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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

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
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
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
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.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
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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