## A First Course in Statistics for Signal AnalysisThis essentially self-contained, deliberately compact, and user-friendly textbook is designed for a first, one-semester course in statistical signal analysis for a broad audience of students in engineering and the physical sciences. The emphasis throughout is on fundamental concepts and relationships in the statistical theory of stationary random signals, explained in a concise, yet fairly rigorous presentation. Topics and Features: Fourier series and transforms are developed from scratch, emphasizing the time-domain vs. frequency-domain duality. Basic concepts of probability theory, laws of large numbers, the stability of fluctuations law, and statistical parametric inference procedures are presented. Introduction of the fundamental concept of a stationary random signal and its autocorrelation structure. Many diverse examples as well as end-of-chapter problems and exercises. Developed by the author over the course of several years of classroom use, A First Course in Statistics for Signal Analysis may be used by junior/senior undergraduates or graduate students in electrical, systems, computer, and biomedical engineering, as well as the physical sciences. The work is also an excellent resource of educational and training material for scientists and engineers working in research laboratories. |

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### Contents

1 | |

12 Time domain and frequency domain descriptions | 8 |

13 Characteristics of signals | 12 |

14 Problems and exercises | 13 |

Spectral Representation of Deterministic Signals Fourier Series and Transforms | 16 |

22 Approximation of periodic signals by finite Fourier sums | 26 |

23 Aperiodic signals and Fourier transforms | 31 |

24 Basic properties of the Fourier transform | 35 |

54 Problems and exercises | 124 |

Transmission of Stationary Signals through Linear Systems | 127 |

61 The time domain analysis | 128 |

62 Frequency domain analysis and system bandwidth | 136 |

63 Digital signal discretetime sampling | 140 |

64 Problems and exercises | 144 |

Optimization of SignaltoNoise Ratio in Linear Systems | 147 |

72 Filter structure matched to signal | 151 |

25 Fourier transforms of some nonintegrable signals Dirac delta impulse | 37 |

26 Discrete and fast Fourier transforms | 42 |

27 Problems and exercises | 44 |

Random Quantities and Random Vectors | 47 |

31 Discrete continuous and singular random quantities | 48 |

32 Expectations and moments of random quantities | 62 |

33 Random vectors conditional probabilities statistical independence and correlations | 67 |

34 The leastsquares fit regression line | 77 |

35 The law of large numbers and the stability of fluctuations law | 80 |

36 Estimators of parameters and their accuracy confidence intervals | 82 |

37 Problems exercises and tables | 86 |

Stationary Signals | 93 |

42 Estimating the mean and the autocorrelation function ergodic signals | 105 |

43 Problems and exercises | 109 |

Power Spectra of Stationary Signals | 113 |

52 Power spectrum and autocorrelation function | 114 |

53 Power spectra of interpolated digital signals | 121 |

73 The Wiener filter | 154 |

74 Problems and exercises | 156 |

Gaussian Signals Correlation Matrices and Sample Path Properties | 158 |

81 Linear transformations of random vectors | 160 |

82 Gaussian random vectors | 162 |

83 Gaussian stationary signals | 165 |

84 Sample path properties of general and Gaussian stationary signals | 167 |

85 Problems and exercises | 173 |

Discrete Signals and Their Computer Simulations | 175 |

92 Cumulative power spectrum of discretetime stationary signal | 176 |

93 Stochastic integration with respect to signals with uncorrelated increments | 179 |

94 Spectral representation of stationary signals | 184 |

95 Computer algorithms | 188 |

96 Problems and exercises | 196 |

Bibliographical Comments | 197 |

201 | |

### Other editions - View all

A First Course in Statistics for Signal Analysis Wojbor Andrzej Woyczyński No preview available - 2006 |

### Common terms and phrases

analysis approximation autocorrelation function autocorrelation sequence calculation central limit theorem Chapter coefficient complex exponentials complex numbers components conditional probability converge correlation covariance defined delta impulse Dirac delta discrete discrete-time white noise domain equation estimator Example Find the power finite Fourier sums Fourier series Fourier transform frequency fX(x Gaussian random quantity Gaussian random vector impulse response impulse response function input signal joint p.d.f. matrix mean power mean-square output autocorrelation output signal parameter periodic signal power spectrum power spectrum density power transfer function probability distribution Problems and exercises random quantity random signal X(t random vector RC filter sample paths Section shown in Figure signal x(t signal-to-noise ratio simulation spectral density spectral representation stationary random signals stationary signal X(t statistically independent Std(X SX f SX(f SY(f theorem tion transfer function Var(X variable variance white noise yx(t zero zero-mean