a set of
m
input sequences, denoted by
x
i
[
k
]'s, and a set of
n
output sequences,
denoted by
y
j
[
k
]'s. A single-input, single-output DT system is illustrated in
Fig. 2.2(b). As for the CT systems, we will focus on single-input, single-output
DT systems in this topic.
The relationship between the input signal and its output response of a single-
input, single-output system, may it be DT or CT, will be shown by the following
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