# 2.2 Dsp00108-averaging time series  (Page 13/14)

 Page 13 / 14

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. 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.

what is the stm
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?
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What is meant by 'nano scale'?
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LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
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Mahi
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Rafiq
what is Nano technology ?
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?
what king of growth are you checking .?
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why?
what school?
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research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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absolutely yes
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Abigail
for teaching engĺish at school how nano technology help us
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Do somebody tell me a best nano engineering book for beginners?
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NANO
how can I make nanorobot?
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what is fullerene does it is used to make bukky balls
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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
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