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 $z$.

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.

Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest.
Rafiq

Rafiq

what is differents between GO and RGO?

Mahi

what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.?
How this robot is carried to required site of body cell.?
what will be the carrier material and how can be detected that correct delivery of drug is done
Rafiq

Rafiq

if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION

Anam

analytical skills graphene is prepared to kill any type viruses .

Anam

Any one who tell me about Preparation and application of Nanomaterial for drug Delivery

The nanotechnology is as new science, to scale nanometric

brayan

nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale

Damian

Is there any normative that regulates the use of silver nanoparticles?