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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

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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