Iowa Hills Software Digital and Analog Filters

Hilbert Phase
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Hilbert Phase is a term we coined. It is a method for calculating the linearity of phase. We call it Hilbert phase because it is the same method used to display the phase characteristics of a Hilbert Transform filter.

We have all seen the usual phase plots, and then ignored them, simply because
it is virtually impossible to glean any useful information from them. As a
result, we have been paying more attention to group delay plots which tell us
how linear the phase is, and how long it takes for a signal to get through the
device. Unfortunately however, group delay has units of time, not phase.

Hilbert Phase shows how many degrees the phase is varying from perfect linear
phase, and has units of phase.

Here is an example of a Butterworth low pass filter where we show a standard
phase plot and the group delay. The group delay shows us where the phase in non
linear, but it is difficult to ascertain much more information than that from
these plots.

Here we show this filter's unwrapped phase and draw a straight line to determine it's non linearity. The difference between the straight line and the unwrapped phase is the Hilbert Phase.

Calculation of Hilbert Phase

We use the same method of calculation that is used to display the phase characteristics of a Hilbert filter. To start, we need the Laplace transform for a time delayed signal.

To demonstrate the use of this Laplace operator, we use a linear phase FIR
filter. We set '-a' to the filter's group delay. The resulting Hilbert Phase is
a flat line because this filter's phase is linear.

In this example we use an Inverse Chebyshev low pass filter. Its group delay is much less than the FIR filter shown above, so the impulse response moves back in time less. The original phase is not linear and the Hilbert phase shows us the amount of non linearity.

Another way to interpret Hilbert Phase is shown here. As with the Laplace transform, the amount of delay used in the upper path is equivalent to the filter's group delay.

Here are two more examples of Hilbert Phase plots. The first is a specially designed 45 degree Hilbert filter.

This shows a typical Hilbert Phase plot for an analog band pass filter. The group delay plot indicates that the phase isn't linear and the Hilbert Phase plot tells us the amount of deviation from linearity in degrees.

In the Iowa Hills filter programs, we use Hilbert phase to show two distinct types of phase behavior.

For most filters, whether it be an opamp filter, IIR filter, RF filter, or an ordinary FIR filter, the Hilbert phase shows the amount of non linearity in the phase, just as in the Butterworth example above. The actual phase of the filter where the Hilbert phase is zero can be anything.

However, for filters such as a Differentiator or Hilbert Transform, the Hilbert Phase plot does not show a deviation from linearity, but rather the phase, adjusted for group delay. In other words, we show the phase results directly from the transfer function after the application of the Laplace delay operator.

These calculations are identical except that when we are only interested in nonlinearity, and not the absolute Hilbert phase value, we modify the results so that the Hilbert Phase starts at zero in the filter's pass band. As an example, if the filter has an inverted output, as some opamp filters have, then the absolute Hilbert phase would start at 180 degrees. Since we only want to show phase non linearity, we subtract out the 180 degree offset so the Hilbert phase plot starts at zero.

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