Digital Signal Processing Reference
In-Depth Information
h(0)
Y(z)
X(z)
Σ
Σ
Σ
z
−
1
z
−
1
z
−
1
h(12)
h(11)
Σ
Σ
h(10)
z
−
1
z
−
1
h(22)
h(21)
Figure 6.4
Transpose of the cascade connection shown in Figure 6.3.
Example 6.3: Polyphase Form
This realization is based on the polyphase decomposition of the FIR transfer
function and is illustrated by choosing the following example:
h(
1
)z
−
1
h(
2
)z
−
2
h(
3
)z
−
3
h(
4
)z
−
4
H
1
(z)
=
h(
0
)
+
+
+
+
h(
5
)z
−
5
h(
6
)z
−
6
h(
7
)z
−
7
h(
8
)z
−
8
+
+
+
+
(6.7)
This can be expressed as the sum of two subfunctions, shown below:
=
h(
0
)
h(
8
)z
−
8
h(
2
)z
−
2
h(
4
)z
−
4
H
1
(z)
+
+
+
+
h(
1
)z
−
1
h(
7
)z
−
7
h(
3
)z
−
3
h(
5
)z
−
5
+
+
+
=
h(
0
)
h(
8
)z
−
8
h(
2
)z
−
2
h(
4
)z
−
4
+
+
+
z
−
1
h(
1
)
h(
7
)z
−
6
h(
3
)z
−
2
h(
5
)z
−
4
+
+
+
+
(6.8)
=
h(
0
)
h(
8
)z
−
8
and
h(
2
)z
−
2
h(
4
)z
−
4
Let us denote
A
0
(z)
+
+
+
=
h(
1
)z
−
1
h(
7
)z
−
7
h(
3
)z
−
3
h(
5
)z
−
5
A
1
(z)
+
+
+
z
−
1
h(
1
)
h(
7
)z
−
6
h(
3
)z
−
2
h(
5
)z
−
4
=
+
+
+
Since the polynomials in the square brackets contain only even-degree terms,
we denote
A
0
(z)
A
0
(z
2
)
and
A
1
(z)
z
−
1
A
1
(z
2
)
. Hence we express
H
1
(z)
=
=
=
A
0
(z
2
)
z
−
1
A
1
(z
2
)
. A block diagram showing this realization is presented in
Figure 6.5(a), where the two functions
A
0
(z
2
)
and
A
1
(z
2
)
are subfilters connected
in parallel.
+
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