<< Chapter < Page Chapter >> Page >
Describes the complex exponential function for discrete time.


Complex exponentials are some of the most important functions in our study of signals and systems. Their importance stems from their status as eigenfunctions of linear time invariant systems; as such, it can be both convenient and insightful to represent signals in terms of complex exponentials. Before proceeding, you should be familiar with complex numbers.

The discrete time complex exponential

Complex exponentials

The complex exponential function will become a critical part of your study of signals and systems. Its general discrete form is written as

z n
where z is a complex number. Recalling the polar expression of complex numbers, z can be expressed in terms of its magnitude z and its angle (or argument) ω in the complex plane: z z j ω . Thus z n z n j ω n . In the context of complex exponentials, ω is referred to as frequency. For the time being, let's consider complex exponentials for which z 1 .

These discrete time complex exponentials have the following property, which will become evident through discussion of Euler's formula.

j ω n j ω 2 n
Given this property, if we have a complex exponential with frequency ω 2 , then this signal "aliases" to a complex exponential with frequency ω , implying that the equation representations of discrete complex exponentials are not unique.

Euler's formula

The mathematician Euler proved an important identity relating complex exponentials to trigonometric functions. Specifically, he discovered the eponymously named identity, Euler's formula, which states that for any real number x ,

e j x = cos ( x ) + j sin ( x )

which can be proven as follows.

In order to prove Euler's formula, we start by evaluating the Taylor series for e z about z = 0 , which converges for all complex z , at z = j x . The result is

e j x = k = 0 ( j x ) k k ! = k = 0 ( - 1 ) k x 2 k ( 2 k ) ! + j k = 0 ( - 1 ) k x 2 k + 1 ( 2 k + 1 ) ! = cos ( x ) + j sin ( x )

because the second expression contains the Taylor series for cos ( x ) and sin ( x ) about t = 0 , which converge for all real x . Thus, the desired result is proven.

Choosing x = ω n , we have:

e j ω n = cos ( ω n ) + j sin ( ω n )

which breaks a discrete time complex exponential into its real part and imaginary part. Using this formula, we can also derive the following relationships.

cos ( ω n ) = 1 2 e j ω n + 1 2 e - j ω n
sin ( ω n ) = 1 2 j e j ω n - 1 2 j e - j ω n

Real and imaginary parts of complex exponentials

Now let's return to the more general case of complex exponentials, z n . Recall that z n z n j ω n . We can express this in terms of its real and imaginary parts:

Re { z n } = z n ω n
Im { z n } = z n ω n

We see now that the magnitude of z establishes an exponential envelope to the signal, with ω controlling rate of the sinusoidal oscillation within the envelope.

If z 1 , we have the case of a decaying exponential envelope.
If z 1 , we have the case of a growing exponential envelope.
If z 1 , we have the case of a constant envelope.

Discrete complex exponential demonstration

Interact (when online) with a Mathematica CDF demonstrating the Discrete Time Complex Exponential. To Download, right-click and save target as .cdf.

Discrete time complex exponential summary

Discrete time complex exponentials are signals of great importance to the study of signals and systems. They can be related to sinusoids through Euler's formula, which identifies the real and imaginary parts of complex exponentials. Eulers formula reveals that, in general, the real and imaginary parts of complex exponentials are sinusoids multiplied by real exponentials.

Questions & Answers

what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Arjuns collection. OpenStax CNX. Aug 10, 2011 Download for free at http://cnx.org/content/col11344/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Arjuns collection' conversation and receive update notifications?