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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 Dept. Of Electrical and Computer Engineering University of Florida, Gainesville, FL 32611, USA. Outline. Motivation – QR technology for landmine detection Temporally uncorrelated interference model

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

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  1. 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 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 MS Thesis

  3. 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) MS Thesis

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

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

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

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

  10. ML Estimate • Maximum likelihood estimate • The only difference MS Thesis

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

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

  13. 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 MS Thesis

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

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

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

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

  19. Properties of Capon (2) • is uncorrelated with MS Thesis

  20. Properties of Capon (3) • is of beta distribution MS Thesis

  21. Numerical Example Empirical results obtained through 10000 trials MS Thesis

  22. Numerical Example Estimates based on real data 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 MS Thesis

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

  25. Unbiased Capon • Bias of Capon is known • Modify Capon to be unbiased MS Thesis

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

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

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

  30. Alternative LS • Steps • Obtain initial estimate by model mismatched ML (M3L) • Estimate parameters of AR process MS Thesis

  31. Alternative LS multichannel Prony estimate • Whiten data in time domain • Obtain improved estimate of based on • Go back to (2) and iterate until converge, i.e., MS Thesis

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

  33. Step (4) of ALS MS Thesis

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

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

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

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

  38. Numerical Example constant signal MS Thesis

  39. Numerical Example constant signal MS Thesis

  40. Numerical Example BPSK signal MS Thesis

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