Digital Signal Processing Reference
In-Depth Information
C
0
b
0
1
w
1
[n]
y
1
[n]
x[n]
a
1
1
z
-1
b
11
a
2
1
z
-1
b
0
2
w
2
[n]
y
2
[n]
x[n]
y[n]
a
1
2
z
-1
b
12
a
2
2
z
-1
b
0
3
w
3
[n]
y
3
[n]
x[n]
a
1
3
z
-1
b
13
a
2
3
z
-1
(e)
Figure 3.24
Continued
the filter is stable for all the above cases as all its poles remain inside the unit circle. The matrix of
coefficients for all the four sections from MATLAB
is:
0.0046
0.0015
0.0046
1
1.6853
0.7290
1
1.5730
1
1
1.7250
0.8573
1
1.7199
1
1
1.7542
0.9488
1
1.7509
1
1
1.7705
0.9882
Each row lists three coefficients of
b
and three coefficients of
a
for its respective section. Based on
themaximumof absolute values of the coefficients for each section, 2 bits are required for the integer
part of respective Q-format. The filter is analyzed for 12-bit and 8-bit precision, and all the
coefficients are converted into Q2.10 and Q2.6 format for the two formats. The pole-zero plots for
these four sections for both the cases are shown in Figure 3.27. In both cases the designs using
cascaded SoS are stable, so the overall system remains stable even for 8-bit quantization. It is
therefore important to first analyze the stability of the system while selecting the word lengths for
fixed-point implementation of IIR filters.
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