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 1)INTRODUCTION 2)OPTIMAL SECOND-ORDER ESTIMATION LARGE-ERROR SMALL-ERROR 3)QUADRATIC EXTENDED KALMAN FILTER 4)SOME ASYMPTOTIC RESULTS 5)CONCLUSIONS

3 OUTLINE 1)INTRODUCTION 2)OPTIMAL SECOND-ORDER ESTIMATION LARGE-ERROR SMALL-ERROR 3)QUADRATIC EXTENDED KALMAN FILTER 4)SOME ASYMPTOTIC RESULTS 5)CONCLUSIONS

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

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

6 ESTIMATION PERFORMANCE BAYESIAN CASE : PRIORLIKELIHOOD DEPENDS ON AND ESTIMATION ERROR SELF-NOISE MEASUREMENT NOISE

7 CLASSICAL ESTIMATION CRITERIA 1)MMSE: 2)MVU: 3)ML: GENERALLY, NOT REALIZABLE !! ML MVU MMSE small-error OPTIMALITY : DIFFICULT !!

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

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

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

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

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

13 OUTLINE 1)INTRODUCTION 2)OPTIMAL SECOND-ORDER ESTIMATION LARGE-ERROR SMALL-ERROR 3)QUADRATIC EXTENDED KALMAN FILTER 4)SOME ASYMPTOTIC RESULTS 5)CONCLUSIONS

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

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

16 M OPTIMIZATION: GEOMETRIC INTERPRETATION (min MSE) (min VAR) (min BIAS 2 ) MVMB MMSE

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

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

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

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

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

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

23 OUTLINE 1)INTRODUCTION 2)OPTIMAL SECOND-ORDER ESTIMATION LARGE-ERROR SMALL-ERROR 3)QUADRATIC EXTENDED KALMAN FILTER 4)SOME ASYMPTOTIC RESULTS 5)CONCLUSIONS

24 DELTA MEASURE NOT INFORMATIVEVERY INFORMATIVE LARGE-ERRORSMALL-ERROR

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 (N SS =2) K=12 NUISANCE PARAM. 2REC MODULATION M=16 OBSERVATIONS (N SS =4) K=12 NUISANCE PARAM.

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

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

31 OUTLINE 1)INTRODUCTION 2)OPTIMAL SECOND-ORDER ESTIMATION LARGE-ERROR SMALL-ERROR 3)QUADRATIC EXTENDED KALMAN FILTER 4)SOME ASYMPTOTIC RESULTS 5)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) MEASUREMENT EQUATION STATE EQUATION LINEAR GAUSSIAN LINEAR GAUSSIAN MVU BAYESIAN MMSE

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

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

35 OUTLINE 1)INTRODUCTION 2)OPTIMAL SECOND-ORDER ESTIMATION LARGE-ERROR SMALL-ERROR 3)QUADRATIC EXTENDED KALMAN FILTER 4)SOME ASYMPTOTIC RESULTS 5)CONCLUSIONS

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

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

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

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

40 OUTLINE 1)INTRODUCTION 2)OPTIMAL SECOND-ORDER ESTIMATION LARGE-ERROR SMALL-ERROR 3)QUADRATIC EXTENDED KALMAN FILTER 4)SOME ASYMPTOTIC RESULTS 5)CONCLUSIONS

41 CONCLUSIONS 1.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.) 2.IN THAT CASE, SECOND-ORDER ESTIMATORS CAN EXPLOIT THE 4TH ORDER INFO. ON THE NUISANCE PARAMETERS  KURTOSIS MATRIX K

42 FURTHER RESEARCH 1. IN MULTIUSER ESTIMATION PROBLEMS… CONSTANT MODULUS PROPERTY STATISTICAL DEPENDENCE IN CODED TRANSMISSIONS 2. ACQUISITION OPTIMIZATION 3. ESTIMATION AND DETECTION THEORY CONNECTION