[music-dsp] Hilbert/Freq.shifter

[Olli Niemitalo]

I've been playing with 90 degree phase difference IIR allpass filter pairs
for a while now. Seems like the cheapest structure is one that has the
first allpass filter delayed by 1 sample, and both allpass filters are of
the same order, and both of the allpass filters consists of cascaded
normalized second-order allpass sections of form:

  H(z) = (a^2 - z^-2) / (1 - a^2 z^-2)

It has poles located at a and -a and zeros at 1/a and -1/a. One section
like that can be implemented efficiently with one multiplication and some 
other operations.

I calculated coefficients (using Differential Evolution) for a filter
pair with both allpasses being cascades of 4 of that kinda sections,
meaning total 8 multiplications. The characteristics are:

  Sampling frequency: 44100Hz. Band: 20-22030Hz. Maximum absolute phase
  difference deviation from 90 degrees in that band: 0.7 degrees. 

The "a" coefficients are:

  filter1: 
    0.6923877778065, 0.9360654322959, 0.9882295226860, 0.9987488452737

  filter2: 
    0.4021921162426, 0.8561710882420, 0.9722909545651, 0.9952884791278

This is purely theoretical, so it would be cool if someone would actually
implement that filter and tell me how it works!

-olli

p.s. remember to delay filter1 by 1 sample and to do a^2 before using the
coefficients. I think filter2 is the +90 degree one.


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