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Describes the complex exponential function for discrete time.

Introduction

Complex exponentials are some of the most important functions in our study of signals and systems. Their importance stems from their status as eigenfunctions of linear time invariant systems; as such, it can be both convenient and insightful to represent signals in terms of complex exponentials. Before proceeding, you should be familiar with complex numbers.

The discrete time complex exponential

Complex exponentials

The complex exponential function will become a critical part of your study of signals and systems. Its general discrete form is written as

z n
where z is a complex number. Recalling the polar expression of complex numbers, z can be expressed in terms of its magnitude z and its angle (or argument) ω in the complex plane: z z j ω . Thus z n z n j ω n . In the context of complex exponentials, ω is referred to as frequency. For the time being, let's consider complex exponentials for which z 1 .

These discrete time complex exponentials have the following property, which will become evident through discussion of Euler's formula.

j ω n j ω 2 n
Given this property, if we have a complex exponential with frequency ω 2 , then this signal "aliases" to a complex exponential with frequency ω , implying that the equation representations of discrete complex exponentials are not unique.

Euler's formula

The mathematician Euler proved an important identity relating complex exponentials to trigonometric functions. Specifically, he discovered the eponymously named identity, Euler's formula, which states that for any real number x ,

e j x = cos ( x ) + j sin ( x )

which can be proven as follows.

In order to prove Euler's formula, we start by evaluating the Taylor series for e z about z = 0 , which converges for all complex z , at z = j x . The result is

e j x = k = 0 ( j x ) k k ! = k = 0 ( - 1 ) k x 2 k ( 2 k ) ! + j k = 0 ( - 1 ) k x 2 k + 1 ( 2 k + 1 ) ! = cos ( x ) + j sin ( x )

because the second expression contains the Taylor series for cos ( x ) and sin ( x ) about t = 0 , which converge for all real x . Thus, the desired result is proven.

Choosing x = ω n , we have:

e j ω n = cos ( ω n ) + j sin ( ω n )

which breaks a discrete time complex exponential into its real part and imaginary part. Using this formula, we can also derive the following relationships.

cos ( ω n ) = 1 2 e j ω n + 1 2 e - j ω n
sin ( ω n ) = 1 2 j e j ω n - 1 2 j e - j ω n

Real and imaginary parts of complex exponentials

Now let's return to the more general case of complex exponentials, z n . Recall that z n z n j ω n . We can express this in terms of its real and imaginary parts:

Re { z n } = z n ω n
Im { z n } = z n ω n

We see now that the magnitude of z establishes an exponential envelope to the signal, with ω controlling rate of the sinusoidal oscillation within the envelope.

If z 1 , we have the case of a decaying exponential envelope.
If z 1 , we have the case of a growing exponential envelope.
If z 1 , we have the case of a constant envelope.

Discrete complex exponential demonstration

ComplexExponentialDemo
Interact (when online) with a Mathematica CDF demonstrating the Discrete Time Complex Exponential. To Download, right-click and save target as .cdf.

Discrete time complex exponential summary

Discrete time complex exponentials are signals of great importance to the study of signals and systems. They can be related to sinusoids through Euler's formula, which identifies the real and imaginary parts of complex exponentials. Eulers formula reveals that, in general, the real and imaginary parts of complex exponentials are sinusoids multiplied by real exponentials.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
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Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
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
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Source:  OpenStax, Arjuns collection. OpenStax CNX. Aug 10, 2011 Download for free at http://cnx.org/content/col11344/1.1
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