Digital Signal Processing Reference
In-Depth Information
because
x
[
k
] is zero for
k 6 0.
Then, from (11.4),
z
[x[n]*y[n]] =
q
n= 0
[
q
k= 0
x[k]y[n - k]]z
-n
(11.37)
q
q
y[n - k]z
-n
],
=
a
x[k][
a
k= 0
n= 0
where the order of the summations is reversed in the last step. Next, we change vari-
ables on the inner summation, letting
m = (n - k).
Then
n = m + k
and
q
q
y[m]z
-m-k
z
[x[n]*y[n]] =
a
x[k]
B
R
a
k= 0
m=-k
q
q
x[k]z
-k
a
y[m]z
-m
=
a
= X(z)Y(z).
(11.38)
k= 0
m= 0
The lower limit is changed to because
y
[
m
] is zero for
Hence, convolution transforms into multiplication in the
z
-domain. Examples of
convolution are given later in this chapter when we consider linear systems.
Several properties of the
z
-transform have been developed. These properties
are useful in generating tables of
z
-transforms and in applying the
z
-transform to
the solutions of linear difference equations with constant coefficients. When possi-
ble, we model discrete-time physical systems with linear difference equations with
constant coefficients; hence, these properties are useful in both the analysis and de-
sign of linear time-invariant physical systems. Table 11.4 gives properties for the
z
-transform and includes some properties in addition to those derived.
m =-k
m = 0,
m 6 0.
TABLE 11.4
Properties of the
z
-Transform
Name
Property
1.
Linearity, (11.8)
[a
1
f
1
[n] + a
2
f
2
[n]] = a
1
F
1
(z) + a
2
F
2
(z)
[f[n - n
0
]u[n - n
0
]] = z
-n
0
F(z),
2.
Real shifting, (11.13)
n
0
G 0
n
0
- 1
[f[n + n
0
]u[n]] = z
n
0
[F(z) -
a
f[n]z
-n
]
3.
Real shifting, (11.25)
n= 0
[a
n
f[n]] = F(z/a)
4.
Complex shifting, (11.23)
[nf[n]] =-z
dF(z)
dz
5.
Multiplication by
n
[f[n/k]] = F(z
k
),
6.
Time scaling, (11.33)
k
a positive integer
7.
Convolution, (11.38)
[x[n]*y[n]] = X(z)Y(z)
n
z
z - 1
F(z)
8.
Summation
[
a
f[k]] =
k= 0
9.
Initial value, (11.27)
f[0] =
lim
z:
q
F(z)
f[
q
] =
f[
q
]
10.
Final value, (11.30)
lim
z:1
(z - 1)F(z),
if
exists
