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Hilbert Transform Filters, and Other
Phase Adjusted Band Pass Filters
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Hilbert Transform filters are just one of the several types of Special Phase Adjusted Filters that are possible. A band pass filter is the only filter type that allows us to set the phase to any value we desire. The other three filter types are constrained by the fact their phase is predetermined at zero and ±Pi.

A brief look at the ideal Hilbert Transform.
To start, here is an ideal Hilbert Transform. It's phase response is 90 degrees all frequencies.

Ideal Hilbert Time Domain Plot.png

Notice however, that this waveform starts before t = 0. This is fine for pure mathematical constructs, but a serious problem for engineers that want to implement it in hardware (it is non-causal). Thus, to implement a Hilbert Transform, we must move this waveform to the right. This is the same as saying we will give it delay, which is the same as saying we must change its constant 90 degree phase into a phase with slope. The amount of slope is determined by the amount of delay we give it.

 

Hilbert Transform Filters.
To start, we show the frequency response for a 65 tap, 90 degree, Hilbert Transform Filter.

A Hilbert Filter can be confusing because it would seem that the phase plot should be a constant 90 degrees, but as you can see, the phase for this Hilbert filter is anything but constant. The reason of course, is the delay we are forced to use in order to construct this filter.

The phase plot on the right is what we expect to see, but this is Hilbert Phase, not ordinary phase. Hilbert Phase is the phase of the response after it has been multiplied by the Laplace time delay operator as described here .

Hilbert Phase Plots

A constant 90 degree Hilbert Phase means that a filter will add 90 degrees more phase to a signal than a reference device with the same group delay as the filter. This diagram illustrates this point. In the diagram, the reference device is a delay line. Delay lines have perfectly linear phase at all frequencies starting at DC. The phase at DC is zero, and the slope of the phase is determined by the length of the delay.

As the diagram shows, a Hilbert Transform Filter will have 90 degrees more phase shift at all frequencies in its pass band than the delay line.

Hilbert Phase Diagram.png

One use for a Hilbert Transform Filter is in the generation of single sideband. In order to generate single sideband, we must add a 180 and 0 degree phase shifted version of the signal to itself, as shown in the diagram below. In this type of single sideband generator, the Hilbert Filter must be paired with a delay line that is the same length as the filters group delay (typically (N-1)/2).

90 Degree Single Side Band Diagram

This is a plot of the single side band spectrum using a 65 tap Hilbert Transform filter.

SSB Spectrum Plot.png

 

Phase Adjusted FIR Band Pass Filters.

The Iowa Hills Hilbert Filter Designer synthesizes a number of special band pass filters. Band pass filters are different from low pass, high pass, and notch filters, in that the phase of the filter isn't predetermined at zero and Pi. The phase must be zero at 0 Hz for a low pass and notch, and the phase must be zero at Pi for a high pass and notch (180 deg if these filters invert the signal). Only the band pass, which has a magnitude of zero at DC and Pi, allows us to set its phase to any value in the pass band .

This plot shows how the phase of a 31 tap band pass filter can be adjusted from -135 to +135 degrees. The group delay is 15 for all phases, and the magnitude varies a small amount as the phase is adjusted.

Phase Adjusted FIR Filter Plots

 

This diagram shows how we can use a pair of phase shifted band pass filters to generate single side band. Instead of using a single Hilbert filter with 90 degrees of phase shift, we used a pair of matched filters with a 90 degrees phase difference between them. There are two advantages to this. First, we can generate narrow band SSB, and second, the filter's pass band need not be flat at 0 dB to attain good side band suppression. The two filters simply need to be matched.  

Dual 45 SSB Diagram.png

This is a plot of one of the 45 degree Hilbert's used to generate SSB. Note that the bandwidth is not full width, as is typically the case for Hilbert filters.

Dual 45 degree Single Side Band FIR Filter Response

Single Side Band FIR Filter Performance Plot

 

Differentiator Band Pass Filter

This plot shows the response of a 32 tap differentiator. The magnitude plot is linear (not in dB) and shows a 6 dB per octave slope.

Differentiator Plot

 

Mixed Band Pass Filter

Here is a dual band pass response obtained by mixing a filter's coefficients with a local oscillator.

100 Tap Mixed BPF Response.png

 

Screen shot of the free Hilbert Filter program.  Available on the Download Page

Hilbert Screen Shot.png

The program's test bench allows the user to test these filters in a single sideband generator (an excellent way to test for gain and phase flatness).  Available on the Download Page

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