Statistical Signal Processing of Complex-Valued Data: The Theory of Improper and Noncircular Signals
Complex-valued random signals are embedded in the very fabric of science and engineering, yet the usual assumptions made about their statistical behavior are often a poor representation of the underlying physics. This book deals with improper and noncircular complex signals, which do not conform to classical assumptions, and it demonstrates how correct treatment of these signals can have significant payoffs. The book begins with detailed coverage of the fundamental theory and presents a variety of tools and algorithms for dealing with improper and noncircular signals. It provides a comprehensive account of the main applications, covering detection, estimation, and signal analysis of stationary, nonstationary, and cyclostationary processes. Providing a systematic development from the origin of complex signals to their probabilistic description makes the theory accessible to newcomers. This book is ideal for graduate students and researchers working with complex data in a range of research areas from communications to oceanography.
Why Read This Book
You should read this book if you need a rigorous, unified treatment of complex-valued random signals that violate the usual circularity assumptions — it shows why impropriety matters and how to exploit it for better detection, estimation, and filtering. You will learn augmented-statistics and widely linear methods, plus concrete algorithms and application examples in communications, radar, and audio/speech processing.
Who Will Benefit
Advanced engineers, graduate students, and researchers working on DSP for communications, radar, audio/speech, or sensing who need to model and process improper/noncircular complex signals for improved performance.
Level: Advanced — Prerequisites: Solid probability and random processes, linear algebra (including complex vectors and matrices), basic detection/estimation theory, and familiarity with Fourier/spectral methods and standard DSP concepts.
Key Takeaways
- Characterize improper and noncircular complex signals using augmented statistics and pseudo-covariance measures.
- Derive and implement widely linear estimators and detectors that exploit noncircularity for performance gains.
- Analyze stationary, nonstationary, and cyclostationary complex processes and apply spectral tools tailored to complex data.
- Design and evaluate adaptive filtering algorithms for complex-valued signals, including LMS variants that use augmented representations.
- Formulate ML, CRLB, and hypothesis-testing results for improper processes and apply them to communications and radar problems.
Topics Covered
- 1. Introduction and motivation: improper and noncircular signals
- 2. Mathematical preliminaries: complex vectors, matrices, and second-order statistics
- 3. Properness, impropriety, and measures of noncircularity
- 4. Augmented representations and widely linear processing
- 5. Estimation and detection theory for improper complex signals
- 6. Spectral analysis and the FFT for complex-valued processes
- 7. Stationary, nonstationary, and cyclostationary complex signals
- 8. Adaptive filtering and widely linear adaptive algorithms
- 9. Applications in communications, radar, and audio/speech processing
- 10. Case studies: detection, parameter estimation, and signal separation
- 11. Practical considerations and implementation notes
- Appendices: mathematical tools and reference results
Languages, Platforms & Tools
How It Compares
Covers similar complex-valued topics to D. P. Mandic's work on noncircular/widely linear adaptive filters and complements Steven M. Kay's statistical signal processing texts by focusing specifically on improper/noncircular complex processes.
