<< Chapter < Page Chapter >> Page >
Describes the complex exponential function.

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. Before proceeding, you should be familiar with complex numbers.

The continuous time complex exponential

Complex exponentials

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

A s t
where s σ ω is a complex number in terms of σ , the attenuation constant, and ω the angular frequency.

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

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 t this gives the result

e j ω t = cos ( ω t ) + j sin ( ω t )

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

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

Continuous time phasors

It has been shown how the complex exponential with purely imaginary frequency can be broken up into its real part and its imaginary part. Now consider a general complex frequency s = σ + ω j where σ is the attenuation factor and ω is the frequency. Also consider a phase difference θ . It follows that

e ( σ + j ω ) t + j θ = e σ t cos ( ω t + θ ) + j sin ( ω t + θ ) .

Thus, the real and imaginary parts of e s t appear below.

Re { e ( σ + j ω ) t + j θ } = e σ t cos ( ω t + θ )
Im { e ( σ + j ω ) t + j θ } = e σ t sin ( ω t + θ )

Using the real or imaginary parts of complex exponential to represent sinusoids with a phase delay multiplied by real exponential is often useful and is called attenuated phasor notation.

We can see that both the real part and the imaginary part have a sinusoid times a real exponential. We also know thatsinusoids oscillate between one and negative one. From this it becomes apparent that the real and imaginary parts of thecomplex exponential will each oscillate within an envelope defined by the real exponential part.

If σ is negative, we have the case of a decaying exponential window.
If σ is positive, we have the case of a growing exponential window.
If σ is zero, we have the case of a constant window.
The shapes possible for the real part of a complex exponential. Notice that the oscillations are the result ofa cosine, as there is a local maximum at t 0 .

Complex exponential demonstration

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

Continuous time complex exponential summary

Continuous 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 purely imaginary complex exponentials. Eulers formula reveals that, in general, the real and imaginary parts of complex exponentials are sinusoids multiplied by real exponentials. Thus, attenuated phasor notation is often useful in studying these signals.

Questions & Answers

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
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
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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signals and systems' conversation and receive update notifications?

Ask