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E-raamat: Random Signals and Noise: A Mathematical Introduction

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  • Formaat: 236 pages
  • Ilmumisaeg: 03-Oct-2018
  • Kirjastus: CRC Press Inc
  • Keel: eng
  • ISBN-13: 9781420007770
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Random Signals and Noise: A Mathematical IntroductionSuurem pilt
  • Formaat: 236 pages
  • Ilmumisaeg: 03-Oct-2018
  • Kirjastus: CRC Press Inc
  • Keel: eng
  • ISBN-13: 9781420007770
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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 xvii
1 Elementary Probability Theory
1(30)
1.1 The Probability Function
1(1)
1.2 A Bit of Philosophy
1(1)
1.3 The One-Dimensional Random Variable
2(1)
1.4 The Discrete Random Variable and the PMF
3(1)
1.5 A Bit of Combinatorics
4(3)
1.5.1 An Introductory Example
4(1)
1.5.2 A More Systematic Approach
5(1)
1.5.3 How Many Ways Can N Distinct Items Be Ordered?
6(1)
1.5.4 How Many Distinct Subsets of N Elements Are There?
6(1)
1.5.5 The Binomial Formula
7(1)
1.6 The Binomial Distribution
7(2)
1.7 The Continuous Random Variable, the CDF, and the PDF
9(3)
1.8 The Expected Value
12(5)
1.9 Two Dimensional Random Variables
17(5)
1.9.1 The Discrete Random Variable and the PMF
18(1)
1.9.2 The CDF and the PDF
19(1)
1.9.3 The Expected Value
20(1)
1.9.4 Correlation
21(1)
1.9.5 The Correlation Coefficient
21(1)
1.10 The Characteristic Function
22(2)
1.11 Gaussian Random Variables
24(2)
1.12 Exercises
26(5)
2 An Introduction to Stochastic Processes
31(10)
2.1 What Is a Stochastic Process?
31(2)
2.2 The Autocorrelation Function
33(1)
2.3 What Does the Autocorrelation Function Tell Us?
33(1)
2.4 The Evenness of the Autocorrelation Function
34(1)
2.5 Two Proofs that Rxx (0) > or equal to |Rxx (τ)|
34(2)
2.6 Some Examples
36(2)
2.7 Exercises
38(3)
3 The Weak Law of Large Numbers
41(14)
3.1 The Markov Inequality
41(1)
3.2 Chebyshev's Inequality
42(1)
3.3 A Simple Example
43(2)
3.4 The Weak Law of Large Numbers
45(2)
3.5 Correlated Random Variables
47(2)
3.6 Detecting a Constant Signal in the Presence of Additive Noise
49(1)
3.7 A Method for Determining the CDF of a Random Variable
50(1)
3.8 Exercises
51(4)
4 The Central Limit Theorem
55(18)
4.1 Introduction
55(1)
4.2 The Proof of the Central Limit Theorem
56(3)
4.3 Detecting a Constant Signal in the Presence of Additive Noise
59(2)
4.4 Detecting a (Particular) Non-Constant Signal in the Presence of Additive Noise
61(2)
4.5 The Monte Carlo Method
63(1)
4.6 Poisson Convergence
64(4)
4.7 Exercises
68(5)
5 Extrema and the Method of Lagrange Multipliers
73(16)
5.1 The Directional Derivative and the Gradient
73(1)
5.2 Over-Determined Systems
74(3)
5.2.1 General Theory
74(1)
5.2.2 Recovering a Constant from Noisy Samples
75(1)
5.2.3 Recovering a Line from Noisy Samples
76(1)
5.3 The Method of Lagrange Multipliers
77(6)
5.3.1 Statement of the Result
77(1)
5.3.2 A Preliminary Result
78(2)
5.3.3 Proof of the Method
80(3)
5.4 The Cauchy-Schwarz Inequality
83(1)
5.5 Under-Determined Systems
84(2)
5.6 Exercises
86(3)
6 The Matched Filter for Stationary Noise
89(16)
6.1 White Noise
89(2)
6.2 Colored Noise
91(5)
6.3 The Autocorrelation Matrix
96(1)
6.4 The Effect of Sampling Many Times in a Fixed Interval
97(1)
6.5 More about the Signal to Noise Ratio
98(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


Shlomo Engelberg
Lisainfo e-raamatute kohta Lisainfo e-raamatute kohta
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