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Figure 11. Plot of cos(x) and cos(x)*cos(x).
Plot of cos(x) and cos(x)*cos(x)

The red curve in Figure 11 shows the function cos(x), and the black curve shows the function produced by multiplying cos(x) by cos(x).

Again the sum of products is not zero

If you sum the values of the black curve in Figure 11 over an even number of cycles, the sum will not be zero. Rather, it will be a positive, non-zero value.

Now refer back to the expression for Real(F) in Figure 6 . The real part of the transform is computed by multiplying the time series by a cosine functionhaving a particular frequency and computing the sum of products. If that time series contains a cosine component with the same frequency as the cosinefunction, that component will contribute a non-zero value to the sum of products. Thus, the real part of the transform at that frequency will not bezero.

Product of a sine function and a cosine function

Now consider the time series for case 3 in Figure 9 , which is the product of a sine function and a cosine function having the same frequency. The result ofcomputing this product is shown graphically in Figure 12

Figure 12. Plot of sin(x), cos(x), and sin(x)*cos(x).
Plot of sin(x), cos(x), and sin(x)*cos(x)

The red curve in Figure 12 shows the function cos(x), and the green curve shows the function sin(x). The black curve shows the function produced bymultiplying sin(x) by cos(x).

The sum of the products will be zero

If you sum the values of the black curve over an even number of cycles, the sum will be zero.

Therefore, referring back to Figure 6 , we see that

  • the Real(F) computation measures only the cosine component in the time series at a particular frequency, and
  • the Imag(F) computation measures only the sine component in the time series having the same frequency.

The Real(F) computation in Figure 6 does not produce a non-zero output due to a sine component in the time series having the same frequency. The Imag(F)computation in Figure 6 does not produce a non-zero output due to a cosine component in the time series having the same frequency.

Thus, at a particular frequency, the existence of a cosine component in the target time series produces the real output, and the existence of a sinecomponent in the target time series produces the imaginary output.

Neither sine nor cosine

In reality, the sinusoidal components that make up a time series will not usually be sine functions or cosine functions. Rather, they will be sinusoidalcomponents having the same shape as a sine or cosine, but not having the same value at zero as either a sine function or a cosine function. However, it can beshown that a general sinusoidal function can always be represented by the sum of a sine function and a cosine function having different amplitudes and the samefrequency.

(A proof of the above statement is beyond the scope of this module. You will simply have to accept on faith that a general time series can berepresented as the sum of a potentially infinite number of sine functions and cosine functions of different frequencies and different amplitudes. Itis these cosine and sine functions that constitute the real and imaginary components of the complex frequency spectrum.)

Questions & Answers

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
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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?
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
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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