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Although these identities apply to the products of sine and cosine values for single angles a and b , it is a simple matter to extend them to represent the products of time series consisting of sine andcosine functions. Such an extension is shown in Figure 8 .

Products of sine and cosine functions

In each of the three cases shown in Figure 8 , the function f(n) is a time series produced by multiplying two other time series, which are either sinefunctions or cosine functions.

Figure 8. Products of sine and cosine functions.
1. f(n) = sin(a*n)*sin(b*n) = (1/2)*(cos((a-b)*n)-cos((a+b)*n))2. f(n) = cos(a*n)*cos(b*n) = (1/2)*(cos((a-b)*n)+cos((a+b)*n))3. f(n) = sin(a*n)*cos(b*n) = (1/2)*(sin((a+b)*n)+sin((a-b)*n))

Rewrite and simplify

Figure 9 rewrites and simplifies these three functions for the special case where a=b , taking into account the fact that cos(0) =1 and sin(0) = 0.

Figure 9. Rewrite and simplify.
1. f(n) = sin(a*n)*sin(a*n) = (1/2)-cos(2*a*n)/2 2. f(n) = cos(a*n)*cos(a*n) = (1/2)+cos(2*a*n)/23. f(n) = sin(a*n)*cos(a*n) = sin(2*a*n)/2

What can we learn from these identities?

First you need to recall that the average of the values describing any true sinusoid is zero when the average is computed over an even number of cycles ofthe sinusoid.

(A true sinusoid does not have a bias to prevent it from being centered on the horizontal axis.)

If a time series consists of the sum of two true sinusoids, then the average of the values describing that time series will be zero if the average iscomputed over an even number of cycles of both sinusoids, and very close to zero if the average is computed over a period that is not an even number of cyclesfor either or both sinusoids.

(The average will approach zero as the length of data over which the average is computed increases.)

Product of two sine functions having the same frequency

Let's apply this knowledge to the three cases shown above for a=b . Consider the time series for case 1 in Figure 9 . This case is the product of two sine functions having the same frequency. The result of multiplying the two sinefunctions is shown graphically in Figure 10 .

Figure 10. Plot of sin(x) and sin(x)*sin(x).
Plot of sin(x) and sin(x)*sin(x)

The red curve in Figure 10 shows the function sin(x), and the black curve shows the function produced by multiplying sin(x) by sin(x).

The sum of the product function is not zero

If you sum the values of the black curve 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 Imag(F) in Figure 6 . The imaginary part is computed by multiplying the time series by a sine function and computing the sum of theproducts. If that time series contains a sine component with the same frequency as the sine function, that component will contribute a non-zero value to the sumof products. Thus, the imaginary part of the transform at that frequency will not be zero.

Product of two cosine functions having the same frequency

Now consider the time series for case 2 in Figure 9 . This case is the product of two cosine functions having the same frequency. The result of multiplying twocosine functions having the same frequency is shown graphically in Figure 11 .

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
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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
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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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