The complex exponential is one of the most fundamental and important signal in signal and system analysis. Its importance comes from its functions as a basis for periodic signals as well as being able to characterize linear, time-invariant signals. Before proceeding, you should be familiar with the ideas and functions of complex numbers.

**Basic Exponential**

For all numbers

From this definition, we can prove the following property for exponentials that will be very useful, especially for the complex exponentials discussed in the next section.

**Complex Continuous-Time Exponential**

Now for all complex numbers

The above expressions do not include any
information on phase however. We can further generalize our
above expressions for the exponential to generalize
sinusoids with any phase by making a final substitution for

**Complex Discrete-Time Exponential**

Finally we have reached the last form of the exponential signal that we will be interested in, the discrete-time exponential signal, which we will not give as much detail about as we did for its continuous-time counterpart, because they both follow the same properties and logic discussed above. Because it is discrete, there is only a slightly different notation used to represents its discrete nature