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An Introduction to Statistical Signal Processing

The origins of this book lie in our earlier book Random Processes: A Mathematical Approach for Engineers, Prentice Hall, 1986. This book began as a second edition to the earlier book and the basic goal remains unchanged — to introduce the fundamental ideas and mechanics of random processes to engineers in a way that accurately reflects the underlying mathematics, but does not require an extensive mathematical background and does not belabor detailed general proofs when simple cases suffice to get the basic ideas across. | An Introduction to Statistical Signal Processing f −1(F ) f F Pr(f ∈ F )=P ({ω : ω ∈ F })=P (f −1(F )) ✲ May 5, 2000 ii An Introduction to Statistical Signal Processing Robert M. Gray and Lee D. Davisson Information Systems Laboratory Department of Electrical Engineering Stanford University and Department of Electrical Engineering and Computer Science University of Maryland iv c 1999 by the authors. v to our Families vi Contents Preface xi Glossary xv 1 Introduction 1 2 Probability 11 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Spinning Pointers and Flipping Coins . . . . . . . . . . . . 15 2.3 Probability Spaces . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.1 Sample Spaces . . . . . . . . . . . . . . . . . . . . . 28 2.3.2 Event Spaces . . . . . . . . . . . . . . . . . . . . . . 31 2.3.3 Probability Measures . . . . . . . . . . . . . . . . . . 42 2.4 Discrete Probability Spaces . . . . . . . . . . . . . . . . . . 45 2.5 Continuous Probability Spaces . . . . . . . . . . . . . . . . 56 2.6 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.7 Elementary Conditional Probability . . . . . . . . . . . . . 71 2.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3 Random Objects 85 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.1.1 Random Variables . . . . . . . . . . . . . . . . . . . 85 3.1.2 Random Vectors . . . . . . . . . . . . . . . . . . . . 89 3.1.3 Random Processes . . . . . . . . . . . . . . . . . . . 93 3.2 Random Variables . . . . . . . . . . . . . . . . . . . . . . . 95 3.3 Distributions of Random Variables . . . . . . . . . . . . . . 104 3.3.1 Distributions . . . . . . . . . . . . . . . . . . . . . . 104 3.3.2 Mixture Distributions . . . . . . . . . . . . . . . . . 108 3.3.3 Derived Distributions . . . . . . . . . . . . . . . . . 111 3.4 Random Vectors and Random Processes . . . . . . . . . . . 115 3.5 Distributions of Random Vectors . . . . . . . . .