Phase Splitters (part 1)

I'm going spend my first few posts talking about a class of filter networks whose purpose is to transform an input signal into 2 (or more) output signals with some desired phase difference between them.

Normally such networks will take the form of a pair with output phase differences of:

1.  90° (π/2) known as a 'Hilbert Transformer' - although not strictly speaking a Hilbert transform because the input phase is not preserved.
   
2. 180° (π) which can be used as a low or high pass filter when the output pairs are combined

Other phase relationships are possible, especially if complex filters are used.

As you might expect when talking about phase (and not magnitude) filtering we are referring to networks of all-pass filters:

Figure 1 - All-pass phase-splitter network

Figure 1 shows the input signal(s) being phase-distorted by a pair of appropriately configured all-pass networks such that the phase of output 2 lags that of output 1 by π/2 (i.e. Hilbert Transform). The input phase may be (and probably will be) different to both.

It turns out that there is a particular set of filter coefficients which can serve as the basis for generating any number of pairs of all-pass sections with arbitrary phase split, accurate to an arbitrary bandwith (with theoretical limitiations at DC (0 Hz) and the Nyquist frequency (π Hz)). Furthermore, each section pair so-generated can be cascaded with other alternatively configured pairs to produce more complicated (yet predictable) phase relationships.

The set of generating filter coefficients are subjected to various transformations on the complex plane to produce the different classes of filter pair.

In the next post I will demonstrate some examples of phase difference relationships I created in back in my 'research labs' - i.e. sitting on the couch with an old laptop running GNU Octave and Pure Data (Pd).