1.6 Complex exponentials

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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 $z$.

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.

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research.net
kanaga
Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
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Daniel
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Abigail
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Anassong
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are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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is Bucky paper clear?
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Do you know which machine is used to that process?
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how to fabricate graphene ink ?
for screen printed electrodes ?
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What is lattice structure?
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Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
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what's the easiest and fastest way to the synthesize AgNP?
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Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
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I'm interested in nanotube
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what is nanomaterials​ and their applications of sensors.
how did you get the value of 2000N.What calculations are needed to arrive at it
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