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The Complex Exponential

Module by: Richard Baraniuk. E-mail the author

Summary: Describes the complex exponential function.

Note: You are viewing an old version of this document. The latest version is available here.

The Exponential Basics

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 xx, we easily derive and define the exponential function from the Taylor's series below:

ex=1+x11!+x22!+x33!+… x 1 x 1 1 x 2 2 x 3 3 …
(1)
ex=∑ k =0∞1k!⁢xk x k 0 1 k x k
(2)
We can prove, using the ratio test, that this series does indeed converge. Therefore, we can state that the exponential function shown above is continuous and easily defined.

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.

e x 1 + x 2 =e x 1 ⁢e x 2 x 1 x 2 x 1 x 2
(3)

Complex Continuous-Time Exponential

Now for all complex numbers s s, we can define the complex continuous-time exponential signal as

f⁢t=A⁢es⁢t=A⁢ei⁢ω⁢t ft A s t A ω t
(4)
where AA is a constant, tt is our independent variable for time, and for ss imaginary, s=i⁢ω s ω . Finally, from this equation we can reveal the ever important Euler's Identity (for more information on Euler read this short biography):
A⁢ei⁢ω⁢t=A⁢cosω⁢t+i⁢(A⁢sinω⁢t) A ω t A ω t A ω t
(5)
From Euler's Identity we can easily break the signal down into its real and imaginary components. Also we can see how exponentials can be combined to represent any real signal. By modifying their frequency and phase, we can represent any signal through a superposity of many signals - all capable of being represented by an exponential.

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 ss, s=σ+i⁢ω s σ ω , which leads us to

f⁢t=A⁢es⁢t=A⁢e(σ+i⁢ω)⁢t=A⁢eσ⁢t⁢ei⁢ω⁢t f t A s t A σ ω t A σ t ω t
(6)
where we define SS as the complex amplitude, or phasor, from the first two terms of the above equation as
S=A⁢eσ⁢t S A σ t
(7)
Going back to Euler's Identity, we can rewrite the exponentials as sinusoids, where the phase term becomes much more apparent.
f⁢t=A⁢eσ⁢t⁢(cosω⁢t+i⁢sinω⁢t) f t A σ t ω t ω t
(8)
As stated above we can easily break this formula into its real and imaginary part as follows:
ℜ⁢f⁢t=A⁢eσ⁢t⁢cosω⁢t f t A σ t ω t
(9)
ℑ⁢f⁢t=A⁢eσ⁢t⁢sinω⁢t f t A σ t ω t
(10)

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

f⁢n=B⁢es⁢n⁢T=B⁢ei⁢ω⁢n⁢T fn B s n T B ω n T
(11)
where n⁢T n T represents the discrete-time instants of our signal.

Euler's Relation

Along with Euler's Identity, Euler also described a way to represent a complex exponential signal in terms of its real and imaginary parts through Euler's Relation:

cosω⁢t=ei⁢w⁢t+e−(i⁢w⁢t)2 ω t w t w t 2
(12)
sinω⁢t=ei⁢w⁢t−e−(i⁢w⁢t)2⁢i ω t w t w t 2
(13)
ei⁢w⁢t=cosω⁢t+i⁢sinω⁢t w t ω t ω t
(14)

Drawing the Complex Exponential

At this point, we have shown how the complex exponential can be broken up into its real part and its imaginary part. It is now worth looking at how we can draw each of these parts. We can see that both the real part and the imaginary part have a sinusoid times a real exponential. We also know that sinusoids oscillate between one and negative one. From this it becomes apparent that the real and imaginary parts of the complex exponential will each oscillate between a window defined by the real exponential part.

Figure 1: The shapes possible for the real part of a complex exponential. Notice that the oscillations are the result of a cosine, as there is a local maximum at t=0 t 0 .
(a) If σσ is negative, we have the case of a decaying exponential window. (b) If σσ is positive, we have the case of a growing exponential window.(c) If σσ is zero, we have the case of a constant window.
Figure 1(a) (compexp1.png)Figure 1(b) (compexp2.png)Figure 1(c) (compexp3.png)

While the σσ determines the rate of decay/growth, the ωω part determines the rate of the oscillations. This is apparent by noticing that the ωω is part of the argument to the sinusoidal part.

Exercise 1

What do the imaginary parts of the complex exponentials drawn above look like?

Solution

They look the same except the oscillation is that of a sinusoid as opposed to a cosinusoid (i.e. it passes through the origin rather than being a local maximum at t=0 t 0 ).

Example 1

The following demonstration allows you to see how the argument changes the shape of the complex exponential. See here for instructions on how to use the demo.

Download LabVIEW Source

The Complex Plane

It becomes extremely useful to view the complex variable ss as a point in the complex plane (the s-plane).

Figure 2: This is the s-plane. Notice that any time ss lies in the right half plane, the complex exponential will grow through time, while any time it lies in the left half plane it will decay.
Figure 2 (compexp4.png)

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