viii CONTENTS Frequency and Time Scaling Number of Samples Complex Frequency-Domain Sequences x(n) Versus Time and X(k) Versus Frequency Sine, Cosine, and θ 27 One-Sided Sequences Combinations of Two-Sided Phasors Time and Spectrum Transformations Transforming Two-Sided Phasor Sequences into One-Sided Sine, Cosine, θ Example 2-1: Nonlinear AmpliÞer Distortion and Square Law Modulator Example 2-2: Analysis of the Ramp Function Spectral Leakage and Aliasing 43 Spectral Leakage Noninteger Values of Time x(n) and Frequency X(k) Example 3-1: Frequency Scaling to Reduce Leakage Aliasing in the Frequency Domain Example 3-2: Analysis of Frequency-Domain Aliasing Aliasing in the Time Domain Smoothing and Windowing 61 Smoothing the Rectangular Window, Without Noise and with Noise Smoothed Sequences Near the Beginning and End Rectangular Window Hamming Window Hanning (Hann) Window Relative Merits of the Three Windows Scaling the Windows Multiplication and Convolution Sequence Multiplication Polynomial Multiplication 77 CONTENTS ix Convolution Discrete Convolution Basic Equation Relating Convolution to Polynomial Multiplication “Fold and Slide” Concept Circular Discrete Convolution (Try to Avoid) Sequence Time and Phase Shift DFT and IDFT of Discrete Convolution Fig 5-6 Compare Convolution and Multiplication Deconvolution Probability and Correlation 95 Properties of a Discrete Sequence Expected Value of x(n) Include Some Additive Noise Envelope Detection of Noisy Sequence Average Power of Noiseless Sequence Power of Noisy Sequence Sequence Averaging Variance Gaussian (Normal) Distribution Cumulative Distribution Correlation and Covariance Autocorrelation Cross-Correlation Autocovariance Cross-Covariance Correlation CoefÞcient The Power Spectrum Finding the Power Spectrum Two-Sided Phasor Spectrum, One-Sided Power Spectrum Example 7-1: The Use of Eq (7-2) Random Gaussian Noise Spectrum Measuring the Power Spectrum Spectrum Analyzer Example Wiener-Khintchine Theorem 113 x CONTENTS System Power Transfer Cross Power Spectrum Example of Calculating Phase Noise The Hilbert Transform 129 The Perfect Hilbert Transformer Example of a Hilbert Transform of an Almost-Square Wave Smoothing of the Example Peaks in Hilbert of Square Wave Mathematics of the Hilbert Transform Analytic Signal Example 8-2: Construction of Analytic Signal Single-Sideband RF Signals SSB Design Basic All-Pass Network −90◦ Cascaded Phase Shift Audio Network Why the −90◦ Network Is Not Equivalent to a Hilbert Transformer Phasing Method SSB Transmitter Filter Method SSB Transmitter Phasing Method SSB Receiver Filter Method SSB Receiver Appendix: Additional Discrete-Signal Analysis and Design Information 153 Discrete Derivative State-Variable Solutions Using the Discrete Derivative to Solve a Time Domain Discrete Differential Equation Glossary 163 Index 171 PREFACE The Introduction explains the scope and motivation for the title subject My association with the Engineering Department of Collins Radio Co., later Rockwell Collins, in Cedar Rapids, Iowa, and my education at the University of Iowa have been helpful background for the topics covered The CD accompanying the book includes the Mathcad V.14 Academic Edition, which is reproduced by permission This software is fully functional, with no time limitation for its use, but cannot be upgraded For technical support, more information about purchasing Mathcad, or upgrading from previous editions, see http://www.ptc.com Mathcad is a registered trademark of Parametric Technology Corporation (PTC), http://www.ptc.com PTC owns both the Mathcad software program and its documentation Both the program and documentation are copyrighted with all rights reserved No part of the program or its documentation may be produced, transmitted, transcribed, stored in a retrieval system, or translated into any language in any form without the written permission of PTC William E Sabin xi ... Appendix: Additional Discrete- Signal Analysis and Design Information 153 Discrete Derivative State-Variable Solutions Using the Discrete Derivative to Solve a Time Domain Discrete Differential... Square Wave Mathematics of the Hilbert Transform Analytic Signal Example 8-2: Construction of Analytic Signal Single-Sideband RF Signals SSB Design Basic All-Pass Network −90◦ Cascaded Phase Shift... Discrete Convolution Basic Equation Relating Convolution to Polynomial Multiplication “Fold and Slide” Concept Circular Discrete Convolution (Try to Avoid) Sequence Time and Phase Shift DFT and