Discrete-Signal Analysis and Design

A clear, step-by-step approach to practical uses of discrete-signal analysis and design, especially for communications and radio engineers

This book provides an introduction to discrete-time and discrete-frequency signal processing, which is rapidly becoming an important, modern way to design and analyze electronics projects of all kinds. It presents discrete-signal processing concepts from the perspective of an experienced electronics or radio engineer, which is especially meaningful for practicing engineers, technicians, and students. The approach is almost entirely mathematical, but at a level that is suitable for undergraduate curriculums and also for independent, at-home study using a personal computer.

Coverage includes:

• First principles, including the Discrete Fourier Transform (DFT)

• Sine, cosine, and theta

• Spectral leakage and aliasing

• Smoothing and windowing

• Multiplication and convolution

• Probability and correlation

• Power spectrum

• Hilbert transform

The accompanying CD-ROM includes Mathcad® v.14 Academic Edition, which is reproduced with permission and has no time limitation for use, providing users with a sophisticated and world-famous tool for a wide range of applied mathematics capabilities.

Discrete-Signal Analysis and Design is written in an easy-to-follow, conversational style and supplies readers with a solid foundation for more advanced literature and software. It employs occasional re-examination and reinforcement of particularly important concepts, and each chapter contains self-study examples and full-page Mathcad® Worksheets, worked-out and fully explained.

Preface xi

Introduction 1

Goals of the Book

Discrete Signals

Advantages of Discrete-Signal Analysis and Design

DFT and IDFT

Mathcad Program

MATLAB and Less Expensive Approaches

Multisim Program from National Instruments Co.

Mathtype Program

LabVIEW

Search Engines

Personal Productivity Software Capability

1 First Principles 9

Sequence Structure in the Time and Frequency Domains

Two-Sided Time and Frequency

Discrete Fourier Transform

Inverse Discrete Fourier Transform

Frequency and Time Scaling

Number of Samples

Complex Frequency-Domain Sequences

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

2 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 Amplifier Distortion and Square Law Modulator

Example 2-2: Analysis of the Ramp Function

3 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

4 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

5 Multiplication and Convolution 77

Sequence Multiplication

Polynomial Multiplication

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

6 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 Coefficient

7 The Power Spectrum 113

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

System Power Transfer

Cross Power Spectrum

Example of Calculating Phase Noise

8 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

William E. Sabin, MSEE, Life Member IEEE, has worked at a professional engineering level in the electronics industry for forty years in almost all areas of signal processing, including analog, discrete, and digital. He has coedited three books on the subject of radio systems and circuits and is the author of about forty technical articles in electronics journals.

“Discrete-Signal analysis and Design is written in an easy-to-follow, conversational style and supplies readers with a solid foundation for more advanced literature and software.  It employs the occasional re-examination and reinforcement of particularly important concepts and each chapter contains self-study examples and full-page Mathcad Worksheets, worked-out and fully explained.”  (International Journal Microstructure & Materials Properties, 2009)