Digital Signal Processing Reference
In-Depth Information
x[n] = z
n
,
For the complex exponential excitation
the convolution sum yields
the system response:
y
ss
[n] =
q
k=-
q
h[k]x[n - k] =
q
k=-
q
h[k]z
n-k
q
= z
n
a
h[k]z
-k
.
(10.87)
k=-
q
In (10.81), the value of
z
1
is not constrained and can be considered to be the variable
z
. From (10.81) with
X = 1,
and from (10.87),
= z
n
q
k=-
q
y
ss
[n] = H(z)z
n
h[k]z
-k
,
and we see that the impulse response and the transfer function of a discrete-time
LTI system are related by
q
h[k]z
-k
.
H(z) =
a
(10.88)
k=-
q
This equation is the desired result. Table 10.1 summarizes the results developed in
this section.
Those readers familiar with the bilateral
z
-transform will recognize in
(10.88) as the
z
-transform of
h
[
n
]. Furthermore, with in (10.88) is
the discrete-time Fourier transform of
h
[
n
]. We see then that both the
z
-transform
(covered in Chapter 11) and the discrete-time Fourier transform (covered in
Chapter 12) appear naturally in the study of discrete-time LTI systems.
In practice, it is more common in describing an LTI system to specify the
transfer function rather than the impulse response
h
[
n
]. However, we can rep-
resent an LTI system with either of the block diagrams given in Figure 10.22, with
and
h
[
n
] related by (10.88).
H(z)
z = e
jÆ
, H(e
jÆ
)
H(z)
H(z)
TABLE 10.1
Input-Output Functions for an LTI System
q
h[k]z
-k
H(z) =
a
k=-
q
Xz
n
:XH(z
1
)z
n
, X = ƒXƒ e
jf
ƒXƒ cos
(Æ
1
n + f) : ƒXƒ ƒH(e
jÆ
1
) ƒ cos
(Æ
1
n + f + u
H
)
x
[
n
]
y
[
n
]
x
[
n
]
y
[
n
]
h
[
n
]
H
[
z
]
H
[
z
]
h
[
k
]
z
k
k
Figure 10.22
LTI system.
