Yi Jiang

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Array Signal Processing in the Know Waveform and
Steering Vector Case

• Yi Jiang
• Dept. Of Electrical and Computer Engineering
• University of Florida, Gainesville, FL 32611,
  USA

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Outline

• Motivation QR technology for landmine detection
• Temporally uncorrelated interference model
• Maximum likelihood estimate
• Capon estimate
• Statistical performance analysis
• Numerical examples
• Temporally correlated interference and noise
• Alternative Least Squares method
• Numerical examples

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Motivation

• Quadrupole Resonance -- a promising technology
  for explosive detection.

• Characteristic response of N-14 in the TNT is a
  known-waveform signal up to an unknown scalar.

• Challenge -- strong radio frequency interference
  (RFI)

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Motivation

• Main antenna receives QR signal plus RFI
• Reference antennas receive RFI only

• Signal steering vector known

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Motivation

• Both spatial and temporal information available
  for interference suppression

• Signal estimation mandatory for detection

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Related Work

• DOA estimation for known-waveform signals
• Li, et al, 1995, Zeira, et al, 1996,
  Cedervall, et al, 1997 Swindlehurst, 1998,
  etc.
• Temporal information helps improve
• Estimation accuracy
• Interference suppression capability
• Spatial resolution

• Exploiting both temporal and spatial information
  for interference suppression and signal parameter
  estimation not fully investigated yet

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Problem Formulation

• Simple Data model
• Conditions
• Array steering vector known with no error
• Signal waveform known with no
  error
• Noise vectors i.i.d.
• Task
• To estimate signal complex-valued amplitude

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Capon Estimate (1)

• Find a spatial filter (step 1)

• Filter in spatial domain (step 2)

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Capon Estimate (2)

• Filter in temporal domain (step 3)

• Combine all three steps together

correlation between received data and signal
waveform
(signal waveform power)
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ML Estimate

• Maximum likelihood estimate

• The only difference

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R vs. T
annoying cross terms
ML removes cross terms by using temporal
information
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Cramer-Rao Bound

• Cramer-Rao Bound (CRB) ---- the best possible
  performance bound for any unbiased estimator

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Properties of ML (1)

• Lemma 1

Key for statistical performance analyses

• Unbiased

• is of complex Wishart distribution
• Wishart distribution is a generalization of
  chi-square distribution

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Properties of ML (2)

• Mean-Squared Error

Define
Fortunately is of Beta distribution
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Properties of ML (3)

• Remarks
• ML is always greater than CRB (as expected)
• ML is asymptotically efficient for large snapshot
  number
• ML is NOT asymptotically efficient for high SNR

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Numerical Example
Threshold effect
ML estimate is asymptotically efficient for large
L
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Numerical Example
ML estimate is NOT asymptotically efficient for
high SNR No threshold effect
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Properties of Capon (1)

• Recall

• Find more about their relationship

(Matrix Inversion Lemma)
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Properties of Capon (2)

• is uncorrelated with

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Properties of Capon (3)

• is of beta distribution

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Numerical Example
Empirical results obtained through 10000 trials
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Numerical Example
Estimates based on real data
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Numerical Example
Capon can has even smaller MSE than unbiased CRB
for low SNR Error floor exists for Capon for
high SNR
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Numerical Example
Capon is asymptotically efficient for large
snapshot number
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Unbiased Capon

• Bias of Capon is known

• Modify Capon to be unbiased

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Numerical Example
Unbiased Capon converges to CRB faster than
biased Capon
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Numerical Example
Unbiased Capon has lower error floor than biased
Capon for high SNR
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New Data Model

• Improved data model
• Model interference and noise as AR process

i.i.d.

• Define

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New Feature

• Potential gain improvement of interference
  suppression by exploiting temporal correlation of
  interference

• Difficulty too much parameters to estimate
• Minimize

w.r.t
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Alternative LS

• Steps
• Obtain initial estimate by model mismatched ML
  (M3L)

1. Estimate parameters of AR process

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Alternative LS
multichannel Prony estimate

1. Whiten data in time domain

1. Obtain improved estimate of based on

1. Go back to (2) and iterate until converge, i.e.,

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Step (4) of ALS

• Two cases
• Damped/undamped sinusoid
• Let

• Arbitrary signal
• Let

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Step (4) of ALS
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Step (4) of ALS

• Lemma.
• For large data sample, minimizing

is asymptotically equivalent to minimizing

• Base on the Lemma.

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Discussion

• ALS always yields more likely estimate than SML

• Order of AR can be estimated via general
  Akaike information criterion (GAIC)

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Numerical Example

• Generate AR(2) random process

decides spatial correlation decides temporal
correlation Decides spectral peak location
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Numerical Example
constant signal SNR -10 dB
Only one local minimum around
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Numerical Example
constant signal
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Numerical Example
constant signal
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Numerical Example
BPSK signal