Appendix A.  The Arithmetic of Complex Numbers

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Appendix A. The Arithmetic of Complex Numbers

To understand digital signal processing, we have to get comfortable using complex numbers. The first step toward this goal is learning to manipulate complex numbers arithmetically. Fortunately, we can take advantage of our knowledge of real numbers to make this job easier. Although the physical significance of complex numbers is discussed in Chapter 8, the following discussion provides the arithmetic rules governing complex numbers.

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To get started, real numbers are those positive or negative numbers we're used to thinking about in our daily lives. Examples of real numbers are 0.3, –2.2, 5.1, etc. Keeping this in mind, we see how a real number can be represented by a point on a one-dimensional axis, called the real axis, as shown in Figure A-1.

Figure A-1. The representation of a real number as a point on the one-dimensional real axis.

We can, in fact, consider that all real numbers correspond to all of the points on the real axis line on a one-to-one basis.

A complex number, unlike a real number, has two parts: a real part and an imaginary part. Just as a real number can be considered to be a point on the one-dimensional real axis, a complex number can be treated as a point on a complex plane as shown in Figure A-2. We'll use this geometrical concept to help us understand the arithmetic of complex numbers.[]

[] The complex plane representation of a complex number is sometimes called an Argand diagram—named after the French mathematician Jean Robert Argand (1768–1825).

Figure A-2. The phasor representation of the complex number C = R + jI on the complex plane.

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A complex number C is represented in a number of different ways in the literature, such as

Equation A-1'

Equation A-1''

Equation A-1'''

Equations (A-1'') and (A-1''') remind us that the complex number C can also be considered the tip of a phasor on the complex plane, with magnitude M, in the direction of ø degrees relative to the positive real axis as shown in Figure A-2. (We'll avoid calling phasor M a vector because the term vector means different things in different contexts. In linear algebra, vector is the term used to signify a one-dimensional matrix. On the other hand, in mechanical engineering and field theory, vectors are used to signify magnitudes and directions, but there are vector operations (scalar or dot product, and vector or cross-product) that don't apply to our definition of a phasor.) The relationships between the variables in this figure follow the standard trigonometry of right triangles. Keep in mind that C is a complex number, and the variables R, I, M, and ø are all real numbers. The magnitude of C, sometimes called the modulus of C, is

Equation A-2

and, by definition, the phase angle, or argument, of C is the arctangent of I/R, or

Equation A-3

The variable ø in Eq. (A-3) is a general angle term. It can have dimensions of degrees or radians. Of course, we can convert back and forth between degrees and radians using p radians = 180°. So, if ør is in radians and ød is in degrees, then we can convert ør to degrees by the expression

Equation A-4

Likewise, we can convert ød to radians by the expression

Equation A-5

The exponential form of a complex number has an interesting characteristic that we need to keep in mind. Whereas only a single expression in rectangular form can describe a single complex number, an infinite number of exponential expressions can describe a single complex number; that is, while, in the exponential form, a complex number C can be represented by C = Mejø, it can also be represented by

Equation A-6

where n = ±1, ±2, ±3, . . . and ø is in radians. When ø is in degrees, Eq. (A-6) is in the form

Equation A-7

Equations (A-6) and (A-7) are almost self-explanatory. They indicate that the point on the complex plane represented by the tip of the phasor C remains unchanged if we rotate the phasor some integral multiple of 2p radians or an integral multiple of 360°. So, for example, if C = Mej(20°), then

Equation A-8

The variable ø, the angle of the phasor in Figure A-2, need not be constant. We'll often encounter expressions containing a complex sinusoid that takes the form

Equation A-9

Equation (A-9) represents a phasor of magnitude M whose angle in Figure A-2 is increasing linearly with time at a rate of w radians each second. If w = 2p, the phasor described by Eq. (A-9) is rotating counterclockwise at a rate of 2p radians per secondone revolution per second—and that's why w is called the radian frequency. In terms of frequency, Eq. (A-9)'s phasor is rotating counterclockwise at w = 2pf radians per second, where f is the cyclic frequency in cycles per second (Hz). If the cyclic frequency is f = 10 Hz, the phasor is rotating 20p radians per second. Likewise, the expression

Equation A-9'

represents a phasor of magnitude M that rotates in a clockwise direction about the origin of the complex plane at a negative radian frequency of –w radians per second.