Elements of Detection and Signal Design

Due to a steady flow of requests over several years, Springer-Verlag now provides a corrected reprint of this text. It is designed to serve as a text for a first semester graduate level course for students in digital communication systems. As a pre­ requisite, it is presumed that the reader has an understanding of basic probability and stochastic processes. The treatment of digital communications in this book is intended to serve as an introduction to the subject. Part one is a development of the elements of statistical communication theory and radar detection. The text begins with a general model of a communication system which is extensively developed and the performance analyses of various conventional systems. The first part also serves as introductory material for the second part of the text which is a comprehensive study of the theory of transmitter optimization for coherent and noncoherent digital commu­ nication systems, that is, the theory of signal design.

One Elements of Detection.- 1 Introduction.- 2 A mathematical model.- 3 General decision-theory concepts.- 4 Binary detection systems minimizing the average risk.- 4.1 Binary decision functions.- Example 4.1 Specialization to a signal space of two elements.- 4.2 Vector model.- 4.3 Coherent binary phase modulation.- 5 Minimax decision-rule concepts.- 6 Radar detection theory.- 6.1 Radar-system-design philosophy.- 6.2 Vector model.- 7 Binary composite-hypothesis testing.- Example 7.1 Detection of one pulse of known arrival time.- Example 7.2 Detection with the complex-envelope representation.- Example 7.3 Binary noncoherent communication system.- 8 Detection and communication in colored noise.- 8.1 Detection in colored noise.- 8.2 Coherent binary communication in colored noise.- 8.3 Noncoherent binary communication in colored noise.- 9 Detecting a stochastic signal in noise.- 9.1 Detection of a random vector.- 9.2 Detection of a stochastic process in noise.- 10 M-ary digital communication systems.- 10.1 Coherent M-ary communication.- 10.2 Noncoherent M-ary communication.- Two Signal Design.- 11 Introduction.- 12 Problem statement for coherent channels.- 12.1 Descriptipn in the time domain.- 12.2 Reduction to finite-dimensional euclidean space.- 12.3 Bandwidth considerations.- 13 Signal design when the dimensionality of the signal set is restricted to 2.- 13.1 Optimal signal selection in two dimensions.- 13.2 Communication efficiency and channel capacity for two-dimensional signal sets.- 13.3 Partial ordering of the class of two-dimensional signal sets.- 13.4 The dependence of some suboptimal signal sets on the signal-to-noise ratio.- 14 General theory.- 14.1 Introduction.- 14.2 Convex-body considerations: small signal-to-noise ratios.- 14.3 Linearly dependent versus linearly independent signal sets.- 14.4 Gradient of the probability of detection.- 14.5 Signal sets whose convex hull does not include the origin.- 14.6 The admissible ? space.- 14.7 Series expansions and asymptotic approximations.- 15 Optimality for coherent systems when dimensionality is not specified: regular simplex coding.- 15.1 Necessary (first-order) considerations for optimality.- 15.2 Uniqueness of the regular simplex satisfying necessary conditions for all signal-to-noise ratios.- 15.3 Global optimality of the regular simplex for large signal-to-noise ratios.- 15.4 Sufficient (second-order) conditions for optimality.- 15.5 Maximizing the minimum distance.- 16 Optimality for coherent systems when the dimensionality is restricted to D ? M ? 2.- 16.1 Necessary (first-order) conditions.- 16.2 Sufficient (second-order) conditions.- 16.3 Choosing the largest of several local maxima.- 16.4 Five signal vectors in three dimensions.- 17 Optimality for Coherent Systems when the Dimensionality is Restricted to D ? M ? K, where K ? M/2.- 17.1 Necessary (first-order) conditions.- 17.2 Choosing the largest of several local extrema.- 17.3 The effect of dimensionality on system performance.- 18 Additional Solutions for Three-Dimensional Signal Structures.- 19 Signal-Design Concepts for Noncoherent Channels.- 19.1 Necessary (first-order) conditions for noncoherent optimality.- 19.2 Evaluation of probability of error for the orthogonal noncoherent signal structure.- 19.3 Sufficient (second-order) conditions for noncoherent optimality.- 19.4 Global optimality when M = 2.- APPENDIX A Summary of conditional gaussian probability density functions.- APPENDIX B Karhunen-Loeve expansion.- APPENDIX C Modified Bessel function of the first kind.- APPENDIX E Summary of tetrachoric series.- APPENDIX F Chi-Squared distribution.