### Digital Signal Processing: An Introduction with MATLAB and Applications

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Wavelets The wavelet transform is an analysis tool that has a relatively short history. It is a daunting task to write about the people who have contributed to wavelets, since anyone who is left out could be reading this now! The wavelet transform is an important tool with many applications, such as compression. I have no doubt that future generations of DSP teachers will rank it second only to the Fourier transform in terms of importance. Advertising by cellular phone marketers, which tries to explain to a nontechnical audience that digital signal processing is better than analog, is an example of the growing public awareness.

Actually, the ads do not try to say how digital is better than analog, but they do point out problems with wireless analog phones. Acknowledgments This book was only possible with the support of many people. I would like to thank Dr. I would also like to thank Dr. Magdy Bayoumi at the University of Louisiana, who taught me much about research as well as how to present research results to an audience.

My students over the years have been quite helpful in pointing out any problems within the text. I would especially like to thank Evelyn Brannock and Ferrol Blackmon for their help reviewing the material.

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I would also like to acknowledge Drs. Preface xxiii Kim King and Raj Sunderraman for their help. Finally, I could not have written this book without the support and understanding of my wife. There is always a driving need to make things better and DSP provides many techniques for doing this. For example, people enjoy music and like to download new songs. However, with slow Internet connection speeds typically 56 kilobits per second for a dial-up modem , downloading a song could take hours.

How is this possible? First, it is important to know about the original song a signal , and how it is represented digitally.

## Digital Signal Processing with Examples in MATLAB®

This knowledge leads to an algorithm to remove data that the user will not miss. All of this is part of Digital Signal Processing. In the ancient world, people used numbers to count things. Ratios soon followed, since some things were more valuable than others. If you were an ancient pig farmer, you might want a loaf of bread, but would you be willing to trade a pig for a loaf of bread? Maybe you would be willing to trade a pig for a loaf of bread and three ducks. The point is, ratios developed as a way to compare dissimilar things. With ratios, come fractions.

A duck may be worth three loaves of 1 This could work the other way, too. If you had a roasted duck, you might divide it into 3 equal sections, and trade one of the sections for a bread loaf. Ratios lead the way to fractions, and our use of the decimal point to separate the whole part of the number from the fractional part.

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Zero is a useful number whose invention is often credited to the Arabs, though there is much debate on this point. Its origins may not ever be conclusive. It is one of the symbols we use in our base 10 system, along with the numerals 1 through 9. Imagine if we used Roman numerals instead! The concept of zero must have been revolutionary in its time, because it is an abstract idea.

A person can see 1 duck or 3 loaves of bread, but how do you see 0 of something? Still, it is useful, even if just a counting system is used.

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There is no symbol for ten like there is for 0—9 at least, not in the decimal number system. Instead, we use a combination of symbols, i.

In any number system, we have this idea of placement, where the digits on the left are greater in magnitude than the digits on the right. This is also true in number systems that computers use, i. Computers use base 2, binary, internally. By viewing the two values that work with these transistors as a logical 0 ground and a logical 1 e. For example, if a binary value appears as 0.

To the computer, such a value given to a digital logical circuit would be close enough to 0. Introduction 1. For example, which is easier for you to remember, or ? As you can see, hexadecimal provides an easier way for people to work with computers, and it translates to and from binary very easily. In fact, one hexadecimal digit represents four bits.

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With a group of four bits, the only possibilities are: , , , , , , , , , , , , , , , and These 16 possible combinations of 4 bits map to the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The word size is dependent on the architecture e. Typically, the word size is a multiple of a byte 8 bits , and hexadecimal numbers work nicely with such machines. For example, a machine with a 1-byte word size would work with data sizes of 2 hexadecimal digits at a time. A bit word size means 4 hexadecimal digits are used. Here is an example demonstrating word size.

We can do both of the above multiplications, but most people have an immediate answer for the one on the left, but need a minute or two for the one on the right. While they might have the single-digit multiplication answers memorized, they use an algorithm for the multiple digit multiplications, i. In a sense, we have a 1-digit word size when performing this calculation. Similarly, a computer will multiply or add, subtract, etc. For example, suppose that a computer has a word size of 8 bits.

If it needed to increment a bit number, it would add one to the low 8 bits, then add the carry to the high 8 bits. How would you represent this? A negative amount is what we use today, e. With our number system, no matter what extremely large number someone comes up with, you could still add 1 to it. This is often seen in formulas as a way of covering all numbers.

A radix point is the name given to the symbol separating the whole part from the fractional part—a period in the U.

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Recall that to convert a binary number to decimal, one would multiply each bit in sequence by a multiple of 2, working from the right to the left. One can start here and work left, then start again at the radix point and work right. We take the whole part, divide it by 2, and keep track of the result and its remainder. Then we repeat this with the result until the result becomes zero. Read the remainders back, from bottom to top, and that is our binary number.

Therefore, decimal 4 equals in binary. We can precede our answer by 0 to avoid confusion with a negative binary number, i. Introduction 5 For a fractional decimal number, multiply by 2 and keep the whole part.

For example, say we want to convert. Our answer for this is. We can put the answers together and conclude that 4. For example, functions like the Fourier transform fft and ifft return their results as complex numbers. This topic is covered in detail later in this book, but for the moment you can think of the Fourier transform as a function that converts data to an interesting form. Complex numbers provide a convenient way to store two pieces of information, either x and y coordinates, or a magnitude length of a 2D vector and an angle how much to rotate the vector. This information has a physical interpretation in some contexts, such as corresponding to the magnitude and phase angle of a sinusoid for a given frequency, when returned from the Fourier transform.

The square root operator is supposed to return the positive root. How can we take a positive number, multiply it by itself, and end up with a negative number? To deal with this, we have the imaginary quantity j. Some people prefer i instead. To specify the real or imaginary part, we use functions of the same name, i.