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286 Chapter 6 IMPLEMENTATION OF DISCRETE-TIME FILTERS
Now, if each coefficient is quantized to B fraction bits (i.e., total register
length is B + 1), then,
Therefore,
|∆h(n)| ≤ 1 2 2−B
|∆H(e jω )|≤ 1 2 2−B M = M 2 2−B (6.81)
Thus, the change in frequency response depends not only on the number
of bits used but also on the length M of the filter. For large M and small
b, this difference can be significant and can destroy the desirable behavior
of the filter, as we see in the following example.
□ EXAMPLE 6.29 An order-30 lowpass FIR filter is designed using the firpm function. This and
other FIR filter design functions will be discussed in Chapter 7. The resulting
filter coefficients are symmetric and are shown in Table 6.2. We will consider
these coefficients as essentially unquantized. The coefficients are quantized to
16 bits (15 fractional plus 1 sign bit) and to 8 bits (7 fractional and 1 sign bit).
The resulting filter frequency responses and pole-zero plots are determined and
compared. These and other relevant steps are shown in the following MATLAB
script.
TABLE 6.2 Unquantized FIR filter coefficients used in
Example 6.29
k b k k
0 0.000199512328641 30
1 −0.002708453461401 29
2 −0.002400461099957 28
3 0.003546543555809 27
4 0.008266607456720 26
5 0.000012109690648 25
6 −0.015608300819736 24
7 −0.012905580320708 23
8 0.017047710292001 22
9 0.036435951059014 21
10 0.000019292305776 20
11 −0.065652005307521 19
12 −0.057621325403582 18
13 0.090301607282890 17
14 0.300096964940136 16
15 0.400022084144842 15
% The following function computes the filter
% coefficients given in Table 6.2.
b = firpm(30,[0,0.3,0.5,1],[1,1,0,0]);
w = [0:500]*pi/500; H = freqz(b,1,w); magH = abs(H);
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