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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

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar 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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