<< 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

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
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
Prasenjit Reply
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.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
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!
QuizOver.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