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350 Chapter 7 FIR FILTER DESIGN
Frequency Samples: M=40,T1=0.5
Impulse Response
1
0.2
Hr(k)
0.5
h(n)
0.1
0
0
0 0.2 0.3 1
frequency in π units
−0.1
0 10 20 30 40
n
Amplitude Response
Magnitude Response
1
0
Hr(w)
0.5
Decibels
30
0
0 0.2 0.3 1
frequency in π units
0 0.2 1
frequency in π units
FIGURE 7.27 Optimum frequency design method: T 1 =0.5
From the plots in Figure 7.28, we observe that the optimum stopband attenuation
is 43 dB. It is obvious that to further increase the attenuation, we will
have to vary more than one sample in the transition band.
□
Clearly, this method is superior in that by varying one sample we
can get a much better design. In practice the transition bandwidth is
generally small, containing either one or two samples. Hence we need to
optimize at most two samples to obtain the largest minimum stopband
attenuation. This is also equivalent to minimizing the maximum side-lobe
magnitudes in the absolute sense. Hence this optimization problem is also
called a minimax problem. This problem is solved by Rabiner et al. [24],
and the solution is available in the form of tables of transition values.
A selected number of tables are also available in [23, Appendix B]. This
problem can also be solved in MATLAB, but it would require the use of
the Optimization toolbox. We will consider a more general version of this
problem in the next section. We now illustrate the use of these tables in
the following examples.
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