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(As mentioned earlier, this process would work even if the time series contained sinusoids other than cosine functions. However, the explanationwould be more complicated.)

What about the other sinusoidal components?

Every other sinusoidal component in the time series (whose frequency doesn't match the selected frequency), will produce a new time series containing two sinusoids when multiplied by the sine function or the cosinefunction.

The frequency of one of the sinusoids in the new time series will be the sum of the frequencies of the sinusoidal component and the sine or cosine function.The frequency of the other sinusoid will be the difference in the frequencies between the sinusoidal component and the sine or cosine function.

As you saw earlier, when this difference is very small, the frequency of the new sinusoid will be very near to zero.

The average value for non-matching components

Ideally, the average value of the product should be zero when the frequency of the original sinusoidal component is different from the sine or cosinefunction by which it is multiplied. The computed average of this time series will converge on zero with the quality of the estimate depending on the numberof points in the average.

Measurement error

However, (and this is very important), when the frequency of the original sinusoid is very close to the frequency of the sine or cosine function,the convergence on zero will be poor even for a large number of points in the average.

Thus, the computation at those frequencies very near to the frequency of an actual sinusoidal component in the raw data will produce a non-zeroaverage value even when there is no sinusoidal component in the raw data at those frequencies. This is a form of measurement error.

Let's see some data

With that as a preface, lets look at some graphs ( Figure 10 and Figure 11 ) resulting from spectral analyses. (These two figures show two different views of the same data.)

Figure 10. Spectra of five different sinusoids of different lengths.
missing image

Five sinusoids, same frequency, different lengths

Figure 10 shows the individual spectra computed for five different sinusoids, each having the same frequency, but different lengths. The combination ofsampling rate and frequency was such that each sinusoid had 32 samples per cycle.

Starting at the top in Figure 10 , the lengths of the five sinusoids were 80, 160, 240, 320, and 400 samples. (The lengths of the five sinusoids were multiples of 80 samples.)

Extend to 400 samples for computation

As mentioned earlier, for the cases where the actual length of the sinusoid was less than 400 samples, the length was extended to 400 samples by appendingan appropriate number of samples having zero values.

(This made it easy to compute and plot the spectrum for every sinusoid over the same frequency range with the same number of points in each plot.)

The spectrum was computed and plotted for each sinusoid at 400 individual frequency points between zero and the folding frequency.

The actual averaging window

Even though the Fourier transform program averaged across 400 samples in all cases, the effective averaging length was equal to the length of the sinusoid. All product points outside that length had a value of zero andcontributed nothing to the average one way or the other.

Questions & Answers

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
Almas
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
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William
currently
William
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
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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