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SAMPLE COVARIANCE BASED PARAMETER ESTIMATION FOR DIGITAL COMMUNICATIONS

SAMPLE COVARIANCE BASED PARAMETER ESTIMATION FOR DIGITAL COMMUNICATIONS. Javier Villares Piera Advisor: Gregori Vázquez Grau Signal Processing for Communications Group Dept. of Signal Processing and Communications Technical University of Catalunya (UPC). OUTLINE. INTRODUCTION

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SAMPLE COVARIANCE BASED PARAMETER ESTIMATION FOR DIGITAL COMMUNICATIONS

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  1. SAMPLE COVARIANCE BASED PARAMETER ESTIMATION FOR DIGITAL COMMUNICATIONS Javier Villares Piera Advisor: Gregori Vázquez Grau Signal Processing for Communications Group Dept. of Signal Processing and Communications Technical University of Catalunya (UPC)

  2. OUTLINE • INTRODUCTION • OPTIMAL SECOND-ORDER ESTIMATION • LARGE-ERROR • SMALL-ERROR • QUADRATIC EXTENDED KALMAN FILTER • SOME ASYMPTOTIC RESULTS • CONCLUSIONS

  3. OUTLINE • INTRODUCTION • OPTIMAL SECOND-ORDER ESTIMATION • LARGE-ERROR • SMALL-ERROR • QUADRATIC EXTENDED KALMAN FILTER • SOME ASYMPTOTIC RESULTS • CONCLUSIONS

  4. PROBLEM STATEMENT PARAMETERS OBSERVATION ADDITIVE GAUSSIAN NOISE MULTIPLICATIVE NON-GAUSSIAN NOISE ESTIMATE FROM KNOWING STATISTICS ON AND THE PARAMETERIZATION OF THE PROBLEM

  5. ESTIMATION PERFORMANCE SELF-NOISE DEPENDS ON AND MEASUREMENT NOISE ESTIMATION ERROR DETERMINISTIC CASE : LIKELIHOOD

  6. ESTIMATION PERFORMANCE SELF-NOISE DEPENDS ON AND MEASUREMENT NOISE ESTIMATION ERROR BAYESIAN CASE : LIKELIHOOD PRIOR

  7. CLASSICAL ESTIMATION CRITERIA • MMSE: • MVU: • ML: GENERALLY, NOT REALIZABLE !! DIFFICULT !! OPTIMALITY : ML MVU MMSE small-error small-error

  8. SMALL-ERROR VS. LARGE-ERROR THRESHOLD ML OBSERVATION LENGTH INCREASES CRB SNR LARGE-ERROR SMALL-ERROR BAYESIAN ESTIMATORS DETERMINISTIC ESTIMATORS (ML = MVU = MMSE  CRB)

  9. ESTIMATION WITH NUISANCE UNKNOWNS NUISANCE PARAMETERS ? UNCONDITIONAL LIKELIHOOD CONDITIONAL LIKELIHOOD • CML x CONTINUOUS, • DETERMINISTIC • Low-SNR UML • GML • x GAUSSIAN

  10. ESTIMATION WITH NUISANCE UNKNOWNS NUISANCE PARAMETERS ? UNCONDITIONAL LIKELIHOOD CONDITIONAL LIKELIHOOD QUADRATIC • CML x CONTINUOUS, • DETERMINISTIC • Low-SNR UML • GML • x GAUSSIAN

  11. QUADRATIC ML-BASED ESTIMATORS COMPARISON CML GML MCRB (x known) Low-SNR UML Higher -order

  12. GAUSSIAN ASSUMPTION IN COMMUNICATIONS ? -1 1 -1 1 BPSK alphabet (higher-order info) Gaussian assumption (mean and variance info)

  13. OUTLINE • INTRODUCTION • OPTIMAL SECOND-ORDER ESTIMATION • LARGE-ERROR • SMALL-ERROR • QUADRATIC EXTENDED KALMAN FILTER • SOME ASYMPTOTIC RESULTS • CONCLUSIONS

  14. SECOND-ORDER ESTIMATOR ESTIMATOR COEFFICIENTS ? WITH SAMPLE COVARIANCE VECTOR

  15. ESTIMATOR OPTIMIZATION • OPTIMUM b: • OPTIMUM M: TRADE-OFF 1) 2) 3) MMSE MVU MVMB

  16. M OPTIMIZATION: GEOMETRIC INTERPRETATION (min MSE) MMSE MVMB (min VAR) (min BIAS2)

  17. BIAS MINIMIZATION UNBIASED • SIMULATION PARAMETERS • FREQ. ESTIMATION • 2 MSK SYMBOLS • NSS = 2 Max. Freq. Error = 1 Max. Freq. Error = 0.5

  18. VARIANCE ANALYSIS WITH COVARIANCE MATRIX OF FOURTH-ORDER MOMENTS OF y

  19. MATRIX Q() NON-GAUSSIAN INFORMATION WITH IF x GAUSSIAN !! 4TH ORDER CUMULANTS (KURTOSIS MATRIX)

  20. KURTOSIS MATRIX IF x IS CIRCULAR WITH M-PSK 16-QAM 4TH TO 2ND ORDER RATIO 64-QAM GAUSSIAN

  21. QUADRATIC ESTIMATORS COMPARISON MVMB • SIMULATION PARAMETERS • FREQ. ESTIMATION • UNIFORM PRIOR (80% Nyq) • 4 MSK SYMBOLS • NSS= 2 Prior variance MMSE Self-noise min{BIAS2}

  22. ASYMPTOTIC ANALYSIS • SIMULATION PARAMETERS • FREQ. ESTIMATION • UNIFORM PRIOR (80% Nyq) • Es/No = 40dB • MSK modulation • NSS = 2 MVMB MMSE (# of samples)

  23. OUTLINE • INTRODUCTION • OPTIMAL SECOND-ORDER ESTIMATION • LARGE-ERROR • SMALL-ERROR • QUADRATIC EXTENDED KALMAN FILTER • SOME ASYMPTOTIC RESULTS • CONCLUSIONS

  24. LARGE-ERROR SMALL-ERROR NOT INFORMATIVE VERY INFORMATIVE DELTA MEASURE

  25. CLOSED-LOOP ESTIMATION AND TRACKING DISCRIMINATOR or DETECTOR LOOP FILTER SMALL-ERROR (STEADY-STATE)

  26. BIAS MINIMIZATION (SMALL-ERROR) UNBIASED

  27. BEST QUADRATIC UNBIASED ESTIMATOR (BQUE) AND WE OBTAIN THAT 2nd-ORDER FIM LOWER BOUND ON THE VARIANCE OF ANY SECOND-ORDER UNBIASED ESTIMATOR

  28. FREQUENCY ESTIMATION PROBLEM • 2REC MODULATION • M=8 OBSERVATIONS (NSS=2) • K=12 NUISANCE PARAM. • 2REC MODULATION • M=16 OBSERVATIONS (NSS=4) • K=12 NUISANCE PARAM.

  29. CHANNEL ESTIMATION PROBLEM • SIMULATION PARAMETERS • CIR LENGTH 3 SYMB • 100 GAUSSIAN CHANNELS • ROLL-OFF = 0.35 • NSS = 3 • OBS. TIME = 100 SYMB. CONSTANT MODULUS

  30. ANGLE-OF-ARRIVAL ESTIMATION PROBLEM SEPARATION 10º SEPARATION 1º • M-PSK MODULATION • 4 ANTENNA • OBS. TIME = 400 SYMB • M-PSK MODULATION • 4 ANTENNA • OBS. TIME  3000 SYMB

  31. OUTLINE • INTRODUCTION • OPTIMAL SECOND-ORDER ESTIMATION • LARGE-ERROR • SMALL-ERROR • QUADRATIC EXTENDED KALMAN FILTER • SOME ASYMPTOTIC RESULTS • CONCLUSIONS

  32. KALMAN FILTER MOTIVATION CLOSED-LOOP ESTIMATOR - OPTIMUM IN THE STEADY-STATE (SMALL-ERROR) KALMAN FILTER - OPTIMUM IN THE STEADY-STATE (SMALL-ERROR) - OPTIMUM IN ACQUISITION (LARGE-ERROR) MVU BAYESIAN MMSE MEASUREMENT EQUATION LINEAR GAUSSIAN STATE EQUATION LINEAR GAUSSIAN

  33. KALMAN FILTER FORMULATION MEASUREMENT EQUATION ZERO-MEAN NONLINEAR IN  STATE EQUATION NONLINEAR IN  ZERO-MEAN PROBLEM QUADRATIC OBSERVATION SAMPLE COV. VECTOR - NON-GAUSSIAN - DEPENDS ON  NONLINEAR PROBLEM LINEARIZATION (EKF FORMULATION)

  34. ACQUSITION RESULTS • SIMULATION PARAMETERS • M-PSK MODULATION • SNR = 40 dB • 4 ANTENNAS SEPARATION = 0.2 SEPARATION = 0.4

  35. OUTLINE • INTRODUCTION • OPTIMAL SECOND-ORDER ESTIMATION • LARGE-ERROR • SMALL-ERROR • QUADRATIC EXTENDED KALMAN FILTER • SOME ASYMPTOTIC RESULTS • CONCLUSIONS

  36. LOW AND HIGH SNR STUDY: DOA 16-QAM (MULTILEVEL) M-PSK (CONSTANT MODULUS) • SEPARATION 1º • M = 4 ANTENNAS • SMALL-ERROR • SEPARATION 1º • M = 4 ANTENNAS • SMALL-ERROR

  37. LARGE SAMPLE STUDY: DIGITAL COMMUNICATIONS FREQUENCY SYNCHRO. TIMING SYNCHRO. • M-PSK • NSS = 2 • ROLL-OFF = 0.75 • M-PSK • NSS = 2 • ROLL-OFF = 0.75

  38. LARGE SAMPLE RESULTS: DOA SEPARATION 10º SEPARATION 1º • M = 4 ANTENNAS • SMALL-ERROR • M = 4 ANTENNAS • SMALL-ERROR

  39. LARGE SAMPLE RESULTS: DOA SEPARATION 10º SEPARATION 1º • M-PSK ( = 1) • EsNo = 60dB • SMALL-ERROR • M-PSK ( = 1) • EsNo = 60dB • SMALL-ERROR

  40. OUTLINE • INTRODUCTION • OPTIMAL SECOND-ORDER ESTIMATION • LARGE-ERROR • SMALL-ERROR • QUADRATIC EXTENDED KALMAN FILTER • SOME ASYMPTOTIC RESULTS • CONCLUSIONS

  41. CONCLUSIONS • IN SECOND-ORDER ESTIMATION, THE GAUSSIAN ASSUMPTION DOES NOT APPLY FOR • MEDIUM SNR • HIGH SNR WITH CONSTANT MODULUS NUISANCE UNKNOWNS, • IF THE OBSERVED VECTOR IS SHORT IN THE PARAMETER DIMENSION (DOA vs. FREQ.) • IN THAT CASE, SECOND-ORDER ESTIMATORS CAN EXPLOIT THE • 4TH ORDER INFO. ON THE NUISANCE PARAMETERS •  KURTOSIS MATRIX K

  42. FURTHER RESEARCH • IN MULTIUSER ESTIMATION PROBLEMS… • CONSTANT MODULUS PROPERTY • STATISTICAL DEPENDENCE IN CODED TRANSMISSIONS • ACQUISITION OPTIMIZATION • ESTIMATION AND DETECTION THEORY CONNECTION

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