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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

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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