Databases Reference
In-Depth Information
f
k
a
0
k
Delay
Delay
a
1
b
1
f
k
-1
g
k
-1
Delay
Delay
a
2
b
2
f
k
-2
g
k
-2
F I GU R E 12 . 13
A discrete system.
let us assume the sequences are all one-sided; that is, they are only nonzero for nonnegative
values of the subscript:
∞
∞
h
n
z
−
n
∞
g
n
z
−
n
f
m
z
−
m
=
(118)
n
=
0
n
=
0
m
=
0
Equating like powers of
z
, we obtain:
g
0
=
h
0
f
0
g
1
=
f
0
h
1
+
f
1
h
0
g
2
=
f
0
h
2
+
f
1
h
1
+
f
2
h
0
.
n
g
n
=
f
m
h
n
−
m
m
=
0
Thus, the output sequence is a result of the discrete convolution of the input sequence with the
impulse response.
Most of the discrete linear systems we will be dealing with will be made up of delay
elements, and their input-output relations can be written as constant coefficient difference
equations. For example, for the system shown in Figure
12.13
, the input-output relationship
can be written in the form of the following difference equation:
g
k
=
a
0
f
k
+
a
1
f
k
−
1
+
a
2
f
k
−
2
+
b
1
g
k
−
1
+
(119)
b
2
g
k
−
2
The transfer function of this system can be easily found by using the
shifting theorem
.The
shifting theorem states that if the Z-transform of a sequence
{
f
n
}
is
F
(
z
)
, then the Z-transform
of the sequence shifted by some integer number of samples
n
0
is
z
−
n
0
F
(
z
)
.
