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 log 2 (M) MACsM × 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 Bank Weighted Overlap Add Filter Bank ≈Mlog 2 (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 (p ρ (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 WeightedOverlapAdd :

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.