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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

where we get a research paper on Nano chemistry....?
Maira Reply
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
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
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
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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