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The ideal average value

Ideally, the average value of the new time series will be equal to the constant value of the sinusoid with zero frequency. This is because, ideally,the average value of the other sinusoid will be zero.

Product of sinusoids with different frequencies

The product of any pair of sinusoids that do not have the same frequency will produce a new time series containing the sum of two sinusoids. One of the newsinusoids will have a frequency that is the sum of the frequencies of the two original sinusoids. The other sinusoid will have a frequency that is thedifference between the frequencies of the two original sinusoids.

Ideal average value is zero

Ideally, the average value of the new time series in this case will be equal to zero, because ideally the average value of each of the sinusoids that make upthe time series will be zero.


As we will see later, we don't always achieve the ideal.

Examples of products ofsinusoids

Let's examine some time series produced by multiplying sinusoids. Figure 1 , Figure 2 , and Figure 3 show the results of multiplying sinusoids having the same and different frequencies. Consider first the plots in Figure 1 .

Figure 1. Products of sinusoids.
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Multiplying sinusoids with same frequency

The top plot in Figure 1 shows a sinusoid whose frequency and sampling rate are such that it has 32 samples per cycle. The second plot from the top in Figure 1 is identical to the top plot. (To simplify the explanation, these two sinusoids are also cosine functions.)

The third plot down from the top in Figure 1 shows the product of these two sinusoids, which have the same frequency. If you examine the third plot, youwill notice several important characteristics.

A double-frequency sinusoid

By matching the peaks, you can determine that the frequency of the sinusoid in the third plot is double the frequency of each of the top two plots. (This is the sum of the frequencies of the two sinusoids that were multipliedtogether.)

Half the amplitude with a positive bias

Next, you will notice that the amplitude of the sinusoid in the third plot is half that of each of the first two plots. In addition, the entire sinusoid inthe third plot is in the positive value range.

The sum of two sinusoids

The third plot is actually the sum of two sinusoids. One of the sinusoids has a frequency of zero, giving a constant value of 0.5. This constant value of 0.5is added to all the values in the other sinusoid, causing it to be plotted in the positive value region.

Later on, we will compute the average value of the time series in the third plot. Ideally, that average value will be the constant value produced by thezero-frequency sinusoid.

Product of sinusoids with different frequencies

Now consider the bottom two plots in Figure 1 . The fourth plot down from the top is a cosine function whose frequency is almost, but not quite the same asthe frequency of the sinusoid in the top plot. The sinusoid in the top plot has 32 samples per cycle while the sinusoid in the fourth plot has 31 samples percycle.

Questions & Answers

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
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
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
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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