Digital Signal Processing Reference
In-Depth Information
to either one of the algorithms
x(n)
∗
h(n)
or
h(n)
∗
x(n)
. For example
y(
0
)
=
0
.
2
y(
1
)
=
0
.
4
+
0
.
1
(
0
.
2
)
(
0
.
1
)
4
(
0
.
2
)
y(
2
)
=
0
.
6
+
(
0
.
1
)(
0
.
4
)
+
(
0
.
1
)
4
(
0
.
4
)
(
0
.
1
)
9
(
0
.
2
)
y(
3
)
=
0
.
8
+
(
0
.
1
)(
0
.
6
)
+
+
·
·
·
Recollect that we have obtained two different equations for finding the output
due to a given input. They are the convolution sum (2.6) and the linear difference
equation (2.2), which are repeated below.
∞
y(n)
=
x(k)h(n
−
k)
(2.53)
k
=
0
N
M
y(n)
=−
a(k)y(n
−
k)
+
b(k)x(n
−
k)
(2.54)
k
=
k
=
1
0
In Equation (2.53), the product of the input sequence and the current and previ-
ous values of the unit impulse response are added, whereas in Equation (2.54)
the previous values of the output and present and past values of the input are
multiplied by the fixed coefficients and added. The transfer function
H(z)
for
the first case is given by
H(z)
=
n
=
0
h(n)z
−
n
, and for the second case, we use
the
z
transform for both sides to get
N
M
a(k)z
−
k
Y(z)
b(k)z
−
k
X(z)
Y(z)
=−
+
k
=
1
k
=
0
Y(z)
1
a(k)z
−
k
N
M
b(k)z
−
k
X(z)
+
=
k
=
1
k
=
0
k
=
0
b(k)z
−
k
Y(z)
X(z)
=
H(z)
=
1
+
k
=
1
a(k)z
−
k
So we can derive the transfer function
H(z)
from the linear difference equation
(2.54), which defines the input-output relationship.
We can also obtain the linear difference equation defining the input-output
relationship, from the transfer function
H(z)
, simply by reversing the steps as
follows. Given the transfer function
H(z)
,weget
Y(z)
[1
+
k
=
1
a(k)z
−
k
]
=
Search WWH ::
Custom Search