Digital Signal Processing Reference
In-Depth Information
The solution of a constant-coefficient linear difference equation is given by
[eq(10.49)]
y[n] = y
c
[n] + y
p
[n].
Recall also that the forced response is of the mathematical form as the system
input
x
[
n
]. Hence, if
x
[
n
] is bounded, is also bounded. If all roots of the char-
acteristic equation satisfy the relation each term of the natural response
is also bounded. Thus, the necessary and sufficient condition that a causal
discrete-time LTI system is BIBO stable is that
y
p
[n]
y
p
[n]
ƒ z
i
ƒ 6 1,
y
c
[n]
ƒ z
i
ƒ 6 1.
We now illustrate the
determination of stability with an example.
EXAMPLE 10.11
Stability of a discrete system
Suppose that a causal system is described by the difference equation
y[n] - 1.25y[n - 1] + 0.375y[n - 2] = x[n].
From (10.48) and (10.54), the system characteristic equation is
z
2
- 1.25z + 0.375 = (z - 0.75)(z - 0.5) = 0.
A MATLAB program that calculates these roots is
p=[1 -1.25 .375];
r=roots(p)
results: r=0.75 0.5
This system is stable, since the magnitude of each root is less than unity. The natural response
is given by
y
c
[n] = C
1
(0.75)
n
+ C
2
(0.5)
n
.
This function approaches zero as
n
approaches infinity.
Consider a second causal system described by the difference
y[n] - 2.5y[n - 1] + y[n - 2] = x[n].
The system characteristic equation is given by
z
2
- 2.5z + 1 = (z - 2)(z - 0.5).
The system is unstable, because the root
z
1
= 2
is greater than unity. The natural response is
given by
y
c
[n] = C
1
(2)
n
+ C
2
(0.5)
n
.
C
1
(2)
n
.
The instability is evident in the term
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