Yi Jiang MS Thesis 1 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

Yi Jiang MS Thesis 2 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

Yi Jiang MS Thesis 3 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)

Yi Jiang MS Thesis 4 Motivation Main antenna receives QR signal plus RFI Reference antennas receive RFI only Signal steering vector known

Yi Jiang MS Thesis 5 Motivation Both spatial and temporal information available for interference suppression Signal estimation mandatory for detection

Yi Jiang MS Thesis 6 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

Yi Jiang MS Thesis 7 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

Yi Jiang MS Thesis 8 Capon Estimate (1) Find a spatial filter (step 1) Filter in spatial domain (step 2)

Yi Jiang MS Thesis 9 Capon Estimate (2) Combine all three steps together Filter in temporal domain (step 3) (signal waveform power) correlation between received data and signal waveform

Yi Jiang MS Thesis 10 ML Estimate Maximum likelihood estimate The only difference

Yi Jiang MS Thesis 11 R vs. T annoying cross terms ML removes cross terms by using temporal information

Yi Jiang MS Thesis 12 Cramer-Rao Bound Cramer-Rao Bound (CRB) ---- the best possible performance bound for any unbiased estimator

Yi Jiang MS Thesis 13 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

Yi Jiang MS Thesis 14 Properties of ML (2) Mean-Squared Error Define Fortunately is of Beta distribution

Yi Jiang MS Thesis 15 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

Yi Jiang MS Thesis 16 Numerical Example Threshold effect ML estimate is asymptotically efficient for large L

Yi Jiang MS Thesis 17 Numerical Example ML estimate is NOT asymptotically efficient for high SNR No threshold effect

Yi Jiang MS Thesis 18 Properties of Capon (1) Recall Find more about their relationship (Matrix Inversion Lemma)

Yi Jiang MS Thesis 19 Properties of Capon (2) is uncorrelated with

Yi Jiang MS Thesis 20 Properties of Capon (3) is of beta distribution

Yi Jiang MS Thesis 21 Numerical Example Empirical results obtained through trials

Yi Jiang MS Thesis 22 Numerical Example Estimates based on real data

Yi Jiang MS Thesis 23 Numerical Example Capon can has even smaller MSE than unbiased CRB for low SNR Error floor exists for Capon for high SNR

Yi Jiang MS Thesis 24 Numerical Example Capon is asymptotically efficient for large snapshot number

Yi Jiang MS Thesis 25 Unbiased Capon Bias of Capon is known Modify Capon to be unbiased

Yi Jiang MS Thesis 26 Numerical Example Unbiased Capon converges to CRB faster than biased Capon

Yi Jiang MS Thesis 27 Numerical Example Unbiased Capon has lower error floor than biased Capon for high SNR

Yi Jiang MS Thesis 28 New Data Model Improved data model Model interference and noise as AR process i.i.d. Define

Yi Jiang MS Thesis 29 New Feature Potential gain – improvement of interference suppression by exploiting temporal correlation of interference Difficulty – too much parameters to estimate  Minimize w.r.t

Yi Jiang MS Thesis 30 Alternative LS Steps 1)Obtain initial estimate by model mismatched ML (M3L) 2)Estimate parameters of AR process

Yi Jiang MS Thesis 31 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.,

Yi Jiang MS Thesis 32 Step (4) of ALS Two cases: Damped/undamped sinusoid Let Arbitrary signal Let

Yi Jiang MS Thesis 33 Step (4) of ALS

Yi Jiang MS Thesis 34 Step (4) of ALS Lemma. For large data sample, minimizing is asymptotically equivalent to minimizing Base on the Lemma.

Yi Jiang MS Thesis 35 Discussion ALS always yields more likely estimate than SML Order of AR can be estimated via general Akaike information criterion (GAIC)

Yi Jiang MS Thesis 36 Numerical Example Generate AR(2) random process decides spatial correlation decides temporal correlation Decides spectral peak location

Yi Jiang MS Thesis 37 Numerical Example constant signal SNR = -10 dB Only one local minimum around

Yi Jiang MS Thesis 38 Numerical Example constant signal

Yi Jiang MS Thesis 39 Numerical Example constant signal

Yi Jiang MS Thesis 40 Numerical Example BPSK signal