Examples

Figure E.2 shows the amplitude response of a length optimal least-squares FIR lowpass filter, for the case in which the cut-off frequency is one-fourth the sampling rate ().

We see that, although the impulse response is optimal in the least squares sense (in fact optimal under any norm with any error-weighting), the filter is quite poor from an audio perspective. In particular, the stopband gain, in which zero is desired, is only about 10 dB down. Furthermore, increasing the length of the filter does not help, as evidenced by the length 71 result in Fig.E.3.

It is not the case that a length FIR filter is too short for implementing a reasonable audio lowpass filters, as can be seen in Fig.E.4. The optimal Chebyshev lowpass filter in this figure was designed by the matlab statement

hh = remez(L-1,[0 0.5 0.6 1],[1 1 0 0]);

We see that the Chebyshev design has a stopband attenuation better than 60 dB, no corner-frequency resonance, and the error is ``equiripple'' in both stopband (visible) and passband (not visible). Note also that there is a transition band between the passband and stopband (specified in the call to remez as being between normalized frequencies 0.5 and 0.6).

The main problem with the least-squares design examples above is the absence of a transition band specification. That is, the desired filter specification asks for infinite roll-off rate, which is too much. (See Fig.E.25 for an illustration of more practical lowpass-filter design specifications.) With a transition band and a weighting function, least-squares FIR filter design can perform very well in practice.

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.