TRACKING AND KALMAN FILTERING MADE EASY

By ANTHONY DAVIS

CONTENTS PART I TRACKING, PREDICTION, AND SMOOTHING BASICS G–H AND G–H–K FILTERS WHY TRACKING AND PREDICTION ARE NEEDED IN A RADAR G–H FILTERS SIMPLE HERUISTIC DERIVATION OF G–H TRACKING AND PREDICTION EQUATIONS – FILTER OTHER SPECIAL TYPES OF FILTERS IMPORTANT PROPERTIES OF G–H TRACKING FILTERS STEADY-STATE PERFORMANCE FOR CONSTANT- VELOCITY TARGET FOR WHAT CONDITIONS IS THE CONSTANT- VELOCITY ASSUMPTION REASONABLE STEADY-STATE RESPONSE FOR TARGET WITH CONSTANT ACCELERATION TRACKING ERRORS DUE TO RANDOM RANGE MEASUREMENT ERROR BALANCING OF LAG ERROR AND RMS PREDICTION ERROR MINIMIZATION OF TRANSIENT ERROR (BENEDICT–BORDNER FILTER) NEW APPROACH TO FINDING OPTIMUM G–H TRACKING FILTERS (THE CRITICALLY DAMPED G–H FILTER) G–H FILTER EXAMPLES CIRCUIT DIAGRAM OF GENERAL G–H FILTER STABILITY OF CONSTANT G–H FILTER 1210 TRACK INITIATION G–H–K FILTER TRACKING IN MULTIDIMENSIONS SELECTION OF COORDINATES FOR TRACKING FILTER KALMAN FILTER TWO-STATE KALMAN FILTER REASONS FOR USING THE KALMAN FILTER PROPERTIES OF KALMAN FILTER KALMAN FILTER IN MATRIX NOTATION DERIVATION OF MINIMUM-VARIANCE EQUATION FIRST DERIVATION SECOND DERIVATION EXACT DERIVATION OF R-DIMENSIONAL KALMAN FILTER TABLE LOOKUP APPROXIMATION TO THE KALMAN FILTER ASQUITH–FRIEDLAND STEADY-STATE G–H KALMAN FILTER SINGER G–H–K KALMAN FILTER 210 CONVENIENT STEADY-STATE G–H–K FILTER DESIGN CURVES 211 SELECTION OF TRACKING FILTER PRACTICAL ISSUES FOR RADAR TRACKING TRACK INITIATION AND CLUTTER REJECTION USE OF TRACK INITIATION TO ELIMINATE CLUTTER CLUTTER REJECTION AND OBSERVATION-MERGING ALGORITHMS FOR REDUCING TRACK INITIATION LOAD MOVING-TARGET DECTOR OBSERVATION MERGING (REDUNDANCY- ELIMINATION, CLUSTERING) EDITING FOR INCONSISTENCIES COMBINED CLUTTER SUPPRESSION AND TRACK INITIATION TRACK-START AND TRACK-DROP RULES DATA ASSOCIATION NEAREST-NEIGHBOR APPROACH OTHER ASSOCIATION APPROACHES TRACK-WHILE-SCAN SYSTEM TRACKING WITH A CHIRP WAVEFORM CHIRP WAVEFORM, PULSE COMPRESSION, MATCH FILTERING OF CHIRP WAVEFORM, AND PULSE COMPRESSION MATCHED FILTER ALTERNATE WAY TO VIEW PULSE COMPRESSION AND PULSE CODING AFFECT OF TRACKING WITH A CHIRP WAVEFORM RANGE–DOPPLER AMBIGUITY PROBLEM OF CHIRP WAVEFORM EFFECT OF USING CHIRP WAVEFORM ON TRACKER FILTERING ACCURACY PART II LEAST-SQUARES FILTERING, VOLTAGE PROCESSING, ADAPTIVE ARRAY PROCESSING, AND EXTENDED KALMAN FILTER LEAST-SQUARES AND MINIMUM-VARIANCE ESTIMATES FOR LINEAR TIME-INVARIANT SYSTEMS GENERAL LEAST-SQUARES ESTIMATION RESULTS GEOMETRIC DERIVATION OF LEAST-SQUARES SOLUTION ORTHONORMAL TRANSFORMATION AND VOLTAGE-PROCESSING (SQUARE-ROOT) METHOD FOR LSE ADAPTIVE NULLING, THE ORTHONORMAL TRANSFORMATION, AND THE LSE MINIMUM-VARIANCE ESTIMATE FIXED-MEMORY POLYNOMIAL FILTER INTRODUCTION DIRECT APPROACH (USING NONORTHOGONAL MTH-DEGREE POLYNOMIAL FIT) DISCRETE ORTHOGONAL LEGENDRE POLYNOMIAL APPROACH REPRESENTATION OF POLYNOMIAL FIT IN TERMS OF ITS DERIVATIVES (STATE VARIABLE REPRESENTATION OF POLYNOMIAL FIT IN TERMS OF PROCESS DERIVATIVES) REPRESENTATION OF LEAST-SQUARES ESTIMATE IN TERMS OF DERIVATIVE STATE VECTOR VARIANCE OF LEAST-SQUARES POLYNOMIAL ESTIMATE SIMPLE EXAMPLE DEPENDENCE OF COVARIANCE ON L, T, M, AND H SYSTEMATIC ERRORS (BIAS, LAG, OR DYNAMIC ERROR) 510 BALANCING SYSTEMATIC AND RANDOM ESTIMATION ERRORS 511 TREND REMOVAL EXPANDING-MEMORY (GROWING-MEMORY) POLYNOMIAL FILTERS EXTRAPOLATION FROM FIXED-MEMORY FILTER RESULTS RECURSIVE FORM STABILITY TRACK INITIATION VARIANCE REDUCTION FACTOR SYSTEMATIC ERRORS FADING-MEMORY (DISCOUNTED LEAST-SQUARES) FILTER DISCOUNTED LEAST-SQUARES ESTIMATE ORTHOGONAL LAGUERRE POLYNOMIAL APPROACH VARIANCE REDUCTION FACTORS COMPARISON WITH FIXED-MEMORY POLYNOMIAL FILTER BALANCING THE SYSTEMATIC AND RANDOM PREDICTION ERROR GENERAL FORM FOR LINEAR TIME-INVARIANT SYSTEM TARGET DYNAMICS DESCRIBED BY POLYNOMIAL AS A FUNCTION OF TIME LINEAR CONSTANT-COEF?CIENT DIFFERENTIAL EQUATION CONSTANT-COEF?CIENT LINEAR DIFFERENTIAL VECTOR EQUATION FOR STATE VECTOR X(T) CONSTANT-COEF?CIENT LINEAR DIFFERENTIAL VECTOR EQUATION FOR TRANSITION MATRIX MORE GENERAL MODEL CONSISTING OF THE SUM OF THE PRODUCT OF POLYNOMIALS AND EXPONENTIALS GENERAL RECURSIVE MINIMUM-VARIANCE GROWING-MEMORY FILTER (BAYES AND KALMAN FILTERS WITHOUT TARGET PROCESS NOISE) BAYES FILTER KALMAN FILTER (WITHOUT PROCESS NOISE) COMPARISON OF BAYES AND KALMAN FILTERS EXTENSION TO MULTIPLE MEASUREMENT CASE VOLTAGE LEAST-SQUARES ALGORITHMS REVISITED COMPUTATION PROBLEMS ORTHOGONAL TRANSFORMATION OF LEAST-SQUARES ESTIMATE ERROR PHYSICAL INTERPRETATION OF ORTHOGONAL TRANSFORMATION PHYSICAL INTERPRETATION OF U REASONS THE SQUARE-ROOT PROCEDURE PROVIDES BETTER ACCURACY WHEN AND WHY INACCURACIES OCCUR GIVENS ORTHONORMAL TRANSFORMATION THE TRANSFORMATION EXAMPLE SYSTOLIC ARRAY IMPLEMENTATION SYSTOLIC ARRAY CORDIC ALGORITHM HOUSEHOLDER ORTHONORMAL TRANSFORMATION COMPARISON OF HOUSEHOLDER AND GIVENS TRANSFORMATIONS FIRST HOUSEHOLDER TRANSFORMATION SECOND AND HIGHER ORDER HOUSEHOLDER TRANSFORMATIONS GRAM–SCHMIDT ORTHONORMAL TRANSFORMATION CLASSICAL GRAM–SCHMIDT ORTHONORMAL TRANSFORMATION MODI?ED GRAM–SCHMIDT ORTHONORMAL TRANSFORMATION MORE ON VOLTAGE-PROCESSING TECHNIQUES COMPARISON OF DIFFERENT VOLTAGE LEAST-SQUARES ALGORITHM TECHNIQUES QR DECOMPOSITION SEQUENTIAL SQUARE-ROOT (RECURSIVE) PROCESSING EQUIVALENCE BETWEEN VOLTAGE-PROCESSING METHODS AND DISCRETE ORTHOGONAL LEGENDRE POLYNOMIAL APPROACH SQUARE-ROOT KALMAN FILTERS LINEAR TIME-VARIANT SYSTEM DYNAMIC MODEL TRANSITION MATRIX DIFFERENTIAL EQUATION NONLINEAR OBSERVATION SCHEME AND DYNAMIC MODEL (EXTENDED KALMAN FILTER) NONLINEAR OBSERVATION SCHEME NONLINEAR DYNAMIC MODEL BAYES ALGORITHM WITH ITERATIVE DIFFERENTIAL CORRECTION FOR NONLINEAR SYSTEMS DETERMINATION OF UPDATED ESTIMATES HISTORICAL BACKGROUND KALMAN FILTER REVISITED KALMAN FILTER TARGET DYNAMIC MODEL KALMAN’S ORIGINAL RESULTS