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The time series in the bottom plot is the product of the time series in the first and fourth plots.

The sum of two sinusoids

Once again, this time series is the sum of two sinusoids. The frequency of one is the difference between the two original frequencies. The frequency of theother is the sum of the two original frequencies.

However, in this case, the difference frequency is not zero . Rather, it is a very low frequency. What you see in the bottom plot of Figure 1 is a sinusoid whose frequency is the sum of the two original frequencies addedto a sinusoid whose frequency is the difference between the two original frequencies. Because the two original frequencies were almost equal, thefrequency of the second sinusoid is very low.

As you can see, the low-frequency component in the bottom plot in Figure 1 appears to be the beginning of a cosine function whose period is much greaterthan the width of the plot (400 points).

Another view of the same data

Figure 2 shows another view of the bottom two plots from Figure 1 .

Figure 2. Products of sinusoids.
missing image

The difference between Figure 2 and Figure 1 is that while Figure 1 shows only 400 points along the x-axis, Figure 2 shows 1200 points along the x-axis. Thus, the horizontal scale in Figure 2 is significantly compressed relative to the horizontal scale in Figure 1 .

More than one cycle

Figure 2 lets you see a little more than one full cycle of the low-frequency component of the time series produced by multiplying the two sinusoids.

( Figure 2 does not provide a very good representation of the high-frequency component. This is because I plotted 1200 points in a part ofthe screen that is only 400 pixels wide. On my computer, I can expand this to the full screen width. However, I can't publish it at that width, so Ipublished the 400-pixel version.)

Averaging can be problematic in this case

Later on, we will compute the average value of the time series represented by the bottom plot in Figures 1 and 2. Ideally, that average value will be zero.However, you have probably already figured out that a great many data points must be included in the computation of the average to get anything near zero. Aneyeball estimate indicates that about 900 data points are required just to include a single cycle of the low-frequency component.

More examples of the products of sinusoids

Figure 3 shows two additional time series created by multiplying sinusoids.

Figure 3. More products of sinusoids.
missing image

The arrangement in Figure 3 is the same as in Figure 1 . The top plot in Figure 3 is the same sinusoid shown in the top plot of Figure 1 . This is a sinusoid with 32 samples per cycle.

Immediately below the top sinusoid in Figure 3 is another sinusoid. This sinusoid has 24 samples per cycle. As you can see, the frequency of thissinusoid is a little higher than the frequency of the sinusoid in the top plot.

The time series in the third plot down from the top is the product of the time series in the top two plots. Again, this time series is composed of two newsinusoids whose frequencies are the sum of and difference between the two original frequencies.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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