Digital Signal Processing Reference
In-Depth Information
x[n]
y[n]
+
+
+
X
a
+
X
y[n-m]
X
a
2
a
M
+
X
a
M-1
Feed forward part
Figure 7.29
Look-ahead transformation to add M registers in the feedback path of a first-order IIR filter
The generalized IIR filter implementing this difference equation with M registers in the feedback
loop is given in Figure 7.29. This reduces the IPB of the orginal DFG by a factor of M. The difference
equation implements the same transfer function H(z):
X
M
1
i¼
0
a
i
z
i
1
az
1
1
HðÞ¼
az
1
¼
ð
7
:
9
Þ
Þ
M
1
ð
The expression in (7.8) adds an extra M
1 coefficients in the numerator of the transfer function of
the systemgiven in (7.9), requiringM
1multiplications andM
1 additions in its implementation.
These extra computations can be reduced by constraining M to be a power of 2, so that M
¼
2
m
.
This makes the transfer function of (7.9) as:
Y
m
i¼
2
i
þ az
1
1
ð
Þ
1
1
HðÞ¼
az
1
¼
ð
7
10
Þ
:
1
Þ
M
1
az
1
ð
The numerator of the transfer function now can be implemented as a cascade of m FIR filters each
requiring only one multiplier, where m
¼
log
2
M.
Example: This example increases the IPB of a first-order IIR filter by M
8. Applying the
look-ahead transformation of (7.9), the transfer function of a first-order system with a pole at 0.9 is
given as:
¼
1
þ
0
:
9
z
1
þ
0
:
81
z
2
þ
0
:
729
z
3
þ
0
:
6561
z
4
þ
0
:
5905
z
5
þ
0
:
5314
z
6
þ
z
7
1
0
:
4305
z
8
1
1
0
:
9
z
1
¼
0
4783
:
HðÞ¼
Using (7.9), this transfer function can be equivalently written as:
ð
1
þ
0
:
9
z
1
Þ
1
þ
0
:
81
z
2
ð
Þð
1
þ
0
:
6561
z
4
Þ
1
1
0
:
9
z
1
¼
HðÞ¼
1
0
:
4305
z
8
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