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For simplicity, let's assume that the sampling frequency was one sample per second. This causes the sinusoid to have a period of 32 seconds and a frequency of 0.03125 cycles per second.

At a sampling rate of one sample per second, the folding frequency occurs at 0.5 cycles per second.

The computational frequency interval

Dividing the folding frequency by 400 we conclude that the Fourier transform program computed a spectral value every 0.00125 cycles per second. Given thatevery second spectral value is zero, the zero values occur every 0.00250 cycles per second.

Let's compute the average of some products

The top plot in Figure 14 shows the result of multiplying a cosine function having a frequency of 0.03125 cycles per second (the frequency of the sinusoid in the previous spectral analysis experiment) by a sine function having a frequency of 0.02875 cycles per second.

(This replicates one of the steps in the computation of the imaginary value in the Fourier transform).

Figure 14. Average values of sinusoid products.
missing image

The difference between the frequencies of the cosine function and the sine function is 0.00250 cycles per second.

(Note that this frequency difference is the reciprocal of the actual number of samples in the earlier time series, which contained 400 samples.This is also the frequency interval on which the Fourier transform produced zero-valued points for the bottom plot in Figure 11 .)

The average of the product time series

The second plot in Figure 14 shows the average value of the time series in the first plot versus the number of samples included in the averaging window.

(This replicates another step in the computation of the imaginary value in the Fourier transform).

It is very important to note that this average plot goes to zero when 400 samples are included in the average.

Product of two cosine functions

Similarly, the third plot in Figure 14 shows the product of the same cosine function as above and another cosine function having the same frequency as thesine function described above.

(This replicates a step in the computation of the real value in the Fourier transform).

The average of the product time series

The fourth plot in Figure 14 shows the average value of the time series in the third plot.

(This replicates another step in the computation of the real value in the Fourier transform).

This average plot goes to zero at an averaging window of about 200 samples, and again at an averaging window of 400 samples.

Where do the zero values match?

The first point at which both average plots go to zero at the same point on the horizontal axis is at an averaging window of 400 samples.

(Both the real and imaginary values must go to zero in order for the spectral value produced by the Fourier transform to go to zero.)

Zero values in the spectrum for a sinusoid

Thus, the values produced by performing a Fourier transform on a single sinusoid go through zero at regular frequency intervals out from the peak inboth directions. The frequency intervals between the zero values are multiples of the reciprocal of the actual length of the sinusoid on which the transform isperformed.

Questions & Answers

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
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
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
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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