Digital Signal Processing Reference
In-Depth Information
where the last step is required to express the variable
z
in positive exponents. Note
that this transfer function is identical to that derived in Chapter 10, which applies
for only a complex-exponential input signal. This transfer function, (11.44), applies
for any input that has a
z
-transform and, hence, is a generalization of that of
Chapter 10. An example is now given.
Transfer function of a discrete system
EXAMPLE 11.8
We consider again the -filter of (11.14), which is depicted in Figure 11.5(a). (See Figure
10.18.) The filter equation is given by
a
y[n] - (1 - a)y[n - 1] = ax[n].
For this example, we let
a = 0.1;
then,
y[n] - 0.9y[n - 1] = 0.1x[n].
The
z
-transform of this equation yields
(1 - 0.9z
-1
)Y(z) = 0.1X(z),
and the transfer function is
Y(z)
X(z)
=
0.1
1 - 0.9z
-1
=
0.1z
z - 0.9
.
H(z) =
(11.45)
Note that we could have written the transfer function directly from (11.39) and (11.44).
■
y
[
n
]
y
[
n
]
x
[
n
]
y
[
n
1]
x
[
n
]
y
[
n
1]
D
0.1
D
1
0.9
(a)
(b)
Y
(
z
)
X
(
z
)
z
1
0.1
X
(
z
)
Y
(
z
)
0.1
z
z
0.9
0.9
(c)
(d)
Figure 11.5
a
-Filter representation.
