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The sound file Linkwitz-Reilly.kym contains perfect reconstruction filterbanks for 2,3,4, and 5 bands. Linkwitz-Reilly crossover filters are really nothing more than cascaded Butterworth filters, where each Butterworth section has an even number of poles. While the composite filters now have a crossover response that is down by 6 dB, instead of the usual 3 dB, the phase wrap from LPF to HPF is complementary, allowing for a zero-glitch crossover when both filters have the same gain.

However... when constructing more than a simple 2-band crossover, it becomes necessary to add in a compensating all-pass filter in the lower bands, one for each higher LPF/HPF pair of higher bands. So for example, the 3-band LW filter bank needs one APF in the low band; the 4-band LW filter bank needs 2 APF's in the lowest band, and one APF in the next lowest band, and so on.

By the time you reach the level of 5 bands in the filter bank, there are so many 2-pole or 4-pole sections that it almost becomes absurd to do this -- 38 IIR filters in each channel of audio (!!) 76 biquads if you insist on 8-pole filters (!!!) It almost makes more sense to use an FFT and have complete phase linearity at this level of complexity.

However... in order to reach as low in frequency with an FFT as you can get with these IIR systems, you'd have a throughput delay of more than 300 ms and an FFT length at 48 kHz Fsamp of more than 16K samples. So there really is a place for these IIR beasts... The throughput delay with the 5-band filter bank is very low, measured with Smaart to be around 0.02 ms even at 8th order, which has a roll-off of nearly 50 dB/octave !!

Now... just because it is possible to create a perfect reconstructing crossover, doesn't necessarily mean that it makes sense to do so. Also included is a simple example sound called "3-dB Bump" which shows what happens if you just sum a 4-pole LPF with a 4-pole HPF. While not necessarily pretty, I doubt there are many recording venues that can offer a room that is as flat as even this... And if you can tolerate the bumps and dips, it becomes a whole lot cheaper to just add filters together and forego the beautiful Linkwitz-Reilly toplogy.

-- DavidMcClain - 12 Jan 2013

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