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The Scientist & Engineer's Guide to Digital Signal Processing, by Steven W. Smith, Ph.D, provides a practical introduction to Digital Signal Processing. Covering a wide range of topics, this book is an ideal introductory text for those new to DSP, and an excellent reference for more experienced users. Begin today to learn more about this powerful technology! Contents at a Glance FOUNDATIONS Chapter 1. The Breadth and Depth of DSP . . . . . . . . . . . . . . . . . . . 1 Chapter 2. Statistics, Probability and Noise . . . . . . . . . . . . . . . . . . 11 Chapter 3. ADC and DAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Chapter 4. DSP Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 FUNDAMENTALS Chapter 5. Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Chapter 6. Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Chapter 7. Properties of Convolution . . . . . . . . . . . . . . . . . . . . . . . 123 Chapter 8. The Discrete Fourier Transform . . . . . . . . . . . . . . . . . . 141 Chapter 9. Applications of the DFT . . . . . . . . . . . . . . . . . . . . . . . . 169 Chapter 10. Fourier Transform Properties . . . . . . . . . . . . . . . . . . . . 185 Chapter 11. Fourier Transform Pairs . . . . . . . . . . . . . . . . . . . . . . . . 209 Chapter 12. The Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 225 Chapter 13. Continuous Signal Processing . . . . . . . . . . . . . . . . . . . . 243 DIGITAL FILTERS Chapter 14. Introduction to Digital Filters . . . . . . . . . . . . . . . . . . . . 261 Chapter 15. Moving Average Filters . . . . . . . . . . . . . . . . . . . . . . . . . 277 Chapter 16. Windowed-Sinc Filters . . . . . . . . . . . . . . . . . . . . . . . . . 285 Chapter 17. Custom Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Chapter 18. FFT Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 Chapter 19. Recursive Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Chapter 20. Chebyshev Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Chapter 21. Filter Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 APPLICATIONS Chapter 22. Audio Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 Chapter 23. Image Formation and Display . . . . . . . . . . . . . . . . . . . . 373 Chapter 24. Linear Image Processing . . . . . . . . . . . . . . . . . . . . . . . . 397 Chapter 25. Special Imaging Techniques . . . . . . . . . . . . . . . . . . . . . 423 Chapter 26. Neural Networks (and more!) . . . . . . . . . . . . . . . . . . . . 451 Chapter 27. Data Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 Chapter 28. Digital Signal Processors . . . . . . . . . . . . . . . . . . . . . . . 503 Chapter 29. Getting Started with DSPs . . . . . . . . . . . . . . . . . . . . . . . 535 COMPLEX TECHNIQUES Chapter 30. Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 Chapter 31. The Complex Fourier Transform . . . . . . . . . . . . . . . . . . 567 Chapter 32. The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . 581 Chapter 33. The z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643

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CHAPTER

The Breadth and Depth of DSP

Digital Signal Processing is one of the most powerful technologies that will shape science and

engineering in the twenty-first century. Revolutionary changes have already been made in a broad

range of fields: communications, medical imaging, radar & sonar, high fidelity music

reproduction, and oil prospecting, to name just a few. Each of these areas has developed a deep

DSP technology, with its own algorithms, mathematics, and specialized techniques. This

combination of breath and depth makes it impossible for any one individual to master all of the

DSP technology that has been developed. DSP education involves two tasks: learning general

concepts that apply to the field as a whole, and learning specialized techniques for your particular

area of interest. This chapter starts our journey into the world of Digital Signal Processing by

describing the dramatic effect that DSP has made in several diverse fields. The revolution has

begun.

The Roots of DSP

Digital Signal Processing is distinguished from other areas in computer science

by the unique type of data it uses: signals. In most cases, these signals

originate as sensory data from the real world: seismic vibrations, visual images,

sound waves, etc. DSP is the mathematics, the algorithms, and the techniques

used to manipulate these signals after they have been converted into a digital

form. This includes a wide variety of goals, such as: enhancement of visual

images, recognition and generation of speech, compression of data for storage

and transmission, etc. Suppose we attach an analog-to-digital converter to a

computer and use it to acquire a chunk of real world data. DSP answers the

question: What next?

The roots of DSP are in the 1960s and 1970s when digital computers first

became available. Computers were expensive during this era, and DSP was

limited to only a few critical applications. Pioneering efforts were made in four

key areas: radar & sonar, where national security was at risk; oil exploration,

where large amounts of money could be made; space exploration, where the

Chapter 1- The Breadth and Depth of DSP 3

Digital

Signal

Processing

Communication

Theory

Analog

Electronics

Digital

Electronics

Probability

and Statistics

Decision

Theory

Analog

Signal

Processing

Numerical

Analysis

FIGURE 1-2

Digital Signal Processing has fuzzy and overlapping borders with many other

areas of science, engineering and mathematics.

in many fields. As an analogy, DSP can be compared to a previous

technological revolution: electronics. While still the realm of electrical

engineering, nearly every scientist and engineer has some background in basic

circuit design. Without it, they would be lost in the technological world. DSP

has the same future.

This recent history is more than a curiosity; it has a tremendous impact on your

ability to learn and use DSP. Suppose you encounter a DSP problem, and turn

to textbooks or other publications to find a solution. What you will typically

find is page after page of equations, obscure mathematical symbols, and

unfamiliar terminology. It's a nightmare! Much of the DSP literature is

baffling even to those experienced in the field. It's not that there is anything

wrong with this material, it is just intended for a very specialized audience.

State-of-the-art researchers need this kind of detailed mathematics to

understand the theoretical implications of the work.

A basic premise of this book is that most practical DSP techniques can be

learned and used without the traditional barriers of detailed mathematics and

theory. The Scientist and Engineer’s Guide to Digital Signal Processing is

written for those who want to use DSP as a tool, not a new career.

The remainder of this chapter illustrates areas where DSP has produced

revolutionary changes. As you go through each application, notice that DSP

is very interdisciplinary, relying on the technical work in many adjacent

fields. As Fig. 1-2 suggests, the borders between DSP and other technical

disciplines are not sharp and well defined, but rather fuzzy and overlapping.

If you want to specialize in DSP, these are the allied areas you will also

need to study.

The Scientist and Engineer's Guide to Digital Signal Processing4

Telecommunications

Telecommunications is about transferring information from one location to

another. This includes many forms of information: telephone conversations,

television signals, computer files, and other types of data. To transfer the

information, you need a channel between the two locations. This may be

a wire pair, radio signal, optical fiber, etc. Telecommunications companies

receive payment for transferring their customer's information, while they

must pay to establish and maintain the channel. The financial bottom line

is simple: the more information they can pass through a single channel, the

more money they make. DSP has revolutionized the telecommunications

industry in many areas: signaling tone generation and detection, frequency

band shifting, filtering to remove power line hum, etc. Three specific

examples from the telephone network will be discussed here: multiplexing,

compression, and echo control.

Multiplexing

There are approximately one billion telephones in the world. At the press of

a few buttons, switching networks allow any one of these to be connected to

any other in only a few seconds. The immensity of this task is mind boggling!

Until the 1960s, a connection between two telephones required passing the

analog voice signals through mechanical switches and amplifiers. One

connection required one pair of wires. In comparison, DSP converts audio

signals into a stream of serial digital data. Since bits can be easily

intertwined and later separated, many telephone conversations can be

transmitted on a single channel. For example, a telephone standard known

as the T-carrier system can simultaneously transmit 24 voice signals. Each

voice signal is sampled 8000 times per second using an 8 bit companded

(logarithmic compressed) analog-to-digital conversion. This results in each

voice signal being represented as 64,000 bits/sec, and all 24 channels being

contained in 1.544 megabits/sec. This signal can be transmitted about 6000

feet using ordinary telephone lines of 22 gauge copper wire, a typical

interconnection distance. The financial advantage of digital transmission

is enormous. Wire and analog switches are expensive; digital logic gates

are cheap.

Compression

When a voice signal is digitized at 8000 samples/sec, most of the digital

information is redundant. That is, the information carried by any one

sample is largely duplicated by the neighboring samples. Dozens of DSP

algorithms have been developed to convert digitized voice signals into data

streams that require fewer bits/sec. These are called data compression

algorithms. Matching uncompression algorithms are used to restore the

signal to its original form. These algorithms vary in the amount of

compression achieved and the resulting sound quality. In general, reducing the

data rate from 64 kilobits/sec to 32 kilobits/sec results in no loss of sound

quality. When compressed to a data rate of 8 kilobits/sec, the sound is

noticeably affected, but still usable for long distance telephone networks.

The highest achievable compression is about 2 kilobits/sec, resulting in

Chapter 1- The Breadth and Depth of DSP 5

sound that is highly distorted, but usable for some applications such as military

and undersea communications.

Echo control

Echoes are a serious problem in long distance telephone connections.

When you speak into a telephone, a signal representing your voice travels

to the connecting receiver, where a portion of it returns as an echo. If the

connection is within a few hundred miles, the elapsed time for receiving the

echo is only a few milliseconds. The human ear is accustomed to hearing

echoes with these small time delays, and the connection sounds quite

normal. As the distance becomes larger, the echo becomes increasingly

noticeable and irritating. The delay can be several hundred milliseconds

for intercontinental communications, and is particularly objectionable.

Digital Signal Processing attacks this type of problem by measuring the

returned signal and generating an appropriate antisignal to cancel the

offending echo. This same technique allows speakerphone users to hear

and speak at the same time without fighting audio feedback (squealing).

It can also be used to reduce environmental noise by canceling it with

digitally generated antinoise.

Audio Processing

The two principal human senses are vision and hearing. Correspondingly,

much of DSP is related to image and audio processing. People listen to

both music and speech. DSP has made revolutionary changes in both

these areas.

Music

The path leading from the musician's microphone to the audiophile's speaker is

remarkably long. Digital data representation is important to prevent the

degradation commonly associated with analog storage and manipulation. This

is very familiar to anyone who has compared the musical quality of cassette

tapes with compact disks. In a typical scenario, a musical piece is recorded in

a sound studio on multiple channels or tracks. In some cases, this even involves

recording individual instruments and singers separately. This is done to give

the sound engineer greater flexibility in creating the final product. The

complex process of combining the individual tracks into a final product is

called mix down. DSP can provide several important functions during mix

down, including: filtering, signal addition and subtraction, signal editing, etc.

One of the most interesting DSP applications in music preparation is

artificial reverberation. If the individual channels are simply added together,

the resulting piece sounds frail and diluted, much as if the musicians were

playing outdoors. This is because listeners are greatly influenced by the echo

or reverberation content of the music, which is usually minimized in the sound

studio. DSP allows artificial echoes and reverberation to be added during

mix down to simulate various ideal listening environments. Echoes with

delays of a few hundred milliseconds give the impression of cathedral like

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