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DFT Filter Banks
Steven Liddell
Prof. Justin Jonas
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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.
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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
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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.
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FFT vs Filterbanks
•FFT has a higher processing loss => decreases the instruments sensitivity.
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Computational Costs
≈N/2 log2(M) MACs M × N MACs
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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.
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DFT Filter Banks
Polyphase Filter BankWeighted Overlap Add Filter
Bank
≈Mlog2(M)+N MACs
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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))
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Prototype filter copied onto each channel.
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Aliasing
Critically sampled(output data rate 1/16 input data rate)
Over Sampled(output data rate >1/16 input data rate)
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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.
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WOLA Filter Bank
• Weighted Overlap Add:
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• 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.
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Fixed point error in the filter coefficients change the channels’ frequency response.
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• 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.