<< Chapter < Page Chapter >> Page >
Lists the four Fourier transforms and when to use them.

Fourier's daring leap

Fourier postulated around 1807 that any periodic signal (equivalently finite length signal) can be built up as an infinite linear combination of harmonic sinusoidal waves.

i.e. Given the collection

B = { e j 2 π T n t } n = -

any

f ( t ) L 2 [ 0 , T )

can be approximated arbitrarily closely by

f ( t ) = n = - C n e j 2 π T n t .

Now, The issue of exact convergence did bring Fourier much criticism from the French Academy of Science (Laplace,Lagrange, Monge and LaCroix comprised the review committee) for several years after its presentation on 1807. It was notresolved for also a century, and its resolution is interesting and important to understand from a practical viewpoint. See more in the section on Gibbs Phenomena .

Fourier analysis is fundamental to understanding the behavior of signals and systems. This is a result of the fact thatsinusoids are Eigenfunctions of linear, time-invariant (LTI) systems. This is to say that if we pass any particular sinusoid through aLTI system, we get a scaled version of that same sinusoid on the output. Then, since Fourier analysis allows us to redefine thesignals in terms of sinusoids, all we need to do is determinehow any given system effects all possible sinusoids (its transfer function ) and we have a complete understanding of the system. Furthermore, sincewe are able to define the passage of sinusoids through a system as multiplication of that sinusoid by the transfer function atthe same frequency, we can convert the passage of any signal through a system from convolution (in time) to multiplication (in frequency). These ideas are what give Fourier analysis itspower.

Now, after hopefully having sold you on the value of this method of analysis, we must examine exactly what we mean by Fourieranalysis. The four Fourier transforms that comprise this analysis are the Fourier Series , Continuous-Time Fourier Transform , Discrete-Time Fourier Transform and Discrete Fourier Transform . For this document, we will view the Laplace Transform and Z-Transform as simply extensions of the CTFT and DTFT respectively. All of thesetransforms act essentially the same way, by converting a signal in time to an equivalent signal in frequency (sinusoids).However, depending on the nature of a specific signal i.e. whether it is finite- or infinite-length and whether it is discrete- or continuous-time) there is anappropriate transform to convert the signal into the frequency domain. Below is a table of the four Fourier transforms andwhen each is appropriate. It also includes the relevant convolution for the specified space.

Table of fourier representations
Transform Time Domain Frequency Domain Convolution
Continuous-Time Fourier Series L 2 0 T l 2 Continuous-Time Circular
Continuous-Time Fourier Transform L 2 L 2 Continuous-Time Linear
Discrete-Time Fourier Transform l 2 L 2 0 2 Discrete-Time Linear
Discrete Fourier Transform l 2 0 N 1 l 2 0 N 1 Discrete-Time Circular

Questions & Answers

what are components of cells
ofosola Reply
twugzfisfjxxkvdsifgfuy7 it
Sami
58214993
Sami
what is a salt
John
the difference between male and female reproduction
John
what is computed
IBRAHIM Reply
what is biology
IBRAHIM
what is the full meaning of biology
IBRAHIM
what is biology
Jeneba
what is cell
Kuot
425844168
Sami
what is cytoplasm
Emmanuel Reply
structure of an animal cell
Arrey Reply
what happens when the eustachian tube is blocked
Puseletso Reply
what's atoms
Achol Reply
discuss how the following factors such as predation risk, competition and habitat structure influence animal's foraging behavior in essay form
Burnet Reply
cell?
Kuot
location of cervical vertebra
KENNEDY Reply
What are acid
Sheriff Reply
define biology infour way
Happiness Reply
What are types of cell
Nansoh Reply
how can I get this book
Gatyin Reply
what is lump
Chineye Reply
what is cell
Maluak Reply
what is biology
Maluak
what is vertibrate
Jeneba
what's cornea?
Majak Reply
what are cell
Achol
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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