MAMI for Asimolar

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Wideband Source Estimation Solutions to Benchmark
Problem 1
Yao Xie, Jian Li Department of Electrical and
Computer Engineering University of
Florida xieyao_at_dsp.ufl.edu
5-7-2006
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Outline

• Introduction
• Frequency Domain Approaches
• Step 1 Divide into frequency bins
• Step 2 Array beamforming in each bin
• Conclusions

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Problem
20 inch

• Given
• 8 element array at known location
• 2 independent sources at known location
• Sources on a plane z30 inch
• Synthetic array data
• Sampled at 22 KHz
• 132096 samples (6.004 sec.)
• Challenge
• Determine individual frequency spectra of the two
  sources

20 inch
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Time Domain Signal Model

• For a focal point

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Spectra of Raw Data

• Spectra of raw data
• Recorded by the first sensor

Using APES (more later)
Using DFT/FFT
Frequency range 1.3 KHz-4 KHz
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Outline

• Introduction
• Frequency Domain Approaches
• Step 1 Divide into frequency bins
• Discrete Fourier Transform (DFT)
• Filter Bank APES
• Step 2 Array beamforming in each bin
• Conclusions

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Frequency Bins

• Divide data sequences into L non-overlapping
  subsequences
• Use DFT(FFT) or APES
• to convert sub-sequences into frequency domain
• All frequency samples at k ? frequency bin k
  (e.g. the blue blocks consist of the first bin)

M 10 L 100 K 1320 (16.7 Hz/bin)
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APES

• Amplitude and Phase Estimation Method (APES)
• For the m-th subsequence
• Design a p-th order filter to pass a
  frequency component and attenuate other
  components
• Solution

where
where
Li et.al. Efficient mixed-spectrum estimation
with applications to target feature extraction,''
IEEE TSP, Feb. 1996.
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Frequency Signal Model

• Signal Model for frequency bins

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Array Beamforming in Each Bin

• Goal
• Estimate power spectral density in each
  frequency bin
• Signal in each frequency bin
• Narrowband beamforming methods applicable

where is the bandwidth of each frequency bin
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Outline

• Introduction
• Frequency Domain Approaches
• Step 1 Divide into frequency bins
• Step 2 Array beamforming in each bin
• Delay and Sum (DAS)
• Robust Capon Beamforming (RCB)
• Wideband RELAX
• Conclusions

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Delay and Sum (DAS)

• For two sources
• Amplitude estimate
• Beamwidth of DAS beamformer varies with frequency
• At high frequency
• Beamwidth smaller
• higher sidelobe
• At low Frequency
• Beamwidth larger
• lower sidelobe

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DAS

• Fortunately, we do not have any problem

At higher end of the frequency band, sidelobes
dont interfere with the other
At lower end of the frequency band, beam at two
sources dont smear
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DAS
DFT based
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DAS
APES based
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Standard Capon Beamforming

• Find beamforming weight vector by solving
• Pass the signal of interest without distortion
• Minimize the total output power
• where
• Solution
• Drawback Performance degrades significantly when
  the steering vector is imprecise

Capon, High-resolution frequency-wavenumber
spectrum analysis, 1969
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Robust Capon Beamforming (RCB)

• Robust Capon Beamforming
• Robust version of Capon
• Allow true steering vector to be within an
  uncertainty set around the assumed steering
  vector
• Solve an optimization problem to find
• Use estimated true steering vector to obtain
  Capon weight vector
• Estimate waveform
• Obtain frequency components by DFT (optional)

Li. et.al., On robust Capon beamforming and
diagonal loading, IEEE TSP, 2003.
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RCB

• Estimate the true steering vector
• The estimated true steering vector
• Weight vector
• Waveform estimation

Li. et.al., On robust Capon beamforming and
diagonal loading, IEEE TSP, 2003, 51
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RCB
DFT based
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RCB
APES based
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Wideband RELAX (WRELAX)

• Iteratively minimize the cost function
• Target locations are known? modify WRELAX a
  little bit

Assume only one source
Assume two sources
If change of is larger than
Assume two sources
Wang. et.al., Wideband RELAX and wideband
CLEAN for aeroacoustic imaging,JASA, 2004, Feb.
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WRELAX
DFT based
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WRELAX
APES based
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Conclusions

• We have tried frequency domain approaches
• DAS
• RCB
• WRELAX
• Frequency domain approach yields smoother
  spectra estimation
• When obtain frequency bins, APES is better than
  DFT in estimation accuracy
• RCB and Wideband RELAX method have the best
  result
• AR model reduces parameter number and yields
  smooth spectra

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Thank You!