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Suppose, for example, that we have two time series, each of which is composed of two sinusoidal components as follows:

f(x) = cos(ax) + cos (bx) g(x) = cos(cx) + cos(dx)

The product of the two time series is given by:

h(x) = f(x)*g(x) = (cos(ax) + cos (bx)) * (cos(cx) + cos(dx))

where the asterisk (*) means multiplication.

Multiplying this out produces the following:

h(x) = cos(ax)*cos(cx) + cos(ax)*cos(dx)+ cos(bx)*cos(cx) + cos(bx)*cos(dx)

A sum of products of sinusoids

Thus, the time series produced by multiplying any two time series consists of the sum of a (potentially large) number of terms, each of which is the product of two sinusoids.

The product of two sinusoids

We probably need to learn a little about the product of two sinusoids. I will discuss this topic with a little more mathematical rigor in a future module. Inthis module, however, I will simply illustrate the topic using graphs.

Important: The product of two sinusoids is always a new time series, which is the sum of two new sinusoids.

The frequencies of the new sinusoids

The frequencies of the new sinusoids are different from the frequencies of the original sinusoids. Furthermore, the frequency of one of the new sinusoidsmay be zero.

What is a sinusoid with zero frequency?

As a practical matter, a sinusoid with zero frequency is simply a constant value. It plots as a horizontal straight lineversus time.

Think of it this way. As the frequency of the sinusoid approaches zero, the period, (which is the reciprocal of frequency), approaches infinity. Thus, the width of the first lobe of the sinusoid widens, causing every value in thatlobe to be the same as the first value.

This will become a very important concept as we pursue DSP operations.

Sum and difference frequencies

More specifically, when you multiply two sinusoids, the frequency of one of the sinusoids in the new time series is the sum of the frequencies of the two sinusoids that were multiplied together. The frequency of the othersinusoid in the new time series is the difference between the frequencies of the two sinusoids that were multiplied together.

An important special case

For the special case where the two original sinusoids have the same frequency, the difference frequency is zero and one of the sinusoids in the newtime series has a frequency of zero. It is this special case that makes digital filtering and digital spectrum analysis possible.

Many sinusoidal products

When we multiply two time series and compute the average of the resulting time series, we are in effect computing the average of the products of all theindividual sinusoidal components contained in the two time series. That is, the new time series contains the products of (potentially many) individual sinusoids contained in the two original time series. In the end, it all comesdown to computing the average value of products of sinusoids.

Product of sinusoids with same frequency

The product of any pair of sinusoids that have the same frequency will produce a time series containing the sum of two sinusoids. One of the sinusoidswill have a frequency of zero (hence it will have a constant value). The other sinusoid will have a frequency that is double the frequency of theoriginal sinusoids.

Questions & Answers

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
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
how can I make nanorobot?
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
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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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