{"product_id":"complex-valued-nonlinear-adaptive-filters-noncircularity-widely-linear-and-neural-models-hardback-9780470066355","title":"Complex Valued Nonlinear Adaptive Filters; Noncircularity, Widely Linear and Neural Models (Hardback) 9780470066355","description":"\u003cfont face=\"Georgia\"\u003e\r\n\u003cp\u003e\u003cfont size=\"6\"\u003eComplex Valued Nonlinear Adaptive Filters\u003c\/font\u003e\u003cbr\u003e\r\n\u003cfont size=\"5\"\u003eNoncircularity, Widely Linear and Neural Models\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\r\n\r\n\u003cp\u003e\u003cfont size=\"4\"\u003eDanilo P. Mandic (Author), Vanessa Su Lee Goh (Author)\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e9780470066355, Wiley\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003eHardback, published 17 April 2009\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e352 pages\u003cbr\u003e25.2 x 17.3 x 2.5 cm, 0.739 kg\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\r\n\r\n\r\n\u003cp align=\"justify\"\u003e\u003cstrong\u003e\u003cfont size=\"3\"\u003eThis book was written in response to the growing demand for a text that provides a unified treatment of linear and nonlinear complex valued adaptive filters, and methods for the processing of general complex signals (circular and noncircular). It brings together adaptive filtering algorithms for feedforward (transversal) and feedback architectures and the recent developments in the statistics of complex variable, under the powerful frameworks of CR (Wirtinger) calculus and augmented complex statistics. This offers a number of theoretical performance gains, which is illustrated on both stochastic gradient algorithms, such as the augmented complex least mean square (ACLMS), and those based on Kalman filters. This work is supported by a number of simulations using synthetic and real world data, including the noncircular and intermittent radar and wind signals.\u003c\/font\u003e\u003c\/strong\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e\u003cp\u003ePreface xiii\u003c\/p\u003e \u003cp\u003eAcknowledgements xvii\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 The Magic of Complex Numbers 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 History of Complex Numbers 2\u003c\/p\u003e \u003cp\u003e1.1.1 Hypercomplex Numbers 7\u003c\/p\u003e \u003cp\u003e1.2 History of Mathematical Notation 8\u003c\/p\u003e \u003cp\u003e1.3 Development of Complex Valued Adaptive Signal Processing 9\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Why Signal Processing in the Complex Domain? 13\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Some Examples of Complex Valued Signal Processing 13\u003c\/p\u003e \u003cp\u003e2.1.1 Duality Between Signal Representations in R and c 18\u003c\/p\u003e \u003cp\u003e2.2 Modelling in C is Not Only Convenient But Also Natural 19\u003c\/p\u003e \u003cp\u003e2.3 Why Complex Modelling of Real Valued Processes? 20\u003c\/p\u003e \u003cp\u003e2.3.1 Phase Information in Imaging 20\u003c\/p\u003e \u003cp\u003e2.3.2 Modelling of Directional Processes 22\u003c\/p\u003e \u003cp\u003e2.4 Exploiting the Phase Information 23\u003c\/p\u003e \u003cp\u003e2.4.1 Synchronisation of Real Valued Processes 24\u003c\/p\u003e \u003cp\u003e2.4.2 Adaptive Filtering by Incorporating Phase Information 25\u003c\/p\u003e \u003cp\u003e2.5 Other Applications of Complex Domain Processing of Real Valued Signals 26\u003c\/p\u003e \u003cp\u003e2.6 Additional Benefits of Complex Domain Processing 29\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Adaptive Filtering Architectures 33\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Linear and Nonlinear Stochastic Models 34\u003c\/p\u003e \u003cp\u003e3.2 Linear and Nonlinear Adaptive Filtering Architectures 35\u003c\/p\u003e \u003cp\u003e3.2.1 Feedforward Neural Networks 36\u003c\/p\u003e \u003cp\u003e3.2.2 Recurrent Neural Networks 37\u003c\/p\u003e \u003cp\u003e3.2.3 Neural Networks and Polynomial Filters 38\u003c\/p\u003e \u003cp\u003e3.3 State Space Representation and Canonical Forms 39\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Complex Nonlinear Activation Functions 43\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Properties of Complex Functions 43\u003c\/p\u003e \u003cp\u003e4.1.1 Singularities of Complex Functions 45\u003c\/p\u003e \u003cp\u003e4.2 Universal Function Approximation 46\u003c\/p\u003e \u003cp\u003e4.2.1 Universal Approximation in R 47\u003c\/p\u003e \u003cp\u003e4.3 Nonlinear Activation Functions for Complex Neural Networks 48\u003c\/p\u003e \u003cp\u003e4.3.1 Split-complex Approach 49\u003c\/p\u003e \u003cp\u003e4.3.2 Fully Complex Nonlinear Activation Functions 51\u003c\/p\u003e \u003cp\u003e4.4 Generalised Splitting Activation Functions (GSAF) 53\u003c\/p\u003e \u003cp\u003e4.4.1 The Clifford Neuron 53\u003c\/p\u003e \u003cp\u003e4.5 Summary: Choice of the Complex Activation Function 54\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Elements of CR Calculus 55\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Continuous Complex Functions 56\u003c\/p\u003e \u003cp\u003e5.2 The Cauchy–Riemann Equations 56\u003c\/p\u003e \u003cp\u003e5.3 Generalised Derivatives of Functions of Complex Variable 57\u003c\/p\u003e \u003cp\u003e5.3.1 CR Calculus 59\u003c\/p\u003e \u003cp\u003e5.3.2 Link between R- and C-derivatives 60\u003c\/p\u003e \u003cp\u003e5.4 CR-derivatives of Cost Functions 62\u003c\/p\u003e \u003cp\u003e5.4.1 The Complex Gradient 62\u003c\/p\u003e \u003cp\u003e5.4.2 The Complex Hessian 64\u003c\/p\u003e \u003cp\u003e5.4.3 The Complex Jacobian and Complex Differential 64\u003c\/p\u003e \u003cp\u003e5.4.4 Gradient of a Cost Function 65\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Complex Valued Adaptive Filters 69\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Adaptive Filtering Configurations 70\u003c\/p\u003e \u003cp\u003e6.2 The Complex Least Mean Square Algorithm 73\u003c\/p\u003e \u003cp\u003e6.2.1 Convergence of the CLMS Algorithm 75\u003c\/p\u003e \u003cp\u003e6.3 Nonlinear Feedforward Complex Adaptive Filters 80\u003c\/p\u003e \u003cp\u003e6.3.1 Fully Complex Nonlinear Adaptive Filters 80\u003c\/p\u003e \u003cp\u003e6.3.2 Derivation of CNGD using CR calculus 82\u003c\/p\u003e \u003cp\u003e6.3.3 Split-complex Approach 83\u003c\/p\u003e \u003cp\u003e6.3.4 Dual Univariate Adaptive Filtering Approach (DUAF) 84\u003c\/p\u003e \u003cp\u003e6.4 Normalisation of Learning Algorithms 85\u003c\/p\u003e \u003cp\u003e6.5 Performance of Feedforward Nonlinear Adaptive Filters 87\u003c\/p\u003e \u003cp\u003e6.6 Summary: Choice of a Nonlinear Adaptive Filter 89\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Adaptive Filters with Feedback 91\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Training of IIR Adaptive Filters 92\u003c\/p\u003e \u003cp\u003e7.1.1 Coefficient Update for Linear Adaptive IIR Filters 93\u003c\/p\u003e \u003cp\u003e7.1.2 Training of IIR filters with Reduced Computational Complexity 96\u003c\/p\u003e \u003cp\u003e7.2 Nonlinear Adaptive IIR Filters: Recurrent Perceptron 97\u003c\/p\u003e \u003cp\u003e7.3 Training of Recurrent Neural Networks 99\u003c\/p\u003e \u003cp\u003e7.3.1 Other Learning Algorithms and Computational Complexity 102\u003c\/p\u003e \u003cp\u003e7.4 Simulation Examples 102\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Filters with an Adaptive Stepsize 107\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Benveniste Type Variable Stepsize Algorithms 108\u003c\/p\u003e \u003cp\u003e8.2 Complex Valued GNGD Algorithms 110\u003c\/p\u003e \u003cp\u003e8.2.1 Complex GNGD for Nonlinear Filters (CFANNGD) 112\u003c\/p\u003e \u003cp\u003e8.3 Simulation Examples 113\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Filters with an Adaptive Amplitude of Nonlinearity 119\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Dynamical Range Reduction 119\u003c\/p\u003e \u003cp\u003e9.2 FIR Adaptive Filters with an Adaptive Nonlinearity 121\u003c\/p\u003e \u003cp\u003e9.3 Recurrent Neural Networks with Trainable Amplitude of Activation Functions 122\u003c\/p\u003e \u003cp\u003e9.4 Simulation Results 124\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Data-reusing Algorithms for Complex Valued Adaptive Filters 129\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 The Data-reusing Complex Valued Least Mean Square (DRCLMS) Algorithm 129\u003c\/p\u003e \u003cp\u003e10.2 Data-reusing Complex Nonlinear Adaptive Filters 131\u003c\/p\u003e \u003cp\u003e10.2.1 Convergence Analysis 132\u003c\/p\u003e \u003cp\u003e10.3 Data-reusing Algorithms for Complex RNNs 134\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Complex Mappings and Möbius Transformations 137\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1 Matrix Representation of a Complex Number 137\u003c\/p\u003e \u003cp\u003e11.2 The Möbius Transformation 140\u003c\/p\u003e \u003cp\u003e11.3 Activation Functions and Möbius Transformations 142\u003c\/p\u003e \u003cp\u003e11.4 All-pass Systems as Möbius Transformations 146\u003c\/p\u003e \u003cp\u003e11.5 Fractional Delay Filters 147\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 Augmented Complex Statistics 151\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e12.1 Complex Random Variables (CRV) 152\u003c\/p\u003e \u003cp\u003e12.1.1 Complex Circularity 153\u003c\/p\u003e \u003cp\u003e12.1.2 The Multivariate Complex Normal Distribution 154\u003c\/p\u003e \u003cp\u003e12.1.3 Moments of Complex Random Variables (CRV) 157\u003c\/p\u003e \u003cp\u003e12.2 Complex Circular Random Variables 158\u003c\/p\u003e \u003cp\u003e12.3 Complex Signals 159\u003c\/p\u003e \u003cp\u003e12.3.1 Wide Sense Stationarity, Multicorrelations, and Multispectra 160\u003c\/p\u003e \u003cp\u003e12.3.2 Strict Circularity and Higher-order Statistics 161\u003c\/p\u003e \u003cp\u003e12.4 Second-order Characterisation of Complex Signals 161\u003c\/p\u003e \u003cp\u003e12.4.1 Augmented Statistics of Complex Signals 161\u003c\/p\u003e \u003cp\u003e12.4.2 Second-order Complex Circularity 164\u003c\/p\u003e \u003cp\u003e\u003cb\u003e13 Widely Linear Estimation and Augmented CLMS (ACLMS) 169\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e13.1 Minimum Mean Square Error (MMSE) Estimation in c 169\u003c\/p\u003e \u003cp\u003e13.1.1 Widely Linear Modelling in c 171\u003c\/p\u003e \u003cp\u003e13.2 Complex White Noise 172\u003c\/p\u003e \u003cp\u003e13.3 Autoregressive Modelling in c 173\u003c\/p\u003e \u003cp\u003e13.3.1 Widely Linear Autoregressive Modelling in c 174\u003c\/p\u003e \u003cp\u003e13.3.2 Quantifying Benefits of Widely Linear Estimation 174\u003c\/p\u003e \u003cp\u003e13.4 The Augmented Complex LMS (ACLMS) Algorithm 175\u003c\/p\u003e \u003cp\u003e13.5 Adaptive Prediction Based on ACLMS 178\u003c\/p\u003e \u003cp\u003e13.5.1 Wind Forecasting Using Augmented Statistics 180\u003c\/p\u003e \u003cp\u003e\u003cb\u003e14 Duality Between Complex Valued and Real Valued Filters 183\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e14.1 A Dual Channel Real Valued Adaptive Filter 184\u003c\/p\u003e \u003cp\u003e14.2 Duality Between Real and Complex Valued Filters 186\u003c\/p\u003e \u003cp\u003e14.2.1 Operation of Standard Complex Adaptive Filters 186\u003c\/p\u003e \u003cp\u003e14.2.2 Operation of Widely Linear Complex Filters 187\u003c\/p\u003e \u003cp\u003e14.3 Simulations 188\u003c\/p\u003e \u003cp\u003e\u003cb\u003e15 Widely Linear Filters with Feedback 191\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e15.1 The Widely Linear ARMA (WL-ARMA) Model 192\u003c\/p\u003e \u003cp\u003e15.2 Widely Linear Adaptive Filters with Feedback 192\u003c\/p\u003e \u003cp\u003e15.2.1 Widely Linear Adaptive IIR Filters 195\u003c\/p\u003e \u003cp\u003e15.2.2 Augmented Recurrent Perceptron Learning Rule 196\u003c\/p\u003e \u003cp\u003e15.3 The Augmented Complex Valued RTRL (ACRTRL) Algorithm 197\u003c\/p\u003e \u003cp\u003e15.4 The Augmented Kalman Filter Algorithm for RNNs 198\u003c\/p\u003e \u003cp\u003e15.4.1 EKF Based Training of Complex RNNs 200\u003c\/p\u003e \u003cp\u003e15.5 Augmented Complex Unscented Kalman Filter (ACUKF) 200\u003c\/p\u003e \u003cp\u003e15.5.1 State Space Equations for the Complex Unscented Kalman Filter 201\u003c\/p\u003e \u003cp\u003e15.5.2 ACUKF Based Training of Complex RNNs 202\u003c\/p\u003e \u003cp\u003e15.6 Simulation Examples 203\u003c\/p\u003e \u003cp\u003e\u003cb\u003e16 Collaborative Adaptive Filtering 207\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e16.1 Parametric Signal Modality Characterisation 207\u003c\/p\u003e \u003cp\u003e16.2 Standard Hybrid Filtering in R 209\u003c\/p\u003e \u003cp\u003e16.3 Tracking the Linear\/Nonlinear Nature of Complex Valued Signals 210\u003c\/p\u003e \u003cp\u003e16.3.1 Signal Modality characterisation in c 211\u003c\/p\u003e \u003cp\u003e16.4 Split vs Fully Complex Signal Natures 214\u003c\/p\u003e \u003cp\u003e16.5 Online Assessment of the Nature of Wind Signal 216\u003c\/p\u003e \u003cp\u003e16.5.1 Effects of Averaging on Signal Nonlinearity 216\u003c\/p\u003e \u003cp\u003e16.6 Collaborative Filters for General Complex Signals 217\u003c\/p\u003e \u003cp\u003e16.6.1 Hybrid Filters for Noncircular Signals 218\u003c\/p\u003e \u003cp\u003e16.6.2 Online Test for Complex Circularity 220\u003c\/p\u003e \u003cp\u003e\u003cb\u003e17 Adaptive Filtering Based on EMD 221\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e17.1 The Empirical Mode Decomposition Algorithm 222\u003c\/p\u003e \u003cp\u003e17.1.1 Empirical Mode Decomposition as a Fixed Point Iteration 223\u003c\/p\u003e \u003cp\u003e17.1.2 Applications of Real Valued EMD 224\u003c\/p\u003e \u003cp\u003e17.1.3 Uniqueness of the Decomposition 225\u003c\/p\u003e \u003cp\u003e17.2 Complex Extensions of Empirical Mode Decomposition 226\u003c\/p\u003e \u003cp\u003e17.2.1 Complex Empirical Mode Decomposition 227\u003c\/p\u003e \u003cp\u003e17.2.2 Rotation Invariant Empirical Mode Decomposition (RIEMD) 228\u003c\/p\u003e \u003cp\u003e17.2.3 Bivariate Empirical Mode Decomposition (BEMD) 228\u003c\/p\u003e \u003cp\u003e17.3 Addressing the Problem of Uniqueness 230\u003c\/p\u003e \u003cp\u003e17.4 Applications of Complex Extensions of EMD 230\u003c\/p\u003e \u003cp\u003e\u003cb\u003e18 Validation of Complex Representations – Is This Worthwhile? 233\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e18.1 Signal Modality Characterisation in R 234\u003c\/p\u003e \u003cp\u003e18.1.1 Surrogate Data Methods 235\u003c\/p\u003e \u003cp\u003e18.1.2 Test Statistics: The DVV Method 237\u003c\/p\u003e \u003cp\u003e18.2 Testing for the Validity of Complex Representation 239\u003c\/p\u003e \u003cp\u003e18.2.1 Complex Delay Vector Variance Method (CDVV) 240\u003c\/p\u003e \u003cp\u003e18.3 Quantifying Benefits of Complex Valued Representation 243\u003c\/p\u003e \u003cp\u003e18.3.1 Pros and Cons of the Complex DVV Method 244\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix A: Some Distinctive Properties of Calculus in C 245\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix B: Liouville’s Theorem 251\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix C: Hypercomplex and Clifford Algebras 253\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eC. 1 Definitions of Algebraic Notions of Group, Ring and Field 253\u003c\/p\u003e \u003cp\u003eC. 2 Definition of a Vector Space 254\u003c\/p\u003e \u003cp\u003eC. 3 Higher Dimension Algebras 254\u003c\/p\u003e \u003cp\u003eC. 4 The Algebra of Quaternions 255\u003c\/p\u003e \u003cp\u003eC. 5 Clifford Algebras 256\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix D: Real Valued Activation Functions 257\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eD. 1 Logistic Sigmoid Activation Function 257\u003c\/p\u003e \u003cp\u003eD. 2 Hyperbolic Tangent Activation Function 258\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix E: Elementary Transcendental Functions (ETF) 259\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix F: The O Notation and Standard Vector and Matrix Differentiation 263\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eF. 1 The O Notation 263\u003c\/p\u003e \u003cp\u003eF. 2 Standard Vector and Matrix Differentiation 263\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix G: Notions From Learning Theory 265\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eG. 1 Types of Learning 266\u003c\/p\u003e \u003cp\u003eG. 2 The Bias–Variance Dilemma 266\u003c\/p\u003e \u003cp\u003eG. 3 Recursive and Iterative Gradient Estimation Techniques 267\u003c\/p\u003e \u003cp\u003eG. 4 Transformation of Input Data 267\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix H: Notions from Approximation Theory 269\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eAppendix I: Terminology Used in the Field of Neural Networks 273\u003c\/p\u003e \u003cp\u003eAppendix J: Complex Valued Pipelined Recurrent Neural Network (CPRNN) 275\u003c\/p\u003e \u003cp\u003eJ.1 The Complex RTRL Algorithm (CRTRL) for CPRNN 275\u003c\/p\u003e \u003cp\u003eJ.1.1 Linear Subsection Within the PRNN 277\u003c\/p\u003e \u003cp\u003eAppendix K: Gradient Adaptive Step Size (GASS) Algorithms in R 279\u003c\/p\u003e \u003cp\u003eK. 1 Gradient Adaptive Stepsize Algorithms Based on ∂J\/∂μ 280\u003c\/p\u003e \u003cp\u003eK. 2 Variable Stepsize Algorithms Based on ∂J\/∂ε 281\u003c\/p\u003e \u003cp\u003eAppendix L: Derivation of Partial Derivatives from Chapter 8 283\u003c\/p\u003e \u003cp\u003eL. 1 Derivation of ∂e(k)\/∂w n (k) 283\u003c\/p\u003e \u003cp\u003eL. 2 Derivation of ∂e ∗ (k)\/∂ε(k − 1) 284\u003c\/p\u003e \u003cp\u003eL. 3 Derivation of ∂w(k)\/∂ε(k − 1) 286\u003c\/p\u003e \u003cp\u003eAppendix M: A Posteriori Learning 287\u003c\/p\u003e \u003cp\u003eM.1 A Posteriori Strategies in Adaptive Learning 288\u003c\/p\u003e \u003cp\u003eAppendix N: Notions from Stability Theory 291\u003c\/p\u003e \u003cp\u003eAppendix O: Linear Relaxation 293\u003c\/p\u003e \u003cp\u003eO. 1 Vector and Matrix Norms 293\u003c\/p\u003e \u003cp\u003eO. 2 Relaxation in Linear Systems 294\u003c\/p\u003e \u003cp\u003eO.2.1 Convergence in the Norm or State Space? 297\u003c\/p\u003e \u003cp\u003eAppendix P: Contraction Mappings, Fixed Point Iteration and Fractals 299\u003c\/p\u003e \u003cp\u003eP. 1 Historical Perspective 303\u003c\/p\u003e \u003cp\u003eP. 2 More on Convergence: Modified Contraction Mapping 305\u003c\/p\u003e \u003cp\u003eP. 3 Fractals and Mandelbrot Set 308\u003c\/p\u003e \u003cp\u003eReferences 309\u003c\/p\u003e \u003cp\u003eIndex 321\u003c\/p\u003e\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003eSubject Areas: Electronics \u0026amp; communications engineering [\u003ca title=\"See our other books on Electronics \u0026amp; communications engineering\" href=\"https:\/\/freshlyprintedbooks.co.uk\/search?q=%22Electronics%20\u0026amp;%20communications%20engineering%20%5BTJ%5D%22\"\u003eTJ\u003c\/a\u003e]\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\u003c\/font\u003e","brand":"Wiley","offers":[{"title":"Brand New","offer_id":52256966639896,"sku":"9780470066355","price":98.99,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0730\/2037\/5320\/files\/9780470066355.jpg?v=1781275558","url":"https:\/\/freshlyprintedbooks.co.uk\/products\/complex-valued-nonlinear-adaptive-filters-noncircularity-widely-linear-and-neural-models-hardback-9780470066355","provider":"Freshly Printed Books","version":"1.0","type":"link"}