# Java1478-fun with java, how and why spectral analysis works  (Page 7/9)

 Page 7 / 9

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). 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 can chip be made from sand
Eke Reply
is this allso about nanoscale material
Almas
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

### Read also:

#### Get Jobilize Job Search Mobile App in your pocket Now!

Source:  OpenStax, Digital signal processing - dsp. OpenStax CNX. Jan 06, 2016 Download for free at https://legacy.cnx.org/content/col11642/1.38
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Digital signal processing - dsp' conversation and receive update notifications? By OpenStax By OpenStax By OpenStax By OpenStax By Stephen Voron By Cath Yu By OpenStax By Yasser Ibrahim By Katy Keilers By John Gabrieli