Lecture Notes for ECE 361. Fall 1995

Transcription

1 Introduction to Digital Communication Systems Lecture Notes for ECE 361 Fall 1995 Dilip V. Sarwate Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign Urbana, Illinois by Dilip V. Sarwate All rights reserved. No part of this manuscript may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise without the prior written permission of the author.

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3 Introduction to Digital Communication Systems: Lecture Notes for ECE 361, Fall 1995 Note to ECE 361 students These optional lecture notes sold through the IEEE Student Branch are intended as a supplement to the required textbook for ECE 361: Introduction to Digital Communications by M. B. Pursley, (also sold through the IEEE Student Branch.) The lecture notes include the text and diagrams that will appear on the viewgraphs that I will use in teaching ECE 361 in Fall The lecture notes also include much additional material in the form of comments and remarks on the material presented on the viewgraphs, as well as several exercises. This additional material is intended to help you understand the concepts presented on the viewgraphs. In some instances, the comments will remind you of a key fact that will make it clear how Step 6 on the viewgraph follows from Step 5. The comments may have a few details that have been skipped on the viewgraph itself, or (horrors!) they may suggest that you work out the details of some derivation all by yourself. Many of these comments (usually in paraphrased form) will also be made orally during the lectures. Thus, the inclusion of the comments in the lecture notes saves you the trouble of scribbling down these pearls of wisdom. Of course, some of these comments will not be made orally, usually because of sheer forgetfulness on my part but occasionally because of a lack of time. Thus, the inclusion of the comments in these lecture notes ensures that what I forgot to say or did not have time to say is nonetheless available to you. Please keep in mind that, regardless of the reason for the omission, the remarks included in these lecture notes but not said out loud in class are just as important as the ones stated in a ringing voice, and you are urged not to omit such unstated facts when studying for the course. In these lecture notes, the material that will appear on the viewgraphs is printed with borders around it. The additional remarks and comments are distinguished by not having any borders. The lecture notes are also printed with wide outside margins to provide you with space for making your own notes or for doodling when the lectures are more boring than usual. Dilip V. Sarwate Urbana, Illinois August 1995 iii

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5 Introduction to Digital Communication Systems: Lecture Notes for ECE 361, Fall 1995 Contents Introduction to Digital Communication Systems... 1 Analog vs Digital Communication Systems... 1 Modern Digital Communicaton Theory... 3 Block Diagram of a Digital Communication System... 4 Probability Review Notation and Axioms Conditional Probability Bayes Formula Stochastic Independence Random Variables Expectation Gaussian Random Variables Many Random Variables Conditional Distributions Jointly Gaussian Random Variables Limit Theorems Markov s and Chebyshev s Inequalities; the Chernoff Bound The Laws of Large Numbers The Central Limit Theorem Random Processes Definition and Basic Properties Distribution and Density Functions Mean, Autocorrelation and Covariance Stationary and Wide-Sense-Stationary Random Processes Cross-Correlation and Cross-Covariance Functions White Noise Processes Random Processes in Linear Systems Mean-Square Convergence vs Stochastic Mean-Square Convergence Interchange of Expectation and Integration Operations The Mean of the Output Process Cross-correlation between Input and Output Processes Autocorrelation Function of the Output Process Spectral Analysis of Random Processes Power Spectrum of a Random Digital Signal Baseband Digital Communication Systems Binary Communication System Signal Output from Filter Noise Output from Filter Thresholds and Error Probabilities Conditional Error Probabilities Two Criteria for Choosing the Threshold Minimax Threshold Criterion Summary for Minimax Threshold Criterion Minimum Average Error Probability (Bayes ) Criterion Properties of Bayes Threshold Properties of Bayes Error Probability Summary for Minimax and Bayes Criteria Matched Filters The Schwarz Inequality Proof Application to Matched Filter Design Error Probabilities for Orthogonal and Antipodal Signaling Frequency-Domain Interpretation of the Matched Filter Transmission of a Sequence of Bits v

6 Introduction to Digital Communication Systems: Lecture Notes for ECE 361, Fall 1995 Correlation Receivers Comparison of Matched Filter and Correlation Receiver Outputs Summary of Matched Filters and Correlation Receivers Signal Design Pre-whitening Filters Signal Space Concepts Orthogonal and Orthonormal Functions Minimum Mean-Square Error Representations Complete Orthonormal Sets Gram-Schmidt Orthonormalization Procedure Orthonormal Representations Signal Representation Noise Representation Vector Channels Decision Regions Binary Signaling Maximum-Likelihood Decision Regions Bayesian Decision Regions Summary for Binary Signaling M-ary Signaling Preference Regions and Decision Regions Symbol Error Probabilities Two Receiver Implementations Union Bound on Error Probabilities Nearest-Neighbor Approximation for Error Probabilities Bit Error Probabilities Relationship between Bit Error Probability and Symbol Error Probability One-dimensional M-ary Signaling Symbol and Bit Error Probabilities Signal Energy Asymptotic Behavior for Large M Two-dimensional M-ary Signaling Symbol and Bit Error Probabilities Signal Energy Asymptotic Behavior for Large M M-ary Phase-Shift Keying Symbol Error Probabilities Summary for One- and Two-Dimensional M-ary Signaling Multidimensional M-ary Signaling Symbol and Bit Error Probabilities Signal Energy Asymptotic Behavior for Large M Bandwidth of a Digital Signal The Landau-Pollak Theorem M-ary Orthogonal Signaling Probability of Correct Decision and Probability of Error Union Bound on the Probability of Error Probability of Bit Error Asymptotic Behavior for Large M Simplex or Trans-Orthogonal M-ary Signaling Bi-Orthogonal M-ary Signaling Union Bound of the Probability of Error Bit Error Probability Generation of Orthogonal, Transorthogonal, and Bi-orthogonal Signals FSK, PPM, and Hadamard Signal Sets Fast Walsh-Hadamard Transforms Summary for Multidimensional M-ary Signaling Optimality of Linear Receivers on Additive Gaussian Noise Channels vi