# 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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Joseph
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nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
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da
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
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da
Application of nanotechnology in medicine
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Kamaluddeen
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narayan
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Bhagvanji
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Damian
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Professor
I think
Professor
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Alexandre
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Alexandre
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Damian
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LITNING
scanning tunneling microscope
Sahil
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Rafiq
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Rafiq
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Anam
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Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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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?
what king of growth are you checking .?
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