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Unformatted text preview: Appendix B Complex Numbers Complex numbers are used extensively in the modeling of signals and systems for two reasons. The first reason is that complex numbers provide a compact and elegant way to talk simultaneously about the phase and amplitude of sinusoidal signals. Complex numbers are therefore heavily used in Fourier analysis, which represents arbitrary signals in terms of sinusoidal signals. The second reason is that a large class of systems, called linear timeinvariant (LTI) systems, treat signals that can be described as complex exponential functions in an especially simple way. They simply scale the signals. These uses of complex numbers are developed in detail in the main body of this text. This appendix summarizes essential properties of complex numbers themselves. We review complex number arith metic, how to manipulate complex exponentials, Euler’s formula, the polar coordinate representa tion of complex numbers, and the phasor representation of sinewaves. B.1 Imaginary numbers The quadratic equation, x 2 1 = , has two solutions, x = + 1 and x = 1. These solutions are said to be roots of the polynomial x 2 1. Thus, this polynomial has two roots, + 1 and 1. More generally, the roots of the nth degree polynomial, x n + a 1 x n 1 + ··· + a n 1 x + a n , (B.1) are defined to be the solutions to the polynomial equation x n + a 1 x n 1 + ··· + a n 1 x + a n = . (B.2) The roots of a polynomial provide a particularly useful factorization into firstdegree polynomials. For example, we can factor the polynomial x 2 1 as x 2 1 = ( x 1 )( x + 1 ) . 541 542 APPENDIX B. COMPLEX NUMBERS Notice the role of the roots, + 1 and 1. In general, if ( B.1 ) has roots r 1 , ··· , r n , then we can factor the polynomial as follows x n + a 1 x n 1 + ··· + a n 1 x + a n = ( x r 1 )( x r 2 ) ··· ( x r n ) . (B.3) It is easy to see that if x = r i for any i ∈ { 1 , ··· n } , then the polynomial evaluates to zero, so ( B.2 ) is satisfied. This raises the question whether ( B.2 ) always has a solution for x . In other words, can we always find roots for a polynomial? The equation x 2 + 1 = (B.4) has no solution for x in the set of real numbers. Thus, it would appear that not all polynomials have roots. However, a surprisingly simple and clever mathematical device changes the picture dramat ically. With the introduction of imaginary numbers , mathematicians ensure that all polynomials have roots. Moreover, they ensure that any polynomial of degree n has exactly n factors as in ( B.3 ). The n values r 1 , ··· , r n (some of which may be repeated) are the roots of the polynomial. If we try by simple algebra to solve ( B.4 ) we discover that we need to find x such that x 2 = 1 ....
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This note was uploaded on 09/29/2010 for the course EE 20N taught by Professor Ayazifar during the Spring '08 term at University of California, Berkeley.
 Spring '08
 Ayazifar

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