# 2.2 Dsp00108-averaging time series  (Page 3/14)

 Page 3 / 14

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.

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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
what is Nano technology ?
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
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
hi
Loga
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers! By OpenStax By Jams Kalo By Madison Christian By OpenStax By Mariah Hauptman By Rebecca Butterfield By Anonymous User By Richley Crapo By Jonathan Long By OpenStax