TABLE OF CONTENTS Complex exponential signals can also be defined both in continuous time and in discrete time. They have real and imaginary parts with sinusoidal behavior.
The continuous-time complex exponential signal can be defined as follows:
| (1.12) |  |
where C = Ae j?, A, ? ? ?, A > 0 is expressed in polar form, and a = ? + j ? 0, ?, ? 0 ? ? is expressed in rectangular form. Thus, we can write
| (1.13) |  |
If we look at the second part of Equation 1.13, we can see that x( t) represents either a circular or a spiral trajectory in the complex plane, depending whether a is zero, negative, or positive. The term 
describes a unit circle centered at the origin counterclockwise in the complex plane as time varies from t =- ? to t = + ? , as shown in Figure 1.23 for the case ? = 0. The times t k indicated in the figure are the times when the complex point 
has a phase of ?/4 .
Using Euler's relation, we obtain the signal in rectangular form:
| (1.14) |  |
where Re{ x( t)} = Ae ?t cos( ? 0 t + ?) and Im{ x( t)} = Ae ?t sin( ? 0
