Random Signals and Noise: A Mathematical Introduction
Understanding the nature of random signals and noise is critically important for detecting signals and for reducing and minimizing the effects of noise in applications such as communications and control systems. Outlining a variety of techniques and explaining when and how to use them, Random Signals and Noise: A Mathematical Introduction focuses on applications and practical problem solving rather than probability theory.
A Firm Foundation
Before launching into the particulars of random signals and noise, the author outlines the elements of probability that are used throughout the book and includes an appendix on the relevant aspects of linear algebra. He offers a careful treatment of Lagrange multipliers and the Fourier transform, as well as the basics of stochastic processes, estimation, matched filtering, the Wiener-Khinchin theorem and its applications, the Schottky and Nyquist formulas, and physical sources of noise.
Practical Tools for Modern Problems
Along with these traditional topics, the book includes a chapter devoted to spread spectrum techniques. It also demonstrates the use of MATLAB® for solving complicated problems in a short amount of time while still building a sound knowledge of the underlying principles.
A self-contained primer for solving real problems, Random Signals and Noise presents a complete set of tools and offers guidance on their effective application.
Why Read This Book
You will gain a practical, application-focused foundation in how randomness affects signals and how to mitigate it across communications, radar, and audio/speech systems. The book balances clear mathematical tools (Fourier transforms, Lagrange multipliers, stochastic processes) with worked techniques you can apply directly to detection, filtering, and spectral-analysis problems.
Who Will Benefit
Engineers and graduate students with basic signals-and-systems and calculus background who need to analyze, detect, or reduce noise in communications, radar, and audio/speech applications.
Level: Intermediate — Prerequisites: Undergraduate calculus, complex numbers, basic signals & systems (Fourier/DTFT, convolution) and elementary probability; linear algebra familiarity is helpful (an appendix is provided).
Key Takeaways
- Analyze stationary random processes and derive their spectral representations for practical signal detection and noise characterization.
- Design and evaluate linear detectors and matched filters for communications and radar applications using practical criteria.
- Apply spectral-analysis techniques and FFT-based methods to estimate power spectra and identify signal components in noisy data.
- Design optimal linear filters (Wiener) and understand adaptive filtering strategies for tracking and noise suppression.
- Formulate and solve estimation and prediction problems (least-squares, linear MMSE) relevant to speech/audio processing and communications.
Topics Covered
- 1. Introduction: Random Signals and Engineering Context
- 2. Review of Probability Concepts Used in Signal Analysis
- 3. Linear Algebra Essentials (Appendix-style material)
- 4. Fourier Transform, DTFT and Related Tools for Random Signals
- 5. Basics of Stochastic Processes and Stationarity
- 6. Correlation Functions and Spectral Representations
- 7. Spectral Estimation and FFT-Based Methods
- 8. Linear Systems Excited by Random Inputs
- 9. Linear Estimation and Prediction (Wiener Theory)
- 10. Detection Theory and Matched Filtering
- 11. Adaptive Filtering and Practical Algorithms
- 12. Applications: Communications, Radar, and Audio/Speech Examples
- 13. Advanced/Practical Topics: Time-Frequency Considerations and Implementation Notes
- Appendices: Mathematical Tools (Lagrange Multipliers, Transforms, Tables)
Languages, Platforms & Tools
How It Compares
More application-oriented and approachable than Papoulis' Probability and Stochastic Processes, and less specialist/derivation-heavy than Kay's Fundamentals of Statistical Signal Processing — sits between broad theory and hands-on engineering use.
