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SAMPLE COVARIANCE BASED PARAMETER ESTIMATION FOR DIGITAL COMMUNICATIONS

This document provides a comprehensive analysis of covariance-based parameter estimation techniques tailored for digital communications. It covers optimal second-order estimation, distinguishing between large-error and small-error scenarios. The use of Quadratic Extended Kalman Filters is discussed, alongside key asymptotic results. Key concepts such as likelihood estimation, Bayesian approaches, and classical criteria (MMSE, MVU, ML) are analyzed to enhance estimation performance. The study emphasizes the challenges of non-Gaussian noise and the optimization of estimators in communication systems.

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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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