Digital Signal Processing Reference
In-Depth Information
x
[
n
]
y
[
n
]
x
[
n
1]
X
(
z
)
Y
(
z
)
z
X
(
z
)
D
1
z
Figure 11.7
(a)
(b)
Unit advance.
We next investigate causal LTI systems. Consider the system of Figure 11.7(a).
This system is a unit advance, with the output equal to the input advanced by one
discrete-time increment; that is,
y[n] = x[n + 1].
For this derivation, we ignore ini-
tial conditions. Then, the output is given by
Y(z) =
z
[x[n + 1]] = z[X(z) - x[0]] = zX(z),
(11.51)
from Table 11.4. The unit advance has a transfer function of and can be
represented by the block diagram of Figure 11.7(b). In a like manner, it is seen that
the transfer function of
N
cascaded unit advances is
H(z) = z
H(z) = z
N
.
If we allow
N
to be
negative, this transfer function also applies to
ƒNƒ
unit delays. For example, the
z
3
,
transfer function for three cascaded advances is
and the transfer function for
z
-3
.
three cascaded delays is
The unit advance is not causal; the system of Figure 11.7 cannot be realized
physically. Consider the transfer function given by
z
2
+ 0.4z + 0.9
z - 0.6
z
+
0.9
z - 0.6
,
H(z) =
= z +
(11.52)
where we can obtain the last function by dividing the numerator of by its de-
nominator. This system is noncausal, since the system can be represented as a unit ad-
vance in parallel with a second system that is physically realizable. This unit advance
appears because the numerator of is of higher order than the denominator.
It is seen from the preceding development that for a
causal system,
the numer-
ator of the transfer function of (11.44) cannot be of higher order than the
denominator, when the exponents are positive. If the transfer function
H(z)
H(z)
H(z)
H(z)
is
expressed in negative exponents, as in (11.44), that is, as
Á
b
0
+ b
1
z
-1
+ b
N- 1
z
-N+ 1
+ b
N
z
-N
Y(z)
X(z)
=
+
H(z) =
+ a
N
z
-N
,
(11.53)
+
Á
+ a
N- 1
z
-N+ 1
a
0
+ a
1
z
-1
then the system is causal, provided that
a
0
Z 0.
We now relate bounded-input bounded-output (BIBO) stability of causal systems
to the system transfer function. Recall the definition of BIBO stability:
BIBO Stability
A system is stable if the output remains bounded for any bounded input.
