Optimal (but poor if unweighted)
Impulse Response Design
Perhaps the most commonly employed error criterion in signal processing is the least-squares error criterion.
Let denote some ideal filter impulse response, possibly infinitely long, and let denote the impulse response of a length causal FIR filter that we wish to design. The sum of squared errors is given by
where does not depend on . Note that . We can minimize the error by simply matching the first terms in the desired impulse response. That is, the optimal least-squares FIR filter has the following ``tap'' coefficients:
The same solution works also for any norm (§4.10.1). That is, the error
is also minimized by matching the leading terms of the desired impulse response.
In the (least-squares) case, we have, by the Fourier energy theorem (§2.3.8),
Therefore, is an optimal least-squares approximation to when is given by (4.5). In other words, the frequency response of the filter is optimal in the (least-squares) sense.
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 stop-band 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.4.4.
It is not the case that a length FIR filter is too short for implementing a reasonable audio lowpass filter, as can be seen in Fig.4.5. The optimal Chebyshev lowpass filter in this figure was designed by the Matlab statement
hh = firpm(L-1,[0 0.5 0.6 1],[1 1 0 0]);where, in terms of the lowpass design specs defined in §4.2 above, we are asking for
- (pass-band edge frequency)5.5
- (stop-band edge frequency)
We see that the Chebyshev design has a stop-band attenuation better than 60 dB, no corner-frequency resonance, and the error is equiripple in both stop-band (visible) and pass-band (not visible). Note also that there is a transition band between the pass-band and stop-band (specified in the call to firpm 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 filter specification calls for an infinite roll-off rate from the pass-band to the stop-band, and this cannot be accomplished by any FIR filter. (Review Fig.4.2 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. As a rule of thumb, the transition bandwidth should be at least , where is the FIR filter length in samples. (Recall that the main-lobe width of a length rectangular window is (§3.1.2).) Such a rule respects the basic Fourier duality of length in the time domain and ``minimum feature width'' in the frequency domain.
Frequency Sampling Method for FIR Filter Design
Lowpass Filter Design Specifications