Digital Signal Processing Reference
In-Depth Information
Fig. 10.1. Input and output
sequences for Example 10.1.
(a) Input sequence
x
[
k
];
(b) output sequence
y
[
k
].
y
[
k
]
y
[
k
]
11.2
10
8
7.9
6
4
2.3
4.9
4
1.6
2
0.6
k
k
−5 −4 −3 −2 −1
01 234 5
−5 −4 −3 −2 −1
01234 5
(a)
(b)
By iterating from
k
=
0, the output response is computed as follows:
y
[0]
=
0
.
4
y
[
−
1]
=
1
.
6
,
y
[1]
=
0
.
4
y
[0]
+
2
0
=
0
.
64
,
y
[2]
=
0
.
4
y
[1]
+
2
1
=
2
.
256
,
y
[3]
=
0
.
4
y
[2]
+
2
2
=
4
.
902
,
y
[4]
=
0
.
4
y
[3]
+
2
3
=
7
.
961
,
y
[5]
=
0
.
4
y
[4]
+
2
4
=
11
.
184
.
Additional values of the output sequence for
k
>
5 can be similarly evaluated
from further iterations with respect to
k
. The input and output sequences are
plotted in Fig. 10.1 for 0
≤
k
≤
5.
In Chapter 3, we showed that the output response of a CT system, represented by
the differential equation in Eq. (3.1), can be decomposed into two components:
the zero-state response and the zero-input response. This is also valid for the
DT systems represented by the difference equation in Eq. (10.1). The output
response
y
[
k
] can be expressed as
y
[
k
]
=
y
zi
[
k
]
zero
-
input response
+
y
zs
[
k
]
,
zero
-
state response
(10.2)
where
y
zi
[
k
] denotes the
zero-input response
(or the
natural response
) of the
system and
y
zs
[
k
] denotes the
zero-state response
(or the
forced response
)of
the DT system.
The zero-input component
y
zi
[
k
] for a DT system is the response produced by
the system because of the initial conditions, and is not due to any external input.
To calculate the zero-input component
y
zi
[
k
], we assume that the applied input
sequence
x
[
k
]
=
0. On the other hand, the zero-state response
y
zs
[
k
] arises due
to the input sequence and does not depend on the initial conditions of the system.
To calculate the zero-state response
y
zs
[
k
], the initial conditions are assumed
to be zero. Based on Eq. (10.2), a DT system represented by Eq. (10.1) can
be considered as an incrementally linear system (see Section 2.2.1) where the
additive offset is caused by the initial conditions (see Fig. 2.10). If the initial
conditions are zero, the DT system becomes on LTID system. We now solve
Example 10.1 in terms of the zero-input and zero-state components of the output.
Example 10.2
Repeat Example 10.1 to calculate (i) the zero-input response
y
zi
[
k
], (ii) the zero-
state response
y
zs
[
k
], and (iii) the overall output response
y
[
k
] for 0
≤
k
≤
5.
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