bruno putzeys wrote on Sat, 30 October 2010 06:33 |
Geoff Emerick de Fake wrote on Fri, 29 October 2010 15:05 | In all practicality, is that really a necessity? Considering cross-overs, for example, phase needs be linear only one octave away of the corner frequency for a 4th-order filter with an attenuation of 20dB at 0.5Fc (HiPass) or 2Fc (Low-pass). Even with phase completely wrong there and away, the recombination error is less than +/-1dB.
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Phase linearity in crossover filters is besides the point. The primary requirement of a crossover filter (including the contribution of the drive units) is that it has to add to unity magnitude ie. a "recombination error" of exactly none at all.
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This is a very restrictive view. Primary requirements of crossover filters are: best pressure response, best energy response and best overall phase response. Very often, these are conflicting constraints, so compromises have to be made; in that case "none error at all is somewhat abusive.
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Butterworth filters do that, and so do double butterworth aka Linkwitz-Reiley filters.
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Agreed for L-R24, but Butterworth 2nd order and LR12 have terrible phase response (drivers must be wired with opposite polarity) and B3 has incorrect energy response.
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Requirements to the phase shift of the two filters separately only relate to off-axis behaviour. Linkwitz-Riley wins because you can get identical phase shifts from DC to light.
Other than that the only phase requirement one can think of as concerns crossover filters are what the phase shift of the sum is, because anything beyond the physically impossible 1st order crossover will result in an all-pass filter.
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You seem to forget that linear-phase FIR filters have the same benefits AND an overall minimum-phase response.
OK, let's forget about crossovers. Then again: In all practicality, is linear phase really a necessity? Let's consider a HPF on a mic, used for attenuating parasitic noise. What is the point of making sure the remnants of noise are still time-aligned with the useful signal?
Or a parametric EQ; is it important that the frequency range that is boosted remains time-aligned with the rest of the spectrum?
If it was such an issue, one would use a phase equalizer to improve a signal without changing its spectral balance.
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Geoff Emerick de Fake wrote on Fri, 29 October 2010 15:05 | OTOH, I never understood how one could justify calling the attached example a linear-phase filter. Even the comment on the left doesn't make sense to me...
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I don't even see what the problem is. A chebychev filter is a filter in which the magnitude response is allowed to ripple within bounds such as to stretch the "flat" range beyond the asymptotic cut-off. A butterworth filter is a filter which is mathematically maximally flat meaning the first 2n-1 derivatives of the magnitude response at DC are zero. A Bessel filter is one where the group delay response is maximally flat in the same vein and a filter with equiripple group delay is one where the group delay is allowed to vary within narrow bounds so that it remains more or less constant over a wider range. Like chebychev and butterworth are two interpretations of "flat magnitude", bessel and equiripple phase error filters are two approximations of "flat group delay". Provided you allow the definition of linear phase to stretch all the way to its original meaning i.e. approximately constant group delay across the pass band, this new filter is an approximation of the ideal just like a bessel filter is an approximation of the ideal.
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So, in analog, linear-phase means constant over the passband, in digital it means constant from DC to Nyquist. What's the validity of comparison, then? Different design rules!