Digital Signal Processing Reference
In-Depth Information
For an
N
th-order discrete-time LTI causal system, the transfer function can be
expressed as
Á
b
0
z
N
+ b
1
z
N- 1
Y(z)
X(z)
=
+
+ b
N- 1
z + b
N
[eq(11.44)]
H(z) =
,
+
Á
+ a
N- 1
z + a
N
a
0
z
N
+ a
1
z
N- 1
with
a
0
Z 0.
The denominator of this transfer function can be factored as
Á
+ a
N- 1
z + a
N
= a
0
(z - p
1
)(z - p
2
)
Á
(z - p
N
).
a
0
z
N
+ a
1
z
N- 1
+
(11.54)
The zeros of this polynomial are the
poles
of the transfer function, where, by defin-
ition, the poles are those values of
z
for which
H(z)
is unbounded.
First, we assume that
H(z)
has no repeated poles. We can then express the
output
Y(z)
in (11.44) as
+
Á
+ b
N- 1
z + b
N
a
0
(z - p
1
)(z - p
2
)
Á
(z - p
N
)
b
0
z
N
+ b
1
z
N- 1
Y(z) = H(z)X(z) =
X(z)
(11.55)
k
1
z
z - p
1
k
2
z
z - p
2
k
N
z
z - p
N
+ Y
x
(z),
Á
=
+
+
+
where is the sum of the terms, in the partial-fraction expansion, that originate
in the poles of the input function Hence, is the
forced response
.
In the partial fraction expansion of (11.55), it is assumed that the order of the
numerator of is lower than that of the denominator. If the order of the numer-
ator polynomial is equal to or greater than the order of the denominator polynomial,
the partial-fraction expansion will include additional terms. [See (F.1).]
The inverse transform of (11.55) yields
Y
x
(z)
X(z).
Y
x
(z)
H(z)
+
Á
+ k
N
p
n
y[n] = k
1
p
n
+ k
2
p
n
+ y
x
[n] = y
n
[n] + y
x
[n].
(11.56)
The terms of originate in the poles of the transfer function, and is the
natural response
. The natural response is always present in the system output, inde-
pendent of the form of the input signal
x
[
n
]. The factor
y
n
[n]
y
n
[n]
p
i
n
in each term of the nat-
ural response is called a
mode
of the system.
If the input
x
[
n
] is bounded, the forced response will remain bounded,
since is of the functional form of
x
[
n
]. [ has the same poles as ] Thus,
an unbounded output must be the result of at least one of the natural-response
terms, becoming unbounded. This unboundedness can occur only if the mag-
nitude of at least one pole, is greater than unity.
From the preceding discussion, we see the requirement for BIBO stability:
y
x
[n]
y
x
[n]
Y
x
(z)
X(z).
k
i
p
i
n
,
ƒp
i
ƒ ,
An LTI discrete-time causal system is BIBO stable, provided that all poles of the sys-
tem transfer function lie inside the unit circle in the
z
-plane.
The stable region of the
z
-plane is illustrated in Figure 11.8. This conclusion
was also reached in Chapter 10 by a different approach.
