<< Chapter < Page Chapter >> Page >

From complex sinusoids to complex exponentials

Recall the form of a discrete-time complex sinusoid: $x[n]=e^{j(\omega n + \phi)$. As we have already seen, that signal itself is complex-valued, i.e., it has both a real and an imaginary part. But look closely at just the exponent, and you will see that the exponent itself is purely imaginary.

Suppose we let the exponent be complex-valued, say of the form $a+jb$. Then we have $e^{(a+jb)n}=e^{an}e^{jbn}=(e^a)^n e^{jbn}$. So the result is a complex sinusoid multipled by a real exponential signal (whose base is $e^a$).

Complex exponentials, defined

We do not typically represent complex exponentials in the way derived above, but rather express them in the form $x[n]=z^n$, where $z$ is a complex number. Being a complex number, it lies on the complex plane with a magnitude of $|z|$ and an angle of $\angle z$ we define as $\omega$. So then, if we would like to express $x[n]=z^n$ as a combination of a real exponential and a complex sinusoid, as above, we have: $x[n]=z^n=|z|^n e^{j\omega n}$. Below are some plots of complex exponentials for different values of $z$.
Image Image
The real and imaginary parts of a complex exponential $x^n$ for which $|z|\lt 1$.
Image Image
The real and imaginary parts of a complex exponential $x^n$ for which $|z|\gt 1$.

So when the magnitude $|z|$ is greater than 1, we have a signal that oscillates and exponentially grows with time, and if the magnitude is less than 1, it decays over time. And, you guessed it, if the magnitude is exactly equal to 1, it does not grow or decay, but only oscillates. In fact, if the magnitude is 1, the complex exponential is, by definition, simply a complex sinusoid: $|1|^n e^{j\omega n}=e^{j\omega n}$. Therefore you can see that complex sinusoids are a subset of the more general complex exponential signals.

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
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.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
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, Discrete-time signals and systems. OpenStax CNX. Oct 07, 2015 Download for free at https://legacy.cnx.org/content/col11868/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Discrete-time signals and systems' conversation and receive update notifications?