Signals and Systems - Electrical Engineering

20 C H A P T E R 0: From the Ground Up!

0.4 Complex or Real?

Most of the theory of signals and systems is based on functions of a complex variable. Clearly, sig-

nals are functions of a real variable corresponding to time or space (if the signal is two-dimensional,

like an image) so why would one need complex numbers in processing signals? As we will see later,

time-dependent signals can be characterized by means of frequency and damping. These two charac-

teristics are given by complex variables such ass=σ+j(whereσis the damping factor andis

the frequency) in the representation of analog signals in the Laplace transform, orz=rejω(wherer

is the damping factor andωis the discrete frequency) in the representation of discrete-time signals in

the Z-transform. Both of these transformations will be considered in detail in Chapters 3 and 9. The

other reason for using complex variables is due to the response of systems to pure tones or sinusoids.

We will see that such response is fundamental in the analysis and synthesis of signals and systems.

We thus need a solid grasp of what is meant by complex variables and what a function of these is

all about. In this section, complex variables will be connected to vectors and phasors (which are

commonly used in the sinusoidal steady-state analysis of linear circuits).

0.4.1 Complex Numbers and Vectors

A complex numberzrepresents any point(x,y)in a two-dimensional plane byz=x+jy, where

x=Re[z] (real part ofz) is the coordinate in thexaxis andy=Im[z] (imaginary part ofz) is the

coordinate in theyaxis. The symbolj=

√

−1 just indicates thatzneeds to have two components

to represent a point in the two-dimensional plane. Interestingly, a vectorEzthat emanates from the

origin of the complex plane(0, 0)to the point(x,y)with a length

|Ez|=

√

x^2 +y^2 =|z| (0.14)

and an angle

θ=∠Ez=∠z (0.15)

also represents the point(x,y)in the plane and has the same attributes as the complex numberz. The

couple(x,y)is therefore equally representable by the vectorEzor by a complex numberzthat can be

written in a rectangular or in a polar form,

z=x+jy=|z|ejθ (0.16)

where the magnitude|z|and the phaseθare defined in Equations (0.14) and (0.15).

It is important to understand that a rectangular plane or a polar complex plane are identical despite

the different representation of each point in the plane. Furthermore, when adding or subtracting

complex numbers the rectangular form is the appropriate one, while when multiplying or dividing

complex numbers the polar form is more advantageous. Thus, if complex numbersz=x+jy=|z|ej∠z

andv=p+jq=|v|ej∠vare added analytically, we obtain

z+v=(x+p)+j(y+q)