Table of Contents

1 Introduction 1
1.1 Examples of the Blind System Identification 1
1.2 Optimization Based Blind System Identification 4
1.3 Blind Identification of Various System Models 5
1.4 Organization of This Book 6
References 8
Part I Preliminaries
2 Linear Algebra and Polynomial Matrices 11
2.1 Vector Space and Basis 11
2.2 Eigenvalue Decomposition 13
2.3 Singular Value Decomposition 15
2.4 Orthogonal Projection and Oblique Projection 16
2.5 Sum and Intersection of Subspaces 18
2.6 Angles Between Subspaces 19
2.7 Polynomial Matrices and Polynomial Bases 20
2.8 Summary 24
References 24
3 Representation of Linear System Models 25
3.1 Transfer Functions 25
3.1.1 Properties of Coprime Matrix Fraction 26
3.1.2 Verification and Computation of Coprime Matrix Fraction 28
3.2 State Space Models 31
3.3 State Space Realization 38
3.4 HankelMatrix Interpretation 40
3.5 Structured State-Space Models 41
3.5.1 Graph Theory 42
3.5.2 Structured Algebraic System Theory 44
3.6 Summary 47
Reference 48
4 Identification of LTI Systems 49
4.1 Least-Squares Identification 50
4.1.1 Identifiability of a Rational Transfer Function Matrix 50
4.1.2 Least-Squares Identification Method 51
4.2 Subspace Identification 53
4.2.1 Subspace Identification via Orthogonal Projection 55
4.2.2 Subspace Identification via State Estimation 56
4.2.3 Subspace Identification via State Compensation 59
4.2.4 Subspace Identification via Markov Parameter Estimation 61
4.3 Parameterized State-Space Identification 62
4.3.1 Gradient-BasedMethod 63
4.3.2 Difference-of-Convex Programming Method 64
4.4 Summary 69
References 70
Part II Blind System Identification with a Single Unknown Input
5 Blind Identification of SIMO FIR Systems 73
5.1 Structured Subspace Factorization 74
5.1.1 Blind Identification of FIR Filters 75
5.1.2 Blind Identification of a Source Signal 78
5.2 Cross RelationMethod 80
5.3 Least-Squares Smoothing Method 83
5.3.1 Blind FIR Filter Identification 84
5.3.2 Blind Source Signal Estimation 85
5.4 Blind Identification of Time-Varying FIR Systems 86
5.4.1 Input Signal Estimation 87
5.4.2 Time-Varying Filter Identification 88
5.5 Blind Identification of Nonlinear SIMO Systems 90
5.5.1 SIMO-Wiener System Identification 91
5.5.2 Hammerstein-Wiener System Identification 93
5.6 Summary 94
References 95
6 Blind Identification of SISO IIR Systems via Oversampling 97
6.1 Oversampling of FIR and IIR Systems 98
6.1.1 Multirate Identities 98
6.1.2 Multirate Transfer Functions 99
6.1.3 Multirate State-Space Models 103
6.2 Coprime Conditions for Lifted SIMO Systems 104
6.3 Blind Identification of Non-minimum Phase Systems 108
6.4 Blind Identification of Hammerstein Systems 110
6.4.1 Blind Identifiability 111
6.4.2 Blind Identification Approach 112
6.5 Blind Identification of Output Switching Systems 114
6.6 Summary 125
References 126
7 Distributed Blind Identification of Networked FIR Systems 127
7.1 Motivation for the Distributed Blind Identification 127
7.2 Distributed Blind System Identification Using Noise-Free Data 128
7.2.1 Distributed Blind Identification Algorithm 129
7.2.2 Convergence Analysis 131
7.2.3 Numerical Simulation 136
7.3 Distributed Blind System Identification Using Noisy Data 138
7.3.1 Distributed Blind Identification Algorithm 139
7.3.2 Convergence Analysis 140
7.3.3 Numerical Simulation 147
7.4 Recursive Blind Source Equalization Using Noisy Data 148
7.4.1 Direct Distributed Equalization 149
7.4.2 Indirect Distributed Equalization 151
7.4.3 Distributed Blind Equalization with Noise-Free Measurements 152
7.4.4 Distributed Blind Equalization with Noisy Measurements 156
7.4.5 Blind Equalization with a Time-Varying Topology 157
7.4.6 Numerical Simulation 159
7.5 Summary 162
References 163
Part III Blind System Identification with Multiple Unknown Inputs
8 Blind Identification of MIMO Systems 167
8.1 Blind Identification ofMIMO FIR Systems 167
8.1.1 Identifiability Analysis 169
8.1.2 Subspace Blind Identification Method 171
8.2 Blind Identification of Multivariable State-Space Models 173
8.2.1 Identifiability of Two Channel Systems 174
8.2.2 Blind Identification of Characteristic Polynomials 179
8.2.3 Blind Identification of Numerator Polynomial Matrices 183
8.2.4 Numerical Simulation 192
8.3 Summary 197
References 198
9 Blind Identification of Structured State-Space Models 199
9.1 Strong Observability of Structured State-Space Models 199
9.1.1 Maximum Unobservable Subspace 200
9.1.2 State Estimation with Unknown Inputs 202
9.2 Blind Identification of Multivariable State-Space Models 204
9.2.1 Identifiability Analysis 206
9.2.2 Subspace-Based Blind Identification Method 215
9.2.3 Numerical Simulations 220
9.3

Sample Pages Preview

Chapter 1 Introduction
In this introductory chapter, the amazing blind system identification world will be introduced through interesting examples in various research fields. Through the blind identifiability analysis from the optimization and system perspectives, it will be shown that the blind identification is infeasible without exploiting prior knowledge of the system structures or specific patterns of the unknown inputs. Also, the number of unknown inputs in a system model is another factor that may cause the challenge of the blind system identification, based on which the main content of the book is separated into two parts, namely the blind system identification with a single unknown input and the blind system identification with multiple unknown inputs. The contents of this chapter include: 1. providing several motivating examples to show the wide applications of blind system identification; 2. analyzing the challenges for solving blind system identification problems; 3. describing the blind system identification problems of various models. At the end of this chapter, the organization of this book will be explained, which provides a road map for reading this book. 1.1 Examples of the Blind System Identification The blind system identification means to recover the system model as well as the unknownsystem input from the output observations only. Due to the fact that the blind system identification does not require a mass training data, it has many interesting applications, such as medical imaging, wireless communication, image restoration as well as adaptive optics. Blind Identification in Medical Imaging The parallel magnetic resonance imaging (pMRI) is one of popular non-invasive medical imaging methods which can yield high resolution images of the inner structure of a human body, providing solid evidence for medical diagnosis (She et al., 2015). As shown in Fig. 1.1, the brain image in the center represents the true image, while four surrounding images are acquired by their corresponding coils. Each acquired image can be modeled by the convolution between the true image and its associated coil sensitivity function. Since the coil sensitivity functions in different environments may not be the same, the reconstruction of the true image from four acquired images turns out to be a blind system identification (deconvolution) problem. Blind Identification inWireless Communication Awireless communication network includes the uplink communication and downlink communication, as illustrated in Fig. 1.2. In the uplink communication, the emitters are the mobile phones and the receiver is the base station. The corresponding communication process can be described by a multi-input single-output (MISO) system model, where the dispersive channels are different and unknown. In the downlink context, the emitter is the base station and the receivers are the mobile phones; thus, the communication model can be described by a single-input multi-output (SIMO) system. Analogous to the uplink phase, the dispersive channels in the downlink phase are different and unknown as well. To deal with the associated deconvolution problem with unknown channel functions, the symbol sequences generated from the emitter are encoded using words in a public dictionary and transmitted under certain modulation scheme, such as TDM and FDM (Giannakis, 2001). By exploiting the latent patterns of the emitted symbol sequences, blind identification methods are to be developed to restore the true symbol sequences (message) from the measured data at the receiver. Fig. 1.1 Illustration of parallel MRI Fig. 1.2 Illustration of wireless communication Blind Identification in a Heating, Ventilation, and Air Conditioning (HVAC) System The HVAC systems are now popular for green buildings in highly developed cities. The temperature fields in the rooms of a building can be described by a compartment model. As shown in Fig. 1.3, the human walk in and out of the building/rooms will influence the room temperature; therefore, by treating the number of people (occupancy) inside the room as the unknown inputs and the room-temperature measurement as the output observation, blind identification strategies can be developed for the occupancy estimation using the room-temperature observations (Ebadat et al., 2015). Blind Identification in Adaptive Optics The objective of the adaptive optics is to recover the image of celestial body through a telescope (see Fig. 1.4).When the light of a celestial body arrives at the outer layer of the atmosphere, it has a perfect plane wavefront; however, this plane wavefront will be distorted when passing through a turbulent atmosphere.