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 FoundationBefore 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 ProblemsAlong 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.

Preface

xvii

1 Elementary Probability Theory

(30)

1.1 The Probability Function

(1)

1.2 A Bit of Philosophy

(1)

1.3 The One-Dimensional Random Variable

(1)

1.4 The Discrete Random Variable and the PMF

(1)

1.5 A Bit of Combinatorics

(3)

1.5.1 An Introductory Example

(1)

1.5.2 A More Systematic Approach

(1)

1.5.3 How Many Ways Can N Distinct Items Be Ordered?

(1)

1.5.4 How Many Distinct Subsets of N Elements Are There?

(1)

1.5.5 The Binomial Formula

(1)

1.6 The Binomial Distribution

(2)

1.7 The Continuous Random Variable, the CDF, and the PDF

(3)

1.8 The Expected Value

(5)

1.9 Two Dimensional Random Variables

(5)

1.9.1 The Discrete Random Variable and the PMF

(1)

1.9.2 The CDF and the PDF

(1)

1.9.3 The Expected Value

(1)

1.9.4 Correlation

(1)

1.9.5 The Correlation Coefficient

(1)

1.10 The Characteristic Function

(2)

1.11 Gaussian Random Variables

(2)

1.12 Exercises

(5)

2 An Introduction to Stochastic Processes

(10)

2.1 What Is a Stochastic Process?

(2)

2.2 The Autocorrelation Function

(1)

2.3 What Does the Autocorrelation Function Tell Us?

(1)

2.4 The Evenness of the Autocorrelation Function

(1)

2.5 Two Proofs that Rxx (0) > or equal to |Rxx (τ)|

(2)

2.6 Some Examples

(2)

2.7 Exercises

(3)

3 The Weak Law of Large Numbers

(14)

3.1 The Markov Inequality

(1)

3.2 Chebyshev's Inequality

(1)

3.3 A Simple Example

(2)

3.4 The Weak Law of Large Numbers

(2)

3.5 Correlated Random Variables

(2)

3.6 Detecting a Constant Signal in the Presence of Additive Noise

(1)

3.7 A Method for Determining the CDF of a Random Variable

(1)

3.8 Exercises

(4)

4 The Central Limit Theorem

(18)

4.1 Introduction

(1)

4.2 The Proof of the Central Limit Theorem

(3)

4.3 Detecting a Constant Signal in the Presence of Additive Noise

(2)

4.4 Detecting a (Particular) Non-Constant Signal in the Presence of Additive Noise

(2)

4.5 The Monte Carlo Method

(1)

4.6 Poisson Convergence

(4)

4.7 Exercises

(5)

5 Extrema and the Method of Lagrange Multipliers

(16)

5.1 The Directional Derivative and the Gradient

(1)

5.2 Over-Determined Systems

(3)

5.2.1 General Theory

(1)

5.2.2 Recovering a Constant from Noisy Samples

(1)

5.2.3 Recovering a Line from Noisy Samples

(1)

5.3 The Method of Lagrange Multipliers

(6)

5.3.1 Statement of the Result

(1)

5.3.2 A Preliminary Result

(2)

5.3.3 Proof of the Method

(3)

5.4 The Cauchy-Schwarz Inequality

(1)

5.5 Under-Determined Systems

(2)

5.6 Exercises

(3)

6 The Matched Filter for Stationary Noise

(16)

6.1 White Noise

(2)

6.2 Colored Noise

(5)

6.3 The Autocorrelation Matrix

(1)

6.4 The Effect of Sampling Many Times in a Fixed Interval

(1)

6.5 More about the Signal to Noise Ratio

(2)

6.6 Choosing the Optimal Signal for a Given Noise Type

100

(1)

6.7 Exercises

101

(4)

7 Fourier Series and Transforms

105

(20)

7.1 The Fourier Series

105

(3)

7.2 The Functions en(t) Span-A Plausibility Argument

108

(3)

7.3 The Fourier Transform

111

(1)

7.4 Some Properties of the Fourier Transform

112

(3)

7.5 Some Fourier Transforms

115

(4)

7.6 A Connection between the Time and Frequency Domains

119

(1)

7.7 Preservation of the Inner Product

120

(1)

7.8 Exercises

121

(4)

8 The Wiener-Khinchin Theorem and Applications

125

(24)

8.1 The Periodic Case

125

(3)

8.2 The Aperiodic Case

128

(1)

8.3 The Effect of Filtering

129

(1)

8.4 The Significance of the Power Spectral Density

130

(1)

8.5 White Noise

131

(1)

8.6 Low-Pass Noise

131

(1)

8.7 Low-Pass Filtered Low-Pass Noise

132

(1)

8.8 The Schottky Formula for Shot Noise

133

(2)

8.9 A Semi-Practical Example

135

(3)

8.10 Johnson Noise and the Nyquist Formula

138

(2)

8.11 Why Use RMS Measurements

140

(1)

8.12 The Practical Resistor as a Circuit Element

141

(2)

8.13 The Random Telegraph Signal Another Low-Pass Signal

143

(1)

8.14 Exercises

144

(5)

9 Spread Spectrum

149

(16)

9.1 Introduction

149

(1)

9.2 The Probabilistic Approach

150

(1)

9.3 A Spread Spectrum Signal with Narrow Band Noise

151

(2)

9.4 The Effect of Multiple Transmitters

153

(2)

9.5 Spread Spectrum The Deterministic Approach

155

(1)

9.6 Finite State Machines

156

(1)

9.7 Modulo Two Recurrence Relations

157

(1)

9.8 A Simple Example

158

(1)

9.9 Maximal Length Sequences

158

(2)

9.10 Determining the Period

160

(1)

9.11 An Example

161

(1)

9.12 Some Conditions for Maximality

162

(1)

9.13 What We Have Not Discussed

163

(1)

9.14 Exercises

163

(2)

10 More about the Autocorrelation and the PSD

165

(6)

10.1 The "Positivity" of the Autocorrelation

165

(1)

10.2 Another Proof that Rxx(0) > or equal to |Rxx(τ)|

166

(1)

10.3 Estimating the PSD

166

(2)

10.4 The Properties of the Periodogram

168

(1)

10.5 Exercises

169

(2)

11 Wiener Filters

171

(14)

11.1 A Non-Causal Solution

171

(3)

11.2 White Noise and a Low-Pass Signal

174

(1)

11.3 Causality, Anti-Causality and the Fourier Transform

175

(2)

11.4 The Optimal Causal Filter

177

(2)

11.5 Two Examples

179

(2)

11.5.1 White Noise and a Low-Pass Signal

179

(1)

11.5.2 Low-Pass Signal and Noise

180

(1)

11.6 Exercises

181

(4)

A A Brief Overview of Linear Algebra

185

(24)

A.1 The Space CN

185

(1)

A.2 Linear Independence and Bases

186

(1)

A.3 A Preliminary Result

187

(1)

A.4 The Dimension of CN

188

(1)

A.5 Linear Mappings

189

(1)

A.6 Matrices

190

(1)

A.7 Sums of Mappings and Sums of Matrices

191

(1)

A.8 The Composition of Linear Mappings-- Matrix Multiplication

192

(1)

A.9 A Very Special Matrix

193

(1)

A.10 Solving Simultaneous Linear Equations

193

(3)

A.11 The Inverse of a Linear Mapping

196

(1)

A.12 Invertibility

197

(2)

A.13 The Determinant —A Test for Invertibility

199

(1)

A.14 Eigenvcctors and Eigenvalues

200

(2)

A.15 The Inner Product

202

(1)

A.16 A Simple Proof of the Cauchy-Schwarz Inequality

203

(1)

A.17 The Hermitian Transpose of a Matrix

204

(1)

A.18 Some Important Properties of Self-Adjoint Matrices

205

(1)

A.19 Exercises

206

(3)

Bibliography

209

(3)

Index

212

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9780849375545

0849375541

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Table of Contents

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