Complex Valued Nonlinear Adaptive Filters: Noncircularity, Widely Linear and Neural Models (Adaptive and Cognitive Dynam
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 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.
Why Read This Book
You will learn a unified, mathematically rigorous approach to processing general complex signals—both circular and noncircular—using modern tools such as CR (Wirtinger) calculus and augmented complex statistics. The book uniquely ties together linear and nonlinear complex-valued adaptive filters (including neural models), showing practical performance gains across LMS/RLS/Kalman frameworks with simulations and real-world examples.
Who Will Benefit
Advanced graduate students, researchers, and practicing engineers in DSP, communications, radar, and audio/speech who need a deep, practical understanding of complex-valued adaptive filtering and widely linear/noncircular methods.
Level: Advanced — Prerequisites: Undergraduate-level linear algebra and probability/statistics, familiarity with digital signal processing fundamentals (Fourier/FFT, linear filtering), and prior exposure to basic adaptive filters (LMS/RLS). Some comfort with multivariate calculus will help.
Key Takeaways
- Apply CR (Wirtinger) calculus to derive gradient-based algorithms for complex-valued cost functions.
- Design and implement augmented (widely linear) adaptive filters such as the augmented complex LMS (ACLMS) to exploit noncircularity.
- Analyze and predict performance and convergence of stochastic-gradient and Kalman-based complex adaptive algorithms.
- Develop nonlinear complex-valued adaptive models, including complex-valued neural networks, and adapt them for practical DSP tasks.
- Use augmented complex statistics to correctly model and process improper (noncircular) signals in communications, radar, and audio/speech applications.
Topics Covered
- Introduction and motivation: complex-valued signal processing
- Complex random variables and second-order statistics
- Augmented complex statistics and widely linear modelling
- CR (Wirtinger) calculus for complex optimisation
- Linear complex-valued adaptive filters (LMS, NLMS, RLS)
- Augmented/widely-linear adaptive algorithms (ACLMS and variants)
- Nonlinear complex-valued adaptive filters and complex neural models
- Kalman filtering and state-space methods for complex signals
- Performance analysis: mean, mean-square and convergence behaviour
- Applications: communications systems, radar, audio and speech processing
- Practical simulations, case studies and real-world data
- Conclusions, open problems and directions for research
Languages, Platforms & Tools
How It Compares
Covers similar foundational material to Schreier & Scharf's Statistical Signal Processing of Complex-Valued Data but places stronger emphasis on nonlinear (neural) adaptive models and practical adaptive algorithms; complements Haykin's Adaptive Filter Theory, which is broader and more frequently real-valued.
