Transcript Optimizing digital filter banks

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 Bank
Weighted 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.