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.

Questions & Answers

Do somebody tell me a best nano engineering book for beginners?

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.

In this morden time nanotechnology used in many field .
1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc
2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc
3- Atomobile -MEMS, Coating on car etc.
and may other field for details you can check at Google

Azam

anybody can imagine what will be happen after 100 years from now in nano tech world

Prasenjit

after 100 year this will be not nanotechnology maybe this technology name will be change .
maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments

Azam

name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world

Prasenjit

how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?

Damian

silver nanoparticles could handle the job?

Damian

not now but maybe in future only AgNP maybe any other nanomaterials

Azam

Hello

Uday

I'm interested in Nanotube

Uday

this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15

Prasenjit

can nanotechnology change the direction of the face of the world

At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.