Random signals and noise : a mathematical introduction

Title:

Personal Author:

Engelberg, Shlomo

Publication Information:

Boca Raton CRC Press, Taylor& Francis Group 2007

Physical Description:

vii, 216 p. : ill. ; 24 cm.

ISBN:

9780849375545

Subject Term:

Signal theory (Telecommunication)

Statistical communication theory

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010210071	TK5102.5 E55 2007	Open Access Book	Book	Searching... Unknown
Searching... SPACE KL Main Library	30000010117128	TK5102.5 E55 2007	Book	Book	Searching... Unknown
Searching... SPACE Pahang Library	30000010117129	TK5102.5 E55 2007	Open Access Book	Book	Searching... Unknown

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.

Preface	p. xvii
1 Elementary Probability Theory	p. 1
1.1 The Probability Function	p. 1
1.2 A Bit of Philosophy	p. 1
1.3 The One-Dimensional Random Variable	p. 2
1.4 The Discrete Random Variable and the PMF	p. 3
1.5 A Bit of Combinatorics	p. 4
1.5.1 An Introductory Example	p. 4
1.5.2 A More Systematic Approach	p. 5
1.5.3 How Many Ways Can N Distinct Items Be Ordered?	p. 6
1.5.4 How Many Distinct Subsets of N Elements Are There?	p. 6
1.5.5 The Binomial Formula	p. 7
1.6 The Binomial Distribution	p. 7
1.7 The Continuous Random Variable, the CDF, and the PDF	p. 9
1.8 The Expected Value	p. 12
1.9 Two Dimensional Random Variables	p. 17
1.9.1 The Discrete Random Variable and the PMF	p. 18
1.9.2 The CDF and the PDF	p. 19
1.9.3 The Expected Value	p. 20
1.9.4 Correlation	p. 21
1.9.5 The Correlation Coefficient	p. 21
1.10 The Characteristic Function	p. 22
1.11 Gaussian Random Variables	p. 24
1.12 Exercises	p. 26
2 An Introduction to Stochastic Processes	p. 31
2.1 What Is a Stochastic Process?	p. 31
2.2 The Autocorrelation Function	p. 33
2.3 What Does the Autocorrelation Function Tell Us?	p. 33
2.4 The Evenness of the Autocorrelation Function	p. 34
2.5 Two Proofs that R[subscript XX](0) [greater than equal] \| R[subscript XX]([tau])\|	p. 34
2.6 Some Examples	p. 36
2.7 Exercises	p. 38
3 The Weak Law of Large Numbers	p. 41
3.1 The Markov Inequality	p. 41
3.2 Chebyshev's Inequality	p. 42
3.3 A Simple Example	p. 43
3.4 The Weak Law of Large Numbers	p. 45
3.5 Correlated Random Variables	p. 47
3.6 Detecting a Constant Signal in the Presence of Additive Noise	p. 49
3.7 A Method for Determining the CDF of a Random Variable	p. 50
3.8 Exercises	p. 51
4 The Central Limit Theorem	p. 55
4.1 Introduction	p. 55
4.2 The Proof of the Central Limit Theorem	p. 56
4.3 Detecting a Constant Signal in the Presence of Additive Noise	p. 59
4.4 Detecting a (Particular) Non-Constant Signal in the Presence of Additive Noise	p. 61
4.5 The Monte Carlo Method	p. 63
4.6 Poisson Convergence	p. 64
4.7 Exercises	p. 68
5 Extrema and the Method of Lagrange Multipliers	p. 73
5.1 The Directional Derivative and the Gradient	p. 73
5.2 Over-Determined Systems	p. 74
5.2.1 General Theory	p. 74
5.2.2 Recovering a Constant from Noisy Samples	p. 75
5.2.3 Recovering a Line from Noisy Samples	p. 76
5.3 The Method of Lagrange Multipliers	p. 77
5.3.1 Statement of the Result	p. 77
5.3.2 A Preliminary Result	p. 78
5.3.3 Proof of the Method	p. 80
5.4 The Cauchy-Schwarz Inequality	p. 83
5.5 Under-Determined Systems	p. 84
5.6 Exercises	p. 86
6 The Matched Filter for Stationary Noise	p. 89
6.1 White Noise	p. 89
6.2 Colored Noise	p. 91
6.3 The Autocorrelation Matrix	p. 96
6.4 The Effect of Sampling Many Times in a Fixed Interval	p. 97
6.5 More about the Signal to Noise Ratio	p. 98
6.6 Choosing the Optimal Signal for a Given Noise Type	p. 100
6.7 Exercises	p. 101
7 Fourier Series and Transforms	p. 105
7.1 The Fourier Series	p. 105
7.2 The Functions e[subscript n] (t) Span-A Plausibility Argument	p. 108
7.3 The Fourier Transform	p. 111
7.4 Some Properties of the Fourier Transform	p. 112
7.5 Some Fourier Transforms	p. 115
7.6 A Connection between the Time and Frequency Domains	p. 119
7.7 Preservation of the Inner Product	p. 120
7.8 Exercises	p. 121
8 The Wiener-Khinchin Theorem and Applications	p. 125
8.1 The Periodic Case	p. 125
8.2 The Aperiodic Case	p. 128
8.3 The Effect of Filtering	p. 129
8.4 The Significance of the Power Spectral Density	p. 130
8.5 White Noise	p. 131
8.6 Low-Pass Noise	p. 131
8.7 Low-Pass Filtered Low-Pass Noise	p. 132
8.8 The Schottky Formula for Shot Noise	p. 133
8.9 A Semi-Practical Example	p. 135
8.10 Johnson Noise and the Nyquist Formula	p. 138
8.11 Why Use RMS Measurements	p. 140
8.12 The Practical Resistor as a Circuit Element	p. 141
8.13 The Random Telegraph Signal-Another Low-Pass Signal	p. 143
8.14 Exercises	p. 144
9 Spread Spectrum	p. 149
9.1 Introduction	p. 149
9.2 The Probabilistic Approach	p. 150
9.3 A Spread Spectrum Signal with Narrow Band Noise	p. 151
9.4 The Effect of Multiple Transmitters	p. 153
9.5 Spread Spectrum-The Deterministic Approach	p. 155
9.6 Finite State Machines	p. 156
9.7 Modulo Two Recurrence Relations	p. 157
9.8 A Simple Example	p. 158
9.9 Maximal Length Sequences	p. 158
9.10 Determining the Period	p. 160
9.11 An Example	p. 161
9.12 Some Conditions for Maximality	p. 162
9.13 What We Have Not Discussed	p. 163
9.14 Exercises	p. 163
10 More about the Autocorrelation and the PSD	p. 165
10.1 The "Positivity" of the Autocorrelation	p. 165
10.2 Another Proof that R[subscript XX](0) [greater than equal] \| R[subscript XX]([tau])\|	p. 166
10.3 Estimating the PSD	p. 166
10.4 The Properties of the Periodogram	p. 168
10.5 Exercises	p. 169
11 Wiener Filters	p. 171
11.1 A Non-Causal Solution	p. 171
11.2 White Noise and a Low-Pass Signal	p. 174
11.3 Causality, Anti-Causality and the Fourier Transform	p. 175
11.4 The Optimal Causal Filter	p. 177
11.5 Two Examples	p. 179
11.5.1 White Noise and a Low-Pass Signal	p. 179
11.5.2 Low-Pass Signal and Noise	p. 180
11.6 Exercises	p. 181
A A Brief Overview of Linear Algebra	p. 185
A.1 The Space C[superscript N]	p. 185
A.2 Linear Independence and Bases	p. 186
A.3 A Preliminary Result	p. 187
A.4 The Dimension of C[superscript N]	p. 188
A.5 Linear Mappings	p. 189
A.6 Matrices	p. 190
A.7 Sums of Mappings and Sums of Matrices	p. 191
A.8 The Composition of Linear Mappings-Matrix Multiplication	p. 192
A.9 A Very Special Matrix	p. 193
A.10 Solving Simultaneous Linear Equations	p. 193
A.11 The Inverse of a Linear Mapping	p. 196
A.12 Invertibility	p. 197
A.13 The Determinant-A Test for Invertibility	p. 199
A.14 Eigenvectors and Eigenvalues	p. 200
A.15 The Inner Product	p. 202
A.16 A Simple Proof of the Cauchy-Schwarz Inequality	p. 203
A.17 The Hermitian Transpose of a Matrix	p. 204
A.18 Some Important Properties of Self-Adjoint Matrices	p. 205
A.19 Exercises	p. 206
Bibliography	p. 209
Index	p. 212

Available:*

On Order

Summary

Summary

Table of Contents