Phase splitters (part 2)

In the last post I promised to demonstrate some of the phase relationships obtainable from phase splitting parallel networks of all-pass filters - a technique known as polyphase IIR. So lets go ahead and do that. Here I will use GNU Octave to create the filters and produce the graphs.

Remember that all the phase plots in this post can be obtained from the same basic filter coefficients subjected to various transformations in the complex plane.

Hilbert Transform

The first plot demonstrates the classic Hilbert Transform configuration, which produces a phase difference of pi or 90 degrees. The 2 output signals are known as the reference signal and the quadrature signal.

Figure 1 - Phase relationship of the Hilbert Transform

The x-axis shows the frequency in radians (note the discontinuities at 0 and pi radians.)

The y-axis gives the phase difference of the 2 outputs: pi/2 from DC+d to Nyquist-d.

d is around 0.00014 radians in this order-20 set-up, which equates to 1Hz at 44100Hz SR.

The Pi Transformer - 2 filters for the price of 1

The second plot has a phase difference of pi in a slightly lower order (i.e. fewer poles and zeros) implementation of the same basic configuration.

Figure 2 - pi phase relationship

The outputs are exactly out-of-phase between DC (0) and 0.5*Nyquist (pi/2)

If the outputs are added we have a high-pass filter with cut-off frequency pi/2.

If the outputs are subtracted we have a low-pass filter with the same cut-off f.

We can combine the outputs with a simple addition or subtraction to produce a low pass or high pass filter, so we get 2 for the price of 1!

Variable transition frequencies

We can re-transform the transformed filter of figure 2 to produce different transition frequencies:

Figure 3 - pi phase relationship with different transition freq

The spike at pi/4 is due to phase-wrapping: a phase difference of 2*pi is

equivalent to a difference of 0 - i.e both imply exactly in-phase.

Series Combinations

If we combine the Hilbert transformer and Pi transformer in series we see the following phase relationship:

Figure 4 - Hilbert and Pi transformers in series

In this set-up filter output 2's phase lags output 1's phase by pi/2 all the way from 0 - pi/2  radians, at which point the relationship flips and output 2's phase leads output 1's phase by pi/2 radians.

Complex Filters

In this final graph you will see an example with a complex filter pair - whose complex coefficients do not come in conjugate pairs.

Figure 5 - Phase difference in complex filter network

The outputs of the complex filter array can be "decomplexified" (and therefore realized) by passing the each one through a special combination of none other than the Hilbert Transformer itself.

You're probably wondering about now just what are this set of filter coefficients that can produce so many types of filter? Don't worry - I'll get into those details in another post!