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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

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
is it a question of log
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
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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