DFT Filter Banks

Steven Liddell

Prof. Justin Jonas

Channelization

• A common task in radio astronomy is the channelization of a signal onto separate frequency channels.

• The output signal has a decreased bandwidth so the output sample rate can be decrease multirate systems.

Why Channelise a signal?

• Allow computation to be performed on a narrower bandwidth and in parallel.

• Implement the F in an FX correlator.• RFI mitigation.• Spectrum analysis.• Pulsar dedispersion

How to Channelize a Signal

• Analogue filter banks.• Unstable; Would rather use digital signals.

• Fast Fourier Transform.• Fast; Not a great frequency response.

• Digital filter banks• More computation required; Get a good response.• Discrete Fourier Transform (DFT) filter banks.

FFT vs Filterbanks

•FFT has a higher processing loss => decreases the instruments sensitivity.

Computational Costs

≈N/2 log2(M) MACs M × N MACs

DFT Filter Banks

• DFT filter banks arise by modifying the FFT’s windowing function to provide channels with improved stop band attenuation and a narrower transition width.

• The modified window is based on a prototype filter which lends its frequency response to each channel.

• Two architectures of DFT looked at.

DFT Filter Banks

Polyphase Filter BankWeighted Overlap Add Filter

Bank

≈Mlog2(M)+N MACs

The Polyphase Filter Bank• Replace a FFT’s window with a set of

polyphase filters.• Create polyphase filters from a prototype filter:

Prototype filter

Polyphase filters (pρ(n))

Prototype filter copied onto each channel.

Aliasing

Critically sampled(output data rate 1/16 input data rate)

Over Sampled(output data rate >1/16 input data rate)

Wola Filter Bank

• The Weighted Overlap and Add filter bank.• Mathematically identical to polyphase filter.• Implementation different decouple number of

channels from sample rate change factor.

WOLA Filter Bank

• Weighted Overlap Add:

• Fixed point arithmetic leads to a errors in the system.• Quantization error can be modelled as noise injected at

a multiplier.

• Error occurs in both the FIR and FFT so need to balance the number of bits.

Fixed point error in the filter coefficients change the channels’ frequency response.

• Efficient through use of FFT but with good frequency response.

• Easily implemented in parallel hardware.• Inherent sample rate change.• Replacing the stand alone FFT in signal paths requiring

high accuracy channelization.