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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

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
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
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
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
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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