Digital Signal Processing Reference
In-Depth Information
x
[
n
]
y
[
n
]
H
[
z
]
Figure 10.20
LTI system.
The form of the transfer function in (10.79) [and (10.80)], which is a ratio of
polynomials, is called a
rational function
. The transfer function of a discrete-time
LTI system described by a linear difference equation with constant coefficients, as
in (10.48), will
always
be a rational function.
We now consider three examples to illustrate the preceding developments.
Transfer function for a discrete system
EXAMPLE 10.14
In this example, we illustrate the transfer function, using the
a-filter
of Example 10.13. The
difference equation of the
a-filter
is given by
y[n] - (1 - a)y[n - 1] = ax[n].
The coefficients, as given in (10.48), are
a
0
= 1,
a
1
=-(1 - a), b
0
= a.
The filter transfer function is normally given in one of two different ways, from (10.79)
and (10.80):
a
1 - (1 - a)z
-1
=
az
z - (1 - a)
.
H(z) =
This transfer function is first order. Figure 10.21 shows the
a-filter
as a block diagram.
x
[
n
]
y
[
n
]
z
z
(1
)
■
Figure 10.21
a-Filter.
Sinusoidal response for a discrete system
EXAMPLE 10.15
In this example, we calculate the system response of an LTI system with a sinusoidal excita-
tion. Consider the
a-filter
of Example 10.14, with
a = 0.1.
The transfer function is given by
0.1z
z - 0.9
.
H(z) =
Suppose that the system is excited by the sinusoidal signal
x[n] = 5
cos
(0.01n + 20°).
In
e
jÆ
1
= e
j0.01
= 1
l
0.573°
,
(10.83), with
0.1(e
j0.01
)
e
j0.01
- 0.9
(5
l
20°
)
H(z) ƒ
z= e
j0.01
X =
0.5
l
20.573°
0.99995 + j0.0100 - 0.9
=
