Polyphase Filterbanks

Introduction
Polyphase filterbanks (PFB) are a very powerful set of filtering tools that can efficiently perform many multi-rate signal processing tasks. GNU Radio has a set of polyphase filterbank blocks to be used in all sorts of applications. See the documentation for the individual blocks for details about what they can do and how they should be used. Furthermore, there are examples for these blocks in gr-filter/examples:

NOTE: you need the Matplotlib Python module installed to run these examples
 * 1) channelize.py creates an appropriate filter to channelizer 9 channels out of an original signal that is 9000 Hz wide, so each output channel is now 1000 Hz. The code then plots the PSD of the original signal to see the signals in the origina spectrum and then makes 9 plots for each of the channels.
 * 2) chirp_channelize.py is similar to channelize.py but includes a VCO to create a chirp signal
 * 3) decimate.py shows an example of using the PFB decimator
 * 4) interpolate.py shows an example of using the PFB interpolator
 * 5) reconstruction.py includes a PFB channelizer and PFB synthesizer
 * 6) resampler.py demonstrates how to use the PFB resampler
 * 7) synth_filter.py is a simple example of using the PFB synthesizer

PFB Usage
The main issue when using the PFB blocks is defining the prototype filter, which is passed to all of the blocks as a vector of taps. The taps from the prototype filter which get partitioned among the N channels of the channelizer.

An example of creating a set of filter taps for a PFB channelizer is found on line 49 of gr-filter/examples/channelizer.py and reproduced below. Notice that the sample rate is the sample rate at the input to the channelizer while the bandwidth and transition width are defined for the channel bandwidths. This makes a fairly long filter that is then split up between the N channels of the PFB.

self._fs = 9000         # input sample rate self._M = 9             # Number of channels to channelize self._taps = filter.firdes.low_pass_2(1, self._fs, 475.50, 50,                                         attenuation_dB=100,                                          window=filter.firdes.WIN_BLACKMAN_hARRIS)

In this example, the signal into the channelizer is sampled at 9 ksps (complex, so 9 kHz of bandwidth). The filter uses 9 channels, so each output channel will have a bandwidth and sample rate of 1 kHz. We want to pass most of the channel, so we define the channel bandwidth to be a low pass filter with a bandwidth of 475.5 Hz and a transition bandwidth of 50 Hz, but we have defined this using a sample rate of the original 9 kHz. The prototype filter has 819 taps to be divided up between the 9 channels, so each channel uses 91 taps. This is probably over-kill for a channelizer, and we could reduce the amount of taps per channel to a couple of dozen with no ill effects.

The basic rule when defining a set of taps for a PFB block is to think about the filter running at the highest rate it will see while the bandwidth is defined for the size of the channels. In the channelizer case, the highest rate is defined as the rate of the incoming signal, but in other PFB blocks, this is not so obvious.

Two very useful blocks to use are the arbitrary resampler and the clock synchronizer (for PAM signals). These PFBs are defined with a set number of filters based on the fidelity required from them, not the rate changes. By default, the filter_size is set to 32 for these blocks, which is a reasonable default for most tasks. Because the PFB uses this number of filters in the filterbank, the maximum rate of the bank is defined from this (see the theory of a polyphase interpolator for a justification of this). So the prototype filter is defined to use a sample rate of filter_size times the signal's sampling rate.

A helpful wrapper for the arbitrary resampler is found in gr-filter/python/pfb.py, which is exposed in Python as filter.pfb.arb_resampler_ccf and filter.pfb.arb_resampler_fff. This block is set up so that the user only needs to pass it the real number rate as the resampling rate. With just this information, this hierarchical block automatically creates a filter that fully passes the signal bandwidth being resampled but does not pass any out-of-band noise. See the code for this block for details of how the filter is constructed.

Of course, a user can create his or her own taps and use them in the arbitrary resampler for more specific requirements. Some of the UHD examples (gr-uhd/examples) use this ability to create a received matched filter or channel filter that also resamples the signal.

The PFB Arbitrary Resampler Kernel
GNU Radio has a PFB arbitrary resampler block that can be used to resample a signal to any arbitrary and real resampling rate. The resampling feature is one that could easily be useful to other blocks, and so we have extracted the kernel of the resampler into its own class that can be used as such.

The PFB arbitrary resampler is defined in pfb_arb_resampler.h and has the following constructor:

namespace gr { namespace filter { namespace kernel { pfb_arb_resampler_XXX(float rate,                              const std::vector &taps,                               unsigned int filter_size); } /* namespace kernel */ } /* namespace filter */ } /* namespace gr */

Currently, only a 'ccf' and 'fff' version are defined. This kernel, like the block itself, takes in the resampling rate as a floating point number. The taps are passed as the baseband prototype filter, and the quantization error of the filter is determined by the filter_size parameter.

The prototype taps are generated like all other PFB filter taps. Specifically, we construct them generally as a lowpass filter at the maximum rate of the filter. In the case of these resamplers, the maximum rate is actually the number of filters.

A simple example follows. We construct a filter that will pass the entire passband of the original signal to be resampled. To make it easy, we work in normalized sample rates for this. The gain of the filter is set to filter_size to compensate for the upsampling, the sampling rate itself is also set to filter_size, which is assuming that the incoming signal is at a sampling rate of 1.0. We defined the passband to be 0.5 to pass the entire width of the original signal and set a transition band to 0.1. Note that this causes a bit of roll-off outside of the original passband and could lead to introducing some aliasing. More care should be taken to construct the passband and transition width of the filter for the given signal while keeping the total number of taps small. A stopband attenuation of 60 dB was used here, and again, this is a parameter we can adjust to alter the performance and size of the filter.

firdes.low_pass_2(filter_size, filter_size, 0.5, 0.1, 60)

As is typical with the PFB filters, a filter size of 32 is generally an appropriate trade-off of accuracy, performance, and memory. This should provide an error roughly equivalent to the quanization error of using 16-bit fixed point representation. Generally, increasing over 32 provides some accuracy benefits without a huge increase in computational demands.