Digital Signal Processing Reference
In-Depth Information
values of the input, then the system is defined as a
causal
system. This means
that the output does not depend on the future values of the input. We will discuss
these concepts again in more detail in later sections of this chapter. In this topic,
we consider only discrete-time systems that are linear and time-invariant (LTI)
systems.
Another way of defining a system in general is that it is an interconnection
of components or subsystems, where we know the input-output relationship of
these components, and that it is the way they are interconnected that determines
the input-output relationship of the whole system. The model for the DT system
can therefore be described by a circuit diagram showing the interconnection of
its components, which are the delay elements, multipliers, and adders, which are
introduced below. In the following sections we will use both of these definitions
to model discrete-time systems. Then, in the remainder of this chapter, we will
discuss several ways of analyzing the discrete-time systems in the time domain,
and in Chapter 3 we will discuss frequency-domain analysis.
2.1.1 Models of the Discrete-Time System
First let us consider a discrete-time system as an interconnection of only three
basic components: the delay elements, multipliers, and adders. The input-output
relationships for these components and their symbols are shown in Figure 2.1.
The fourth component is the modulator, which multiplies two or more signals
and hence performs a nonlinear operation.
A simple discrete-time system is shown in Figure 2.2, where input signal
x(n)
={
x(
0
), x(
1
), x(
2
), x(
3
)
}
is shown to the left of
v
0
(n)
=
x(n)
. The signal
Delay Element
X
1
(
n
)
Y
1
(
n
)
=
X
1
(
n
−
1)
z
−
1
Multiplier
X
2
(
n
)
Y
2
(
n
)
=
K
X
2
(
n
)
K
X
3
(
n
)
Adder
Y
3
(
n
)
=
X
3
(
n
)
+
X
4
(
n
)
Σ
X
4
(
n
)
Y
5
(
n
)
=
X
5
(
n
)
m
6
(
n
)
Modulator
X
5
(
n
)
Σ
m
6
(
n
)
Figure 2.1
The basic components used in a discrete-time system.
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