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If you know the value of a single real sample and you know its position in the series relative to the origin, you can write equationsthat describe the real and imaginary parts of the transform of that single sample without any requirement to actually perform a Fourier transform.

Those equations are simple sine and cosine equations as a function of the units of the output domain. This is an important concept that contributesgreatly to the implementation of the FFT algorithm.

Transformation of a complex series

The FFT algorithm is an algorithm that transforms a series of complex values in one domain into a series of complex values in another domain. The images inthe figures discussed so far indicate a transformation of a complex function given by f(x) into another complex function given by F(k). There is nothing inthese images to indicate anything about time and frequency.

If the complex part of the input series f(x) is not zero, things get somewhat more complicated. For example, the real and imaginary parts of the transform ofan impulse having both real and imaginary parts are not necessarily cosine and sine curves. This is illustrated in Figure 8 .

Figure 8. Transform of a complex impulse with a shift equal to two sample intervals.
missing image

Figure 8 shows the results of transforming an impulse having both real andimaginary parts and a shift of two sample intervals.

Although both the real and imaginary parts of the transformed result have the shape of a sinusoid, neither is a cosine curve and neither is a sine curve. Bothof the curves are sinusoidal curves that have been shifted along the horizontal output axis moving their peaks and zero crossings away from the origin.

Linearity still applies

Because the Fourier transform is a linear transform, you can transform the real and imaginary parts of the input separately and add the two resultingtransforms. The sum of the two transforms represents the transform of the entire input series including both real and imaginary parts. The program that I willdiscuss later takes advantage of this fact. Once again, the main point is:

Even for a complex input series, if you know the values of the real and imaginary parts of a sample and you know the value of the shiftassociated with that sample, you can write equations that describe the real part and the imaginary part of the transform results.

Can produce the transform of a time series by the adding transforms of the individual samples

That brings us to the crux of the matter. Given an input series consisting of a set of sequential samples taken atuniform sampling intervals, we know how to write equations for the real and imaginary parts that would be produced by performing a Fourier transform oneach of those samples individually.

The input series is the sum of the individual samples

We know that we can consider the input series to consist of the sum of the individual samples, each having a specified value and a different shift. We knowthat the Fourier transform is a linear transform. Therefore, the Fourier transform of an input series is the sum of the transforms of the individualsamples.

Questions & Answers

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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
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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