Finite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 270

FIGURE 7.33
(Top) Magnitude frequency response of an ideal lowpass filter and a typical lowpass filter designed using the
Parks-McClellan algorithm. (Middle) Error between the ideal and practical responses. (Bottom) Weighted error
between the ideal and practical responses.
specifies the stopband magnitude attenuation. In terms of dB value specification, we have
d p dB ¼ 20 log 10 ð 1 þ d p Þ and d s dB ¼ 20 log 10 d s .
The middle graph in Figure 7.33 describes the error between the ideal frequency response and the
actual frequency response. In general, the error magnitudes in the passband and stopband are
different. This makes optimization unbalanced, since the optimization process involves an entire
band. When the error magnitude in a band dominates the other(s), the optimization process may
deemphasize the contribution due to a small magnitude error. To make the error magnitudes
balanced, a weight function can be introduced. The idea is to weight a band with a bigger magnitude
error with a small weight factor and to weight a band with a smaller magnitude error with a big
weight factor. We use a weight factor W p for weighting the passband error and W s for weighting the
stopband error. The bottom graph in Figure 7.33 shows the weighted error, and clearly, the error
magnitudes on both bands are at the same level. Selection of the weighting factors is further
illustrated in the following design procedure.
Optimal FIR Filter Design Procedure for the Parks-McClellan Algorithm
1. Specify the band edge frequencies such as passband and stopband frequencies, passband ripple,
stopband attenuation, filter order, and sampling frequency of the DSP system.
2. Normalize band edge frequencies to the Nyquist limit (folding frequency ¼ f s = 2) and specify the
ideal magnitudes.