Introduction to Digital Signal Proce Introduction to Digital Signal Processing al Processing

492 Pages·2017·16.42 MB·English

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Introduction to
Digital Signal Processing

Theory and Applications
Using MATLAB®

Kathleen A.K. Ossman, Ph.D.
[email protected]
Department of Engineering Education
University of Cincinnati
Cincinnati, OH 45221
© 2017 Kathleen A. K. Ossman
The manuscript: "Introduction to Digital Signal Processing Theory and Applications using
MATLAB®," is protected under the copyright laws of the United States and other countries in
both electronic and printed form.

MATLAB® and Simulink® are registered trademarks of The MathWorks, Inc. and are used with
permission. The MathWorks does not warrant the accuracy of the text or exercises in this book.
This book’s use or discussion of MATLAB® software or related products does not constitute
endorsement or sponsorship by The MathWorks of a particular pedagogical approach or
particular use of the MATLAB® software.

PERMISSIBLE USES
1. You may download any or all of the book in electronic form, for your personal use.
2. You may create one printed copy of any or all of the book, for your personal use.
PROHIBITED USES (Unless approved in writing by the author)
1. Using the electronic files, or printed copies derived therefrom, for any commercial
purpose.
2. Making a copy of any portion of the book, either in electronic or printed form (except as
allowed above).
3. Posting the book on a website or network.
4. All other uses that are prohibited under copyright law.
Table of Contents
Preface vi

1. Introduction 1
MATLAB® Exercise: Introduction to MATLAB 3

2. Signals and Signal Spectra 11
2.1 Analog Signals 11
2.2 Discrete Signals 12
2.3 Digital Signals 13
2.4 Examples of Signals 14
2.5 Spectrum of Periodic Signals 20
2.6 Spectrum of Non-periodic Signals 24
2.7 Signal Bandwidth 26
2.8 Spectrogram 28
2.9 Application: Synthesis of Musical Instruments 30
2.10 Application: Simple Audio Effects 41
Answers to Challenge Questions 45
Chapter 2 Problems 49
MATLAB® Exercise: Sinusoidal Signals and Sound Synthesis 51

3. Sampling and Reconstructing Signals 57
3.1 The Basics of Sampling 57
3.2 A/D Conversion: Terminology and Equations 58
3.3 Types of A/D Converters 60
3.4 Nyquist-Shannon Sampling Theorem 66
3.5 Aliasing 69
3.6 Design of Anti-aliasing Filters 74
3.7 Practical Considerations 77
3.8 Reconstruction 80
3.9 Practical Application for Undersampling 83
Answers to Challenge Questions 84
Chapter 3 Problems 87
MATLAB® Exercise: Sampling, Aliasing, and Reconstruction 89

4. Key Mathematical Concepts 95
4.1 Fixed-point Representation 95
4.2 Floating-point Representation 102
4.3 Fixed-point vs. Floating-point Processors 105
4.4 Discrete Systems 105
4.5 Z-Transforms 107
4.6 Convolution 113
4.7 Inverse Z-Transforms 117
4.8 Correlation 120
Answers to Challenge Questions 124
Chapter 4 Problems 126
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5. Finite Impulse Response (FIR) Filters 128
5.1 Finite Impulse Response (FIR) Digital Filters 128
5.2 Designing Low-Pass FIR Filters 130
5.3 Designing High-Pass, Band-Pass, and Notch FIR Filters 141
5.4 Designing FIR Filters using FDAT 145
5.5 Practical Considerations for FIR Filters 156
5.6 Application: Special Types of FIR Filters 163
5.7 Multi-Rate Digital Signal Processing 175
Answers to Challenge Questions 185
Chapter 5 Problems 189
MATLAB® Exercise: FIR Filters 192

6. Infinite Impulse Response (IIR) Filters 202
6.1 Infinite Impulse Response (IIR) Digital Filters 202
6.2 IIR Filter Design 205
6.3 Designing IIR Filters using FDAT 207
6.4 Filter Realizations 215
6.5 Practical Considerations for IIR Filters 218
6.6 Examples of of IIR Filters 230
6.7 Application: Audio Effects Using IIR Filters 233
Answers to Challenge Questions 247
Chapter 6 Problems 250
MATLAB® Exercise: IIR Filters 252

7. Discrete Fourier Transforms 258
7.1 Discrete Fourier Transforms 258
7.2 Windowing the Data to Enhance Spectral Analysis 270
7.3 Fast Fourier Transforms 272
7.4 Power Spectral Density 283
7.5 Convolution and Correlation using Fast Fourier Transforms 286
7.6 Matched Filters 291
7.7 Algorithms for Focused Spectral Analysis 295
Answers to Challenge Questions 303
Chapter 7 Problems 305
MATLAB® Exercise: Fast Fourier Transforms 308

8. Real-Time Processing Algorithms 313
8.1 Difference Equations and Circular Addressing 313
8.2 Overlap-Add and Overlap-Save Algorithms for Convolution or Correlation 314
8.3 Real-time Spectral Analysis using Fast Fourier Transforms 318
8.4 Sliding Discrete Fourier Transform 329
8.5 Adaptive Filtering 342
Answers to Challenge Questions 347
Chapter 8 Problems 349
MATLAB® Exercise: Introduction to Adaptive FIR filters 351

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9. Introduction to Image Processing 357
9.1 Terminology and Common Image Formats 357
9.2 Image Effects 359
9.3 De-noising Images 374
9.4 Image Compression: JPEG Format 377
Answers to Challenge Questions 405
Chapter 9 Problems 407
MATLAB® Exercise: Image Processing 409

10. Introduction to Radar Processing 414
11.1 Introduction 414
11.2 Performance Metrics 416
11.3 Matched Filters, Chirp Signals, and Pulse Compression 420
11.4 Determining Range 432
11.5 Pulse Doppler Processing to Estimate Target Velocity 455
Chapter 10 Problems 465
MATLAB® Exercise: Radar Range Processing 468

Appendix 471
Table A.1: Fourier Series 472
Table A.2: Fourier Transforms 473
Table A.3: Fourier Transform Properties 474
Table A.4: Z-Transform Properties 474
Table A.5: Z-Transforms 475

Bibliography 476

Index 479

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Preface

This book provides an introduction to the field of digital signal processing (DSP) and includes
several application areas in addition to basic theory.

Pedagogical Approach

 Thorough coverage of sampling, design of digital filters, and Discrete Fourier transforms
is provided with practical considerations included throughout.

 Every DSP concept introduced in the text is followed with clear, carefully written
numerical examples to illustrate the concept.

 Graphics are used to help students visualize concepts rather than becoming lost in the
mathematics.

 MATLAB® is used extensively throughout the textbook both to help students
comprehend digital signal processing concepts and to teach students to use MATLAB® as
a design tool.

 Lab exercises are included at the end of each chapter to provide students with hands on
practice with the material covered in the chapter. These exercises could be used as a lab
manual for a lab course accompanying a DSP lecture course.

 Challenge questions are spread throughout each chapter to allow students to test their
understanding of the material as they are reading the text. Answers are provided at the
end of each chapter.

 The text includes many practical applications for DSP including image processing, music
synthesis, and radar processing to keep students interested in the material.

 Problems are included at the end of each chapter.

Intended Audience
This book is written for electrical and computer engineering students and for engineering
technology students. For engineering students, the text serves as a practical introduction to the
subject with interesting applications. For engineering technology students, the text provides a
basis for developing good applications and projects. Engineering professionals with little or no
background in digital signal processing would find this text to be a useful introduction to DSP as
well providing good instruction on how to use MATLAB® and Simulink®.

About the Author
Kathleen Ossman earned her BSEE and MSEE from the Georgia Institute of Technology and a
Ph.D. in Electrical Engineering from the University of Florida. She is an associate professor at
the University of Cincinnati and has enjoyed teaching electrical engineering courses and labs for
many, many years.
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Chapter Synopses

Chapter 1: Introduction
This chapter introduces a digital signal processing system, list several practical applications, and
provides a brief outline of the rest of the text. The lab exercise at the end of the chapter provides
an introduction to using MATLAB® – inputting data, data formats, plotting signals, and writing
functions.

Chapter 2: Signals and Signal Spectra
This chapter introduces signals, both analog and digital. Periodic and non-periodic signals are
discussed. Signals are represented both in the time domain (very familiar to students) and in the
frequency domain (not so familiar) through several illustrative examples. Fourier series, Fourier
transform, definition of signal bandwidth, and spectrograms are included in the frequency
domain discussion. Applications include using additive or FM synthesis to synthesize various
musical instruments and an introduction to simple audio effects such as tremolo and ring
modulation. The lab exercise at the end of the chapter stresses signal representation in both the
time and frequency domain using MATLAB® and includes some interesting audio applications.

Chapter 3: Sampling and Reconstructing Signals
This chapter deals with an extremely important DSP concept – sampling analog signals. A/D
converters are covered in detail including basic terminology, types of converters (successive
approximation, flash, and sigma-delta), and the importance of matching the input signal to the
A/D converter. The sampling theorem is stated and aliasing is discussed and illustrated through
several examples. The design of analog anti-aliasing filters is covered using a really nice design
tool available for free from Texas Instruments called FilterPro. The process of reconstruction,
recreating an analog signal from digital samples using interpolation filters, is also covered. The
lab exercise at the end of the chapter includes several exercises to illustrate the principles of A/D
conversion, sampling frequency, aliasing, and reconstruction.

Chapter 4: Key Mathematical Concepts
This chapter covers fixed-point and floating-point representation of numbers and compares and
contrasts fixed-point and floating-point processors for various applications. Discrete systems
and Z-transforms are also covered in preparation for the chapters on digital filter design and
analysis. Finally, convolution and correlation are covered in preparation for the chapters on
digital filters, fast Fourier transforms, and radar processing.

Chapter 5: Finite Impulse Response (FIR) Filters
This chapter begins by defining an FIR filter then covers the design of FIR filters using the
common techniques of windowing, optimal design, and custom design. The Filter Design and
Analysis Tool (FDAT), a MATLAB® tool for designing digital filters, is introduced. This
chapter includes multiple examples of many different types of FIR filters. Special types of FIR
filters are introduced including feedforward comb filters, Gaussian filters for blurring images,
and filters for creating chorus and flanging effects for audio signals. Multi-rate processing, the
conversion from one sampling rate to another, is covered at the end of the chapter. The lab
exercise at the end of the chapter provides students with practice using the FDAT and allows
them to experiment with different designs using windowing and optimal design.
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Chapter 6: Infinite Impulse Response (IIR) Filters
This chapter begins by defining an IIR filter then covers the design of IIR filters using the FDAT
to illustrate concepts through several examples. The critically important concept of IIR filter
realization using biquads is covered in detail. FIR and IIR filters are compared and contrasted
for various practical applications. Several common IIR filters including echo filters, reverb
filters, and graphic equalizers are discussed. The lab exercise at the end of the chapter provides
students with additional practice in using the FDAT for IIR filter design and allows them to
experiment with the effect of coefficient wordlength on filter stability and performance for a full
and a biquad realization of filters.

Chapter 7: Discrete Fourier Transforms
This chapter introduces students to one of the more difficult and mathematically challenging
DSP topics, discrete Fourier transforms (DFT). Several illustrative examples are included to
stress the effect of sampling frequency and time duration on spectral resolution and accuracy.
The Fast Fourier Transform (FFT), an efficient computational algorithm for computing a DFT, is
explained. The Radix-2 decimation in time FFT algorithm is explained through equations,
graphs, and the classic butterfly diagram. Flow graphs (unique to this text) are used to clearly
show the operations involved in this FFT algorithm and the memory allocation. The end of the
chapter includes some of the practical applications for FFTs: power spectral density, fast
convolution and correlation, matched filters, and Zoom FFTs for focused analysis. The lab
includes several exercises that are designed to illustrate the effects of sampling frequency and
time duration on the accuracy and resolution of the resulting spectrum.

Chapter 8: Real-Time Processing Algorithms
This chapter covers some useful algorithms for real-time processing of signals including the
sliding Discrete Fourier Transform, the Overlap Add and Overlap Save algorithms for real-time
convolution or filtering, and an introduction to adaptive filters. The first lab exercise requires
students to write their own m-file for an overlap-add processor. The second lab exercise
introduces adaptive filter blocks in Simulink®.

Chapter 9: Introduction to Image Processing
This chapter provides an introduction to image processing and includes filters to create image
effects including contrast enhancement, blurring, sharpening, edge detection and de-noising.
Image compression using the jpeg format is illustrated through several examples which detail the
use of discrete cosine transforms and Huffman coding. In the lab exercise, students explore
image processing concepts and gain experience with 3-D data using MATLAB®.

Chapter 10: Introduction to Radar Processing
This chapter provides an introduction to radar processing. Chirp signals and other pulse signals
are introduced. Designing and using matched filters to process returning pulse signals in order to
determine range and velocity of targets is illustrated through examples and through the lab
exercise at the end of the chapter.

Appendix
The appendix includes reference tables for Fourier series, Fourier Transforms, and Z-Transforms

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Adopting this Text
For schools on the quarter system, Chapters 1-6 can be covered in the first quarter with the
remainder of the text (Chapters 7-10) covered in the second quarter. For schools on the semester
system, Chapters 1-7 and additional topics from one of the later chapters can be covered in a
single semester (Chapters 8-10 are independent of one another). The remaining chapters could
be covered in a second semester with some supplemental material (perhaps project work using
DSP hardware) added.

Acknowledgements
I would like to thank all of my students over that last couple of decades, particularly the
individuals who strongly encouraged me to write a textbook and the ones who provided feedback
on the rough draft of the manuscript. Watching my students progress from freshmen year, often
uncertain as to what engineering is all about, to senior year, prepared and confident to move on
into the workplace, has been a very rich and rewarding experience for me.

Thanks also go to my husband, Bill, and my sons, Chris and Mike, who were very patient with
the late evenings and long weekends spent writing. I’d also like to acknowledge my parents,
Norm and Julie Kurtz, who encouraged me to study hard, take challenging classes, and discover
what I was capable of. My colleagues at the College of Applied Science, University of
Cincinnati, were very supportive of my efforts in writing this text. Three colleagues in
particular, also dear friends, offered valuable advice and encouragement: Claudia Taylor, Brian
Resnick, and Virginia Westheider.

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