Details

Autor(en) / Beteiligte

• Mandic, Danilo P
• Goh, Vanessa Su Lee

Titel

Complex valued nonlinear adaptive filters : noncircularity, widely linear, and neural models [electronic resource]

Auflage

1st ed

Link zum Volltext

• Wiley Online Library - AutoHoldings Books

Beschreibungen/Notizen

• Description based upon print version of record.
• Includes bibliographical references and index.
• Complex Valued Nonlinear Adaptive Filters: Noncircularity, Widely Linear and Neural Models; Series Page; Contents; Preface; Acknowledgements; 1 The Magic of Complex Numbers; 1.1 History of Complex Numbers; 1.1.1 Hypercomplex Numbers; 1.2 History of Mathematical Notation; 1.3 Development of Complex Valued Adaptive Signal Processing; 2 Why Signal Processing in the Complex Domain?; 2.1 Some Examples of Complex Valued Signal Processing; 2.1.1 Duality Between Signal Representations in R and C; 2.2 Modelling in C is Not Only Convenient But Also Natural
• 2.3 Why Complex Modelling of Real Valued Processes?2.3.1 Phase Information in Imaging; 2.3.2 Modelling of Directional Processes; 2.4 Exploiting the Phase Information; 2.4.1 Synchronisation of Real Valued Processes; 2.4.2 Adaptive Filtering by Incorporating Phase Information; 2.5 Other Applications of Complex Domain Processing of Real Valued Signals; 2.6 Additional Benefits of Complex Domain Processing; 3 Adaptive Filtering Architectures; 3.1 Linear and Nonlinear Stochastic Models; 3.2 Linear and Nonlinear Adaptive Filtering Architectures; 3.2.1 Feedforward Neural Networks
• 3.2.2 Recurrent Neural Networks3.2.3 Neural Networks and Polynomial Filters; 3.3 State Space Representation and Canonical Forms; 4 Complex Nonlinear Activation Functions; 4.1 Properties of Complex Functions; 4.1.1 Singularities of Complex Functions; 4.2 Universal Function Approximation; 4.2.1 Universal Approximation in R; 4.3 Nonlinear Activation Functions for Complex Neural Networks; 4.3.1 Split-complex Approach; 4.3.2 Fully Complex Nonlinear Activation Functions; 4.4 Generalised Splitting Activation Functions (GSAF); 4.4.1 The Clifford Neuron
• 4.5 Summary: Choice of the Complex Activation Function5 Elements of CR Calculus; 5.1 Continuous Complex Functions; 5.2 The Cauchy-Riemann Equations; 5.3 Generalised Derivatives of Functions of Complex Variable; 5.3.1 CR Calculus; 5.3.2 Link between R- and C-derivatives; 5.4 CR-derivatives of Cost Functions; 5.4.1 The Complex Gradient; 5.4.2 The Complex Hessian; 5.4.3 The Complex Jacobian and Complex Differential; 5.4.4 Gradient of a Cost Function; 6 Complex Valued Adaptive Filters; 6.1 Adaptive Filtering Configurations; 6.2 The Complex Least Mean Square Algorithm
• 6.2.1 Convergence of the CLMS Algorithm6.3 Nonlinear Feedforward Complex Adaptive Filters; 6.3.1 Fully Complex Nonlinear Adaptive Filters; 6.3.2 Derivation of CNGD using CR calculus; 6.3.3 Split-complex Approach; 6.3.4 Dual Univariate Adaptive Filtering Approach (DUAF); 6.4 Normalisation of Learning Algorithms; 6.5 Performance of Feedforward Nonlinear Adaptive Filters; 6.6 Summary: Choice of a Nonlinear Adaptive Filter; 7 Adaptive Filters with Feedback; 7.1 Training of IIR Adaptive Filters; 7.1.1 Coefficient Update for Linear Adaptive IIR Filters
• 7.1.2 Training of IIR filters with Reduced Computational Complexity
• This 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 stochast
• English