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551

CHAPTER

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Complex Numbers

Complex numbers are an extension of the ordinary numbers used in everyday math. They have

the unique property of representing and manipulating two variables as a single quantity. This fits

very naturally with Fourier analysis, where the frequency domain is composed of two signals, the

real and the imaginary parts. Complex numbers shorten the equations used in DSP, and enable

techniques that are difficult or impossible with real numbers alone. For instance, the Fast Fourier

Transform is based on complex numbers. Unfortunately, complex techniques are very

mathematical, and it requires a great deal of study and practice to use them effectively. Many

scientists and engineers regard complex techniques as the dividing line between DSP as a tool,

and DSP as a career. In this chapter, we look at the mathematics of complex numbers, and

elementary ways of using them in science and engineering. The following three chapters discuss

important techniques based on complex numbers: the complex Fourier transform, the Laplace

transform, and the z-transform. These complex transforms are the heart of theoretical DSP. Get

ready, here comes the math!

The Complex Number System

To illustrate complex numbers, consider a child throwing a ball into the air.

For example, assume that the ball is thrown straight up, with an initial

velocity of 9.8 meters per second. One second after it leaves the child's

hand, the ball has reached a height of 4.9 meters, and the acceleration of

gravity (9.8 meters per second

) has reduced its velocity to zero. The ball

then accelerates toward the ground, being caught by the child two seconds

after it was thrown. From basic physics equations, the height of the ball at

any instant of time is given by:

The Scientist and Engineer's Guide to Digital Signal Processing552

t ' 1± 1&h/4.9

where h is the height above the ground (in meters), g is the acceleration of

gravity (9.8 meters per second

), v is the initial velocity (9.8 meters per

second), and t is the time (in seconds).

Now, suppose we want to know when the ball passes a certain height.

Plugging in the known values and solving for t:

For instance, the ball is at a height of 3 meters twice: (going up)t ' 0.38

and seconds (going down). t ' 1.62

As long as we ask reasonable questions, these equations give reasonable

answers. But what happens when we ask unreasonable questions? For

example: At what time does the ball reach a height of 10 meters? This

question has no answer in reality because the ball never reaches this height.

Nevertheless, plugging the value of into the above equation gives twoh ' 10

answers: and . Both these answers containt ' 1% &1.041 t ' 1& &1.041

the square-root of a negative number, something that does not exist in the world

as we know it. This unusual property of polynomial equations was first used

by the Italian mathematician Girolamo Cardano (1501-1576). Two centuries

later, the great German mathematician Carl Friedrich Gauss (1777-1855)

coined the term complex numbers, and paved the way for the modern

understanding of the field.

Every complex number is the sum of two components: a real part and an

imaginary part. The real part is a real number, one of the ordinary

numbers we all learned in childhood. The imaginary part is an imaginary

number, that is, the square-root of a negative number. To keep things

standardized, the imaginary part is usually reduced to an ordinary number

multiplied by the square-root of negative one. As an example, the complex

number: , is first reduced to: , and then tot ' 1% &1.041 t ' 1% 1.041 &1

the final form: . The real part of this complex number is 1,t ' 1% 1.02 &1

while the imaginary part is . This notation allows the abstract term,1.02 &1

, to be given a special symbol. Mathematicians have long used i to denote

. In comparison, electrical engineers use the symbol, j, because i is used&1

to represent electrical current. Both symbols are common in DSP. In this book

the electrical engineering convention, j, will be used.

For example, all the following are valid complex numbers: , ,1% 2j 1& 2j

, , , etc. All ordinary numbers, such as:&1% 2j 3.14159% 2.7183j (4/3)% (19/2)j

2, 6.34, and -1.414, can be viewed as a complex number with zero for the

imaginary part, i.e., , , and .2% 0j 6.34% 0j &1.414% 0j

Just as real numbers are described as having positions along a number line,

complex numbers are represented by locations in a two-dimensional display

called the complex plane. As shown in Fig. 30-1, the horizontal axis of the

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