Digital Signal Processing Reference
In-Depth Information
10.1 Finite-differen
ce equation representation of LTID systems
As discussed in Section 3.1, an LTIC system can be modeled using a linear,
constant-coefficient differential equation. Likewise, the input-output relation-
ship of a linear DT system can be described using a difference equation, which
takes the following form:
y
[
k
+
n
]
+
a
n
−
1
y
[
k
+
n
−
1]
++
a
0
y
[
k
]
=
b
m
x
[
k
+
m
]
+
b
m
−
1
x
[
k
+
m
−
1]
++
b
0
x
[
k
]
,
(10.1)
where
x
[
k
] denotes the input sequence and
y
[
k
] denotes the resulting out-
put sequence, and coefficients
a
r
(for 0
≤
r
≤
n
−
1), and
b
r
(for 0
≤
r
≤
m
)
are parameters that characterize the DT system. The coefficients
a
r
and
b
r
are constants if the DT system is also time-invariant. For causal signals and
systems analysis, the following
n
initial (or ancillary) conditions must be spec-
ified in order to obtain the solution of the
n
th-order difference equation in
Eq. (10.1):
y
[
−
1]
,
y
[
−
2]
,...,
y
[
−
n
]
.
We now consider an iterative procedure for solving linear, constant-coefficient
difference equations.
Example 10.1
The DT sequence
x
[
k
]=2
ku
[
k
] is applied at the input of a DT system described
by the following difference equation:
y
[
k
+
1]
−
0
.
4
y
[
k
]
=
x
[
k
]
.
By iterating the difference equation from the ancillary condition
y
[
−
1]
=
4,
compute the output response
y
[
k
] of the DT system for 0
≤
k
≤
5.
Solution
Express
y
[
k
+
1]
−
0
.
4
y
[
k
]
=
x
[
k
] as follows:
y
[
k
]
=
0
.
4
y
[
k
−
1]
+
x
[
k
−
1]
=
0
.
4
y
[
k
−
1]
+
2(
k
−
1)
u
(
k
−
1)
.
.
x
[
k
]
=
2
ku
[
k
]
,
which can alternatively be expressed as
0
.
4
y
[
k
−
1]
k
=
0
y
[
k
]
=
0
.
4
y
[
k
−
1]
+
2(
k
−
1)
≥
1
.
k
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