MS Thesis 1Yi Jiang

Yi Jiang

Dept. Of Electrical and Computer Engineering University of Florida,

Gainesville, FL 32611, USA

Array Signal Processing in the Know Waveform and Steering

Vector Case

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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

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

Quadrupole Resonance -- a promising technology for explosive detection.

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)

Combine all three steps together

Filter in temporal domain (step 3)

(signal waveform power)

correlation between receiveddata and signal waveform

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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)

Unbiased

Lemma 1

Key for statistical performance analyses

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

1) Obtain initial estimate by model mismatched ML (M3L)

2) Estimate parameters of AR process

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Alternative LS

multichannel Prony estimate

4) Obtain improved estimate of based on

3) Whiten data in time domain

5) 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