Filter Bank - FFT Filter Banks

FFT Filter Banks

A bank of receivers can be created by performing a sequence of FFTs on overlapping segments of the input data stream. A weighting function (aka window function) is applied to each segment to control the shape of the frequency responses of the filters. The wider the shape, the more often the FFTs have to be done to satisfy the Nyquist sampling criteria (which is what distinguishes a filter bank from a spectrum analyzer). For a fixed segment length, the amount of overlap determines how often the FFTs are done (and vice versa). Also, the wider the shape of the filters, the fewer filters that are needed to span the input bandwidth. Eliminating unnecessary filters (i.e. decimation in frequency) is efficiently done by treating each weighted segment as a sequence of smaller blocks, and the FFT is performed on only the sum of the blocks. This has been referred to as multi-block windowing and weighted pre-sum FFT (see Sampling the DTFT).

A special case occurs when, by design, the length of the blocks is an integer multiple of the interval between FFTs. Then the FFT filter bank can be described in terms of one or more polyphase filter structures where the phases are recombined by an FFT instead of a simple summation. The number of blocks per segment is the impulse response length (or depth) of each filter. The computational efficiencies of the FFT and polyphase structures, on a general purpose processor, are identical.

Synthesis (i.e. recombining the outputs of multiple receivers) is basically a matter of resampling each one at a rate commensurate with the total bandwidth to be created, translating each channel to its new center frequency, and summing the streams of samples. Resampling is an interpolation process, equivalent to inserting multiple zero-valued samples in-between the actual ones and passing the new sequence through a lowpass filter, called the synthesis filter. The net frequency response of each channel is the product of the synthesis filter with the frequency response of the filter bank (analysis filter). Ideally, the frequency responses of adjacent channels add to a constant value at every frequency between the channel centers. That condition is known as perfect reconstruction.