1st Edition

Random Signals and Noise A Mathematical Introduction

By Shlomo Engelberg Copyright 2007
    236 Pages 15 B/W Illustrations
    by CRC Press

    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 MATLABĀ® 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.

    ELEMENTARY PROBABILITY THEORY
    The Probability Function
    A Bit of Philosophy
    The One-Dimensional Random Variable
    The Discrete Random Variable and the PMF
    A Bit of Combinatorics
    The Binomial Distribution
    The Continuous Random Variable, the CDF, and the PDF
    The Expected Value
    Two Dimensional Random Variables
    The Characteristic Function
    Gaussian Random Variables
    Exercises
    AN INTRODUCTION TO STOCHASTIC PROCESSES
    What Is a Stochastic Process?
    The Autocorrelation Function
    What Does the Autocorrelation Function Tell Us?
    The Evenness of the Autocorrelation Function
    Two Proofs that Rxx(0) ≥ |Rxx(t)|
    Some Examples
    Exercises
    THE WEAK LAW OF LARGE NUMBERS
    The Markov Inequality
    Chebyshev's Inequality
    A Simple Example
    The Weak Law of Large Numbers
    Correlated Random Variables
    Detecting a Constant Signal in the Presence of Additive Noise
    A Method for Determining the CDF of a Random Variable
    Exercises
    THE CENTRAL LIMIT THEOREM
    Introduction
    The Proof of the Central Limit Theorem
    Detecting a Constant Signal in the Presence of Additive Noise
    Detecting a (Particular) Non-Constant Signal in the Presence of Additive Noise
    The Monte Carlo Method
    Poisson Convergence
    Exercises
    EXTREMA AND THE METHOD OF LAGRANGE MULTIPLIERS
    The Directional Derivative and the Gradient
    Over-Determined Systems
    The Method of Lagrange Multipliers
    The Cauchy-Schwarz Inequality
    Under-Determined Systems
    Exercises
    THE MATCHED FILTER FOR STATIONARY NOISE
    White Noise
    Colored Noise
    The Autocorrelation Matrix
    The Effect of Sampling Many Times in a Fixed Interval
    More about the Signal to Noise Ratio
    Choosing the Optimal Signal for a Given Noise Type
    Exercises
    FOURIER SERIES AND TRANSFORMS
    The Fourier Series
    The Functions en(t) Span-a Plausibility Argument
    The Fourier Transform
    Some Properties of the Fourier Transform
    Some Fourier Transforms
    A Connection between the Time and Frequency Domains
    Preservation of the Inner Product
    Exercises
    THE WIENER-KHINCHIN THEOREM AND APPLICATIONS
    The Periodic Case
    The Aperiodic Case
    The Effect of Filtering
    The Significance of the Power Spectral Density
    White Noise
    Low-Pass Noise
    Low-Pass Filtered Low-Pass Noise
    The Schottky Formula for Shot Noise
    A Semi-Practical Example
    Johnson Noise and the Nyquist Formula
    Why Use RMS Measurements
    The Practical Resistor as a Circuit Element
    The Random Telegraph Signal-Another Low-Pass Signal
    Exercises
    SPREAD SPECTRUM
    Introduction
    The Probabilistic Approach
    A Spread Spectrum Signal with Narrow Band Noise
    The Effect of Multiple Transmitters
    Spread Spectrum-The Deterministic Approach
    Finite State Machines
    Modulo Two Recurrence Relations
    A Simple Example
    Maximal Length Sequences
    Determining the Period
    An Example
    Some Conditions for Maximality
    What We Have Not Discussed
    Exercises
    MORE ABOUT THE AUTOCORRELATION AND THE PSD
    The "Positivity" of the Autocorrelation
    Another Proof that Rxx(0) ≥ |Rxx(t)|
    Estimating the PSD
    The Properties of the Periodogram
    Exercises
    WIENER FILTERS
    A Non-Causal Solution
    White Noise and a Low-Pass Signal
    Causality, Anti-Causality and the Fourier Transform
    The Optimal Causal Filter
    Two Examples
    Exercises
    APPENDIX: A BRIEF OVERVIEW OF LINEAR ALGEBRA
    The Space CN
    Linear Independence and Bases
    A Preliminary Result
    The Dimension of CN
    Linear Mappings
    Matrices
    Sums of Mappings and Sums of Matrices
    The Composition of Linear Mappings-Matrix Multiplication
    A Very Special Matrix
    Solving Simultaneous Linear Equations
    The Inverse of a Linear Mapping
    Invertibility
    The Determinant-A Test for Invertibility
    Eigenvectors and Eigenvalues
    The Inner Product
    A Simple Proof of the Cauchy-Schwarz Inequality
    The Hermitian Transpose of a Matrix
    Some Important Properties of Self-Adjoint Matrices
    Exercises
    BIBLIOGRAPHY
    INDEX

    Biography

    Shlomo Engelberg