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This module includes a table of common discrete fourier transforms.

Introduction

Once one has obtained a solid understanding of the fundamentals of Fourier series analysis and the General Derivation of the Fourier Coefficients , it is useful to have an understanding of the common signals used in Fourier Series Signal Approximation.

Deriving the coefficients

Consider a square wave f(x) of length 1. Over the range [0,1), this can be written as

x ( t ) = 1 t 1 2 ; - 1 t > 1 2 .

Fourier series approximation of a square wave

Fourier series approximation to sq t . The number of terms in the Fourier sum is indicated in eachplot, and the square wave is shown as a dashed line over two periods.

Real even signals

Given that the square wave is a real and even signal,
  • f ( t ) = f ( - t ) EVEN
  • f ( t ) = f * ( t ) REAL
  • therefore,
  • c n = c - n EVEN
  • c n = c n * REAL

Deriving the coefficients for other signals

The Square wave is the standard example, but other important signals are also useful to analyze, and these are included here.

Constant waveform

This signal is relatively self-explanatory: the time-varying portion of the Fourier Coefficient is taken out, and we are left simply with a constant function over all time.

x ( t ) = 1

Fourier series approximation of a constant wave

Sinusoid waveform

With this signal, only a specific frequency of time-varying Coefficient is chosen (given that the Fourier Series equation includes a sine wave, this is intuitive), and all others are filtered out, and this single time-varying coefficient will exactly match the desired signal.

x ( t ) = c o s ( 2 π t )

Fourier series approximation of a sinusoid wave

Triangle waveform

x ( t ) = t t 1 / 2 1 - t t > 1 / 2
This is a more complex form of signal approximation to the square wave. Because of the Symmetry Properties of the Fourier Series, the triangle wave is a real and odd signal, as opposed to the real and even square wave signal. This means that
  • f ( t ) = - f ( - t ) ODD
  • f ( t ) = f * ( t ) REAL
  • therefore,
  • c n = - c - n
  • c n = - c n * IMAGINARY

Fourier series approximation of a triangle wave

Sawtooth waveform

x ( t ) = t / 2
Because of the Symmetry Properties of the Fourier Series, the sawtooth wave can be defined as a real and odd signal, as opposed to the real and even square wave signal. This has important implications for the Fourier Coefficients.

Fourier series approximation of a sawtooth wave

Dft signal approximation

fourierDiscreteDemo
Interact (when online) with a Mathematica CDF demonstrating the common Discrete Fourier Series. To download, right-click and save as .cdf.

Conclusion

To summarize, a great deal of variety exists among the common Fourier Transforms. A summary table is provided here with the essential information.

Common discrete fourier transforms
Description Time Domain Signal for n Z [ 0 , N - 1 ] Frequency Domain Signal k Z [ 0 , N - 1 ]
Constant Function 1 δ ( k )
Unit Impulse δ ( n ) 1 N
Complex Exponential e j 2 π m n / N δ ( ( k - m ) N )
Sinusoid Waveform c o s ( j 2 π m n / N ) 1 2 ( δ ( ( k - m ) N ) + δ ( ( k + m ) N ) )
Box Waveform ( M < N / 2 ) δ ( n ) + m = 1 M δ ( ( n - m ) N ) + δ ( ( n + m ) N ) sin ( ( 2 M + 1 ) k π / N ) N sin ( k π / N )
Dsinc Waveform ( M < N / 2 ) sin ( ( 2 M + 1 ) n π / N ) sin ( n π / N ) δ ( k ) + m = 1 M δ ( ( k - m ) N ) + δ ( ( k + m ) N )

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
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
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
ya I also want to know the raman spectra
Bhagvanji
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, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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