1st Edition

# Random Signals and Noise A Mathematical Introduction

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

236 Pages 15 B/W Illustrations
by CRC Press

Also available as eBook on:

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