Andreas Richter

Estimation of Radio Channel Parameters: Models and Algorithms

IS ß 2005

Contents

1 Introduction 1 1.1 Background 1 1.2 Overview and Contributions 4 1.3 Notation 5

2 Radio Channel and System Model 6 2.1 Definition of a Ray 6 2.2 Definition of a Propagation Path 8 2.3 Frequency and Temporal Domain Sampling 11

2.3.1 Expressions for the MIMO-Channel 11 2.4 Data Models for Antenna Arrays 16

2.4.1 Stored Beam-Pattern 16 2.4.2 Array Response Factorisation 17 2.4.3 Effective Aperture Distribution Function 19 2.4.4 Separating Radio Channel Model and Antenna Array Model 20 2.4.5 Definition of Data- and Parameter-Dimensions 22

2.5 Dense Multipath Components 23 2.5.1 Model for Dense Multipath Components in the Time Delay Domain 24 2.5.2 Data Model for Parameter Estimation 27 2.5.3 Modeling the DMC in the Spatial and the Time Domain 30 2.5.4 Examples for DMC and Discussion 30

2.6 Complete Radio Channel Model 34 2.7 On Radio Channel Statistics 34

3 Radio Channel Measurement 36 3.1 Broadband Radio Channel Sounding Techniques 36 3.2 MIMO Channel Sounding 39 3.3 Antenna Array Architectures for Channel Sounding Applications 41 3.4 On the Choice of Reference Scenarios 44 3.5 A Cross Array is Not Suitable for Radio Channel Sounding 44 3.6 Parameter Normalization 46 3.7 Incorporation of Sequential Spatial Sampling into the Data Model 46 3.8 Measurements with Missing Apertures 48

4 Limits on Channel Parameter Estimation 51 4.1 Cramer-Rao Lower Bound for the Deterministic Parameters 53

4.1.1 Fisher Information Matrix for the DML Problem 53 4.1.2 Cramer-Rao Lower Bound for the DML Problem 56 4.1.3 Cramer-Rao Lower Bound of Physical Path Parameters 57 4.1.4 Deterministic Cramer-Rao Lower Bounds for several canonical Models 58 4.1.5 Inherent Limits on the Variance of the Deterministic Parameters 66

4.2 Expression for the Jacobian and the Fisher Information Matrix 67 4.2.1 Füll Polarimetrie Model 68 4.2.2 Polarimetrie Model for one Link End 69 4.2.3 Model for Non-polarimetric Measurements 70

4.3 Cramer-Rao lower bound for the Stochastic Model Parameters 71 4.4 Joint DML and SML Problem 74 4.5 Conclusion on the CRLB of the Deterministic Parameters 75

5 Estimation of Path Parameters 77 5.1 Global Maximization Algorithms 78

5.1.1 Basis Functions with Kronecker Structure 84

xi

5.1.2 Efficient Computation ofthe Correlation Function 85 5.1.3 Relation between Correlation function and Multidimensional Power

Spectra 87 5.1.4 Iterative ML Estimation using Parameter Subset Update Techniques 87 5.1.5 Path Parameter Initialization for Iterative Maximum Likelihood

Algorithms 88 5.2 Local Maximization 91

5.2.1 Steepest Descent Method 92 5.2.2 Newton-Raphson Method 93 5.2.3 Gauß-Newton Method 93 5.2.4 Levenberg-Marquardt Method 94 5.2.5 Optimisation of Parameter Subsets 99 5.2.6 Estimation ofthe Covariance Matrix ofthe Parameter Estimates 100 5.2.7 Model Order Selection for Gauß-Newton based Algorithms 101 5.2.8 Problem Conditioning 105 5.2.9 Implementation Issues 106

5.3 Subspace Based Algorithms 109 5.3.1 Signal Subspace Estimation for Polarimetrie Measurements 111 5.3.2 On the Choice of Subarray Sizes for Multidimensional Smoothing 112 5.3.3 Signal Subspace Estimation for Conjugate Centro-Symmetric Data 114 5.3.4 Economy Size Signal Subspace Estimation 115 5.3.5 Economy Size Signal Subspace Estimation for Data in White Noise 115 5.3.6 Economy Size Signal Subspace Estimation for Data in Coloured Noise 118 5.3.7 Subspace Rotation Invariance - ESPRIT 121 5.3.8 LS-ESPRIT 122 5.3.9 Unitary ESPRIT 123 5.3.10 Multidimensional Unitary ESPRIT 123 5.3.11 Data with Hidden Rotational Invariance Structure 125 5.3.12 Unitary ESPRIT for CUBA Configurations 127

6 Estimation of DMC Parameters 131 6.1 Maximum Likelihood Estimation of DMC Parameters 131

6.1.1 Local Search Strategies 131 6.1.2 Direct Approach 132 6.1.3 Averaged Covariance Matrix 133 6.1.4 Approximation ofthe Covariance Matrix with a Diagonal Matrix 133 6.1.5 A Numerically Efficient Algorithm for the Estimation of DMC

Parameters 137 6.1.6 Model Selection for DMC and Noise 144 6.1.7 Estimation ofthe Parameters of Multiple Independent DMC Processes 146 6.1.8 Estimation of an Initial Solution 147 6.1.9 Frequency Domain Smoothing 150 6.1.10 A Comment on the Least Squares Estimation of Parametric Covariance

Matrices 152 6.1.11 A Generator for the DMC Process 152 6.1.12 Implementation Issues 154

6.2 Joint Estimation of Concentrated Propagation Paths and DMC 157 6.2.1 Joint Maximum Likelihood Estimation (RIMAX) 158 6.2.2 Application of Subspace Based Algorithms for Parameter Estimation in

the Presence of DMC 163 6.2.3 Estimation of an initial Solution without a priori Information 163 6.2.4 General Limitations 164

xii

6.3 Conclusions on Parameter Estimation 164 7 Antenna Array Calibration 165 8 Summary and Conclusions 171

8.1 Summary and Conclusion 171 8.2 Further research areas 172

Appendix 173 A List of Frequently Used Symbols 173 B Abbreviations 177 C Some Common Definitions and UsefuI Relations 179 Bibliography 185

xiii