yi jiang ms thesis 1 yi jiang dept. of electrical and computer engineering university of florida,...

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