Elements of Detection and 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.