ComplexNotes - Appendix A Appendix on complex numbers A.1...

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Unformatted text preview: Appendix A Appendix on complex numbers: A.1 complex numbers We begin with a review of several properties of complex numbers, their representation, and some of their basic properties. The use of complex numbers, complex-valued functions, and functions of a complex variable will prove essential for an understanding of the material in this text, so it is important that before proceeding with the rest of this material, some basic notions are well understood. Without the ability to manipulate complex numbers and functions, our treatment of discrete-time system theory would be much more di cult. There are many ways in which complex numbers may be represented. Two representations that will be used extensively in this text are the rectangular form and the polar form. The rectangular form, also called the Cartesian form, represents a complex number z as an ordered pair of real numbers, usually written z = x + jy where x and y are real numbers, with x referred to as the real part of z and y referred to as the imaginary part of z and j = √- 1 . We can write x = < z and y = = z to illustrate taking the real part and the imaginary part of the complex number z . In polar form, we can write z = re jθ , where r > is referred to as the magnitude of the complex number z and θ is the phase or angle of z. We can then express these relationships as r = | z | , and θ = ∠ z, and use Euler's relation e jθ = cos( θ ) + j sin( θ ) , to relate the complex cartesian and polar representations as r = p x 2 + y 2 , and θ = arctan( y/x ) . These relationships can be obtained by considering the real and imaginary parts of a complex number as points in the complex ( x,y ) plane. Then the complex number can be thought of as a vector in the plane from the origin to the point ( x,y ) , with the magnitude of the vector being r and the angle formed from the real line to the vector yielding θ , as in gure A.1. We see that by Euler's relation, we have z = re jθ = ( r cos( θ )) + j ( r sin( θ )) and then we obtain x = r cos( θ ) and y = r sin( θ ) . 83 84 APPENDIX A. APPENDIX ON COMPLEX NUMBERS: θ y ℜ ℑ r x Figure A.1: Vector representation of a complex number, relating the polar and Cartesian forms. Euler's relation can be used to relate the real part and imaginary parts with the magnitude and phase. We can similarly write p x 2 + y 2 = p r 2 (cos( θ ) 2 + sin( θ ) 2 ) = r, and y x = r sin( θ ) r cos( θ ) = tan( θ ) , such that θ = arctan( y/x ) . Complex numbers are simply a useful tool that enables us to describe a wider class of equations than do the real numbers alone. For example, if we consider the equation x 2 + 1 = 0 , and ask for what values of x does this equation have a solution? We nd that when x takes on values from the real numbers, then there are no solutions. However we can learn more about this equation, and about equations involving higher order polynomials in x if we can introduce a solution to this equation. In order to do so, we must now think of the function...
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