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The DFT algorithm

Using my alternative notation described earlier in Figure 2 , the expressions that you must evaluate to determine the frequency spectral content of a target timeseries at a frequency F are shown in Figure 6 (note that I didn't bother to divide by N which is fairly common practice) .

Figure 6. Forward Fourier transform.
Real(F) = S(n=0,N-1)[x(n)*cos(2Pi*F*n)] Imag(F) = S(n=0,N-1)[x(n)*sin(2Pi*F*n)]ComplexAmplitude(F) = Real(F) - j*Imag(F) Power(F) = Real(F)*Real(F) + Imag(F)*Imag(F)

What does this really mean?

Before you panic, let me explain what this means in layman's terms. Given a time series, x(n), you can determine if that time series contains a cosinecomponent or a sine component at a given frequency, F, by doing the following:

  • Create one new time series, cos(n), which is a cosine function with the frequency F.
  • Create another new time series, sin(n), which is a sine function with the frequency F. (The methods needed to create the cosine and sine time series are available in the Math class in the standard Java library.)
  • Multiply x(n) by cos(n) and compute the sum of the products. Save this value, calling it Real(F). This is an estimate of the amplitude, if any, ofthe cosine component with the matching frequency contained in the time series x(n).
  • Multiply x(n) by sin(n) and compute the sum of the products. Save this value, calling it Imag(F). This is an estimate of the amplitude, if any, ofthe sine component with the matching frequency contained in the time series x(n).
  • Consider the values for Real(F) and Imag(F) to be the real and imaginary parts of a complex number.
  • Consider the sum of the squares of the real and imaginary parts to represent the power at that frequency in the time series.

It's that simple

That's all there is to it. For each frequency of interest, you can use this process to compute a complex number, Real(F) - jImag(F), whichrepresents the component of that frequency in the target time series.

(The mathematicians in the audience probably prefer to use the symbol i instead of the symbol j to represent the imaginary part. The use of j forthis purpose comes from my electrical engineering background.)

Similarly, you can compute the sum of the squares of the real and imaginary parts and consider that to be a measure of the power at that frequency in thetime series.

(This is typically the value that you would see being displayed by one of the dancing vertical bars on the front of the equalizer on your stereosystem.)

Normally we are interested in more than one frequency, so we would repeat the above procedure once for each frequency of interest.

(This would produce the set of values that you would likely see being displayed by all of the dancing vertical bars on the font of the equalizeron your stereo system.)

Why does this work?

This works because of the three trigonometric identities shown in Figure 7 .

Figure 7. Three trigonometric identities.
1. sin(a)*sin(b)=(1/2)*(cos(a-b)-cos(a+b)) 2. cos(a)*cos(b)=(1/2)*(cos(a-b)+cos(a+b))3. sin(a)*cos(b)=(1/2)*(sin(a+b)+sin(a-b))

Questions & Answers

are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
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
has a lot of application modern world
Kamaluddeen
yes
narayan
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Digital signal processing - dsp. OpenStax CNX. Jan 06, 2016 Download for free at https://legacy.cnx.org/content/col11642/1.38
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