The Volterra Series and its Application
Modeling of weakly nonlinear systems by means of Volterra series analysis is presented. Necessary conditions for representing nonlinearities by a Volterra series are developed analytically as well as heuristically. A two-condition convergence criterion for Volterra series and a method for determining Volterra transfer functions are established. For systems with multiple nodes, an extension of Volterra series analysis; method of nonlinear currents is developed and applied to a MESFET amplifier. Finally, methods of quantifying nonlinear behavior are discussed.
Why Read This Book
You will learn a rigorous, engineering-focused approach to modeling weakly nonlinear systems using Volterra series so you can predict harmonics, intermodulation, and distortion from first principles. The book pairs analytic convergence criteria and kernel-determination methods with a practical extension (nonlinear currents) and circuit examples, so you can move from theory to real RF/amplifier analysis.
Who Will Benefit
Readers are practicing signal-processing or RF engineers and graduate students who need to model, analyze, or mitigate weak nonlinearities in communication, audio, or radar systems.
Level: Advanced — Prerequisites: Undergraduate-level signals and systems and linear DSP, multivariable calculus, complex variables and Fourier/FFT familiarity; basic circuit theory and semiconductor device concepts for the application chapters.
Key Takeaways
- Apply Volterra series expansions to represent and analyze weakly nonlinear time-invariant systems
- Derive and use a two-condition convergence criterion to assess when Volterra representations are valid
- Determine Volterra transfer functions (kernels) from time- and frequency-domain formulations
- Extend Volterra analysis to multi-node networks via the method of nonlinear currents and apply it to device circuits
- Evaluate and quantify nonlinear behavior in amplifiers—predict harmonics, intermodulation, and distortion metrics
- Implement practical numerical approaches for computing kernels and performing higher-order spectral analysis
Topics Covered
- 1. Introduction and motivation: nonlinear modeling in DSP and RF
- 2. Mathematical preliminaries: multilinear systems, convolution, and notation
- 3. The Volterra series: definitions, causality, and basic properties
- 4. Representability: necessary conditions for Volterra modeling
- 5. Convergence theory: the two-condition convergence criterion
- 6. Determination of Volterra transfer functions: time- and frequency-domain methods
- 7. Frequency-domain tools: higher-order spectra and FFT-based computation
- 8. Numerical methods and practical computation of kernels
- 9. Multi-node systems and the method of nonlinear currents
- 10. Application to MESFET amplifiers: worked circuit example
- 11. Quantifying nonlinear behavior: metrics, measurements, and interpretation
- 12. Examples, limitations, and guidelines for practice
- Appendices: mathematical tools, tables, and reference results
Languages, Platforms & Tools
How It Compares
Whereas standard DSP texts (e.g., Oppenheim & Schafer) focus on linear theory, Dunn concentrates on Volterra-based nonlinear analysis; it complements nonlinear system-identification treatments (e.g., Billings' NARMAX-oriented texts) by emphasizing analytic conditions, kernel derivation, and circuit-level applications rather than black-box model fitting.
