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346 Chapter 7 FIR FILTER DESIGN
Finally, we have
h(n) =IDFT [H(k)] (7.39)
Note that several textbooks (e.g., [18, 23, 24]) provide explicit formulas
to compute h(n), given H(k). We will use MATLAB’s ifft function
to compute h(n) from (7.39).
Basic idea Given the ideal lowpass filter H d (e jω ), choose the filter
length M and then sample H d (e jω )atM equispaced frequencies between
0 and 2π. The actual response H(e jω )isthe interpolation of the samples
H(k) given by (7.34). This is shown in Figure 7.25. The impulse response
is given by (7.39). Similar steps apply to other frequency-selective filters.
Furthermore, this idea can also be extended for approximating arbitrary
frequency-domain specifications.
From Figure 7.25, we observe the following:
1. The approximation error—that is, the difference between the ideal and
the actual response—is zero at the sampled frequencies.
2. The approximation error at all other frequencies depends on the shape
of the ideal response; that is, the sharper the ideal response, the larger
the approximation error.
3. The error is larger near the band edges and smaller within the band.
There are two design approaches. In the first approach, we use the
basic idea literally and provide no constraints on the approximation error;
that is, we accept whatever error we get from the design. This approach
is called a naive design method. In the second approach, we try to
minimize error in the stopband by varying values of the transition band
samples. It results in a much better design called an optimum design
method.
H d (e jω )
1
Ideal Response and
Frequency Samples
H(e jω )
1
Frequency Samples and
Approximated Response
0
k ω 0
0 1 2 3 4 5 6 7 8 9 10 π
0 1 2 3 4 5 6 7 8 9 10 π
FIGURE 7.25 Pictorial description of frequency sampling technique
k ω
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