DISCRETE SIGNAL ANALYSIS AND DESIGN p3

viii CONTENTSFrequency and Time Scaling Number of Samples Complex Frequency-Domain Sequences xn Versus Time and Xk Versus Frequency One-Sided Sequences Combinations of Two-Sided Phasors

viii CONTENTS

Frequency and Time Scaling

Number of Samples

Complex Frequency-Domain Sequences

x(n) Versus Time and X(k) Versus Frequency

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 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 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

Sequence Multiplication

Polynomial Multiplication

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

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

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

x CONTENTS

System Power Transfer

Cross Power Spectrum

Example of Calculating Phase Noise

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

Discrete Derivative

State-Variable Solutions

Using the Discrete Derivative to Solve a Time Domain

Discrete Differential Equation

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 Aca-demic 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 Corpora-tion (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 docu-mentation 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

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