Phase error in Butterworth and Linkwitz-Riley filters
Odd-order Butterworth filters and Linkwitz-Riley ("Butterworth squared") filters have the property that the magnitude of the sum of the low-pass and high-pass outputs is independent of frequency - they are described as "all-pass" filters. This is useful in speaker design, since it gives us the opportunity to remove the dip or excess in response at the crossover frequency that is often seen, particularly with even-order Butterworth filters. The L-R filters have the additional useful property that the total power response of the speakers can also be made independent of frequency as the two parts are in phase at the crossover frequency. Their summed response is, however, not identical to the input, because of the phase rotation around the crossover frequency, which is pi/2 (90o) multiplied by the order of the filter: first order is 90o, second is 180o, and so on.
This means that the filters have poor impulse response, the phase distortion becoming worse with increasing order of the filter. The subjective effect of this error has not been reliably tested to my knowledge, as until now the use of analogue filters has made it extremely difficult to remove other contributions such as vertical response and the effect of driver frequency response from the phase error. Now that digital signal processing is widely available, though, the audibility of a simple phase error at any frequency should be straightforward to investigate. Siegfried Linkwitz has a fascinating article where he presents a circuit which switches in and out the phase error of second and fourth order filters. He challenges the reader to build the circuit and duplicate his experiment (by the way, he doesn't hear any difference, at least at 100Hz).
Filler drivers for second-order filters
Bækgård1 showed that an extra driver and a bandpass filter could modify a second-order Butterworth crossover to give a linear phase response.
A general second-order Butterworth has the form
Now if we add the low pass to the polarity-inverted high-pass (if we don't reverse the polarity of one or the other, we get a null at the crossover frequency).
The absolute magnitude squared of this is
The filter is only all-pass (constant voltage sum) when Q=0.5, which corresponds to the second-order Linkwitz-Riley filter (the square of the first-order Butterworth), while the second-order Butterworth (Q=0.707) has a 3dB peak at w=w0.
Now, if we re-connect one of the drivers so that they both have the same polarity, and subtract the sum of FLP and FHP from unity, we get
which has the form of a standard bandpass filter, with centre frequency w0 and sharpness Q.
This means that we can, in principle, use an extra driver, fed via the above bandpass filter, to correct for the phase error. In practice, there are all sorts of caveats. The driver centres need to be within some fraction of the wavelength for their outputs to sum in phase, worsening the existing problem of getting a midrange and tweeter close enough on a baffle. The filler driver needs to have a reasonably flat response and sufficient power handling over a wide frequency range, since the bandpass has a first-order decay each side of w0.
Filler drivers for fourth-order filters
We can, surprisingly, also use a filler driver with a fourth-order filter. The fourth-order Linkwitz-Riley filter is the square of the second-order Butterworth filter:
This is all-pass, since , but is not linear phase. However,
so the addition of a filler driver with a band-pass filter will transform an all-pass, but not linear phase, fourth-order Linkwitz-Riley filter into an all-pass, linear phase filter. Note that this is only true for the fourth-order L-R, and not other fourth-order filters.
Exactly the same caveats hold for the filler driver as did in the second-order case. Indeed, the advantages of a higher-order crossover (power handling, reduced driver overlap and less stringent demands on driver bandwidth) are lost for the extra driver.
1Bækgård, E, 1997. "A Novel Approach to Linear Phase Loudspeakers Using Passive Crossover Networks". J.Audio Eng. Soc., 25, 284-294
Alex Megann, December 1999