CH30 - CHAPTER 30 Complex Numbers Complex numbers are an...

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551 CHAPTER 30 h & gt 2 2 % vt 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 2 ) 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:
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The Scientist and Engineer's Guide to Digital Signal Processing 552 t 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 2 ), 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 two h 10 answers: and . Both these answers contain t 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.
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CH30 - CHAPTER 30 Complex Numbers Complex numbers are an...

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