The Volterra and Wiener Theories of Nonlinear Systems
This text presents a complete and detailed development of the analysis, design and characterization of non-linear systems using the Volterra and Wiener theories, as well as gate functions, thus yielding new insights and a better comprehension of the subject. The Volterra and Wiener theories are useful in the study of systems in biological, mechanical, and electrical fields.
Why Read This Book
You will gain a rigorous, mathematically precise foundation for modeling and analyzing nonlinear systems using Volterra and Wiener series, with practical pathways to kernel measurement and system characterization. The book blends theoretical depth with concrete techniques (including gate functions and kernel estimation) so you can apply these methods to audio, speech, radar, communications, and biological systems.
Who Will Benefit
Advanced engineers, researchers, and graduate students working on nonlinear signal processing, system identification, or modeling of real-world DSP systems who need a rigorous treatment of Volterra/Wiener methods.
Level: Expert — Prerequisites: Solid grounding in linear systems and signals, Fourier and Laplace transforms, multivariate calculus, probability and stochastic processes, and familiarity with discrete-time DSP and basic filter design.
Key Takeaways
- Formulate and manipulate Volterra series representations of weakly nonlinear, time-invariant systems.
- Derive and compute Wiener kernels via orthogonal functional expansions and understand their statistical interpretation.
- Design and execute experimental kernel-estimation procedures (measurement protocols) for practical systems.
- Analyze nonlinear distortion and intermodulation effects in audio, speech, radar, and communications applications using higher-order spectra.
- Approximate and truncate Volterra models for implementable nonlinear filters and evaluate trade-offs between complexity and accuracy.
- Relate Volterra/Wiener descriptions to other nonlinear model structures (e.g., Hammerstein/Wiener, polynomial models) and apply gate-function techniques for insight and simplification.
Topics Covered
- Introduction: Motivation and scope of Volterra and Wiener theories
- Mathematical preliminaries: functionals, orthogonal expansions, and notation
- The Volterra series: definitions, properties, and convergence issues
- Symmetry, causality, and memory in Volterra kernels
- Wiener theory: orthogonal functional expansions and Wiener kernels
- Gate functions and their role in nonlinear system characterization
- Kernel estimation: measurement methods, experimental design, and noise considerations
- Discrete-time Volterra/Wiener systems and computational implementation
- Frequency-domain representations and higher-order spectra (bispectrum, polyspectra)
- Approximation, truncation, and practical realizations of Volterra filters
- Applications: audio & speech processing, radar signal analysis, and communications
- Connections to adaptive filtering and statistical signal-processing frameworks
- Numerical examples, case studies, and practical pitfalls
- Appendices: mathematical tools, proofs, and reference tables
Languages, Platforms & Tools
How It Compares
Complementary to David Mendel's treatments of higher-order statistics (which emphasize polyspectra), Schetzen is the classic, deeper theoretical source for Volterra/Wiener kernel theory and practical kernel measurement; for system-identification breadth compare with Ljung's System Identification (which is broader but less focused on Volterra/Wiener formalism).
