2.2 Dsp00108-averaging time series  (Page 11/14)

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(I also applied an additional scale factor to the spectral results to compensate for the fact that fewer total samples were included in theaverage for the short samples. This caused the amplitude of the peak in the spectrum to be nominally the same in all five cases.)

A horizontally-expanded plot

As you can see in Figure 10 , there isn't much in the spectra to the right of about 50 spectral points. That is as it should be since the single sinusoid ineach time series was at the low end of the spectrum.

Figure 11 shows the same data as Figure 10 with only the first 50 frequency points plotted on the horizontal axis. The remaining 350 frequency points weresimply ignored. This provides a much better view of the structure of the peaks in the different spectra.

Figure 11. Spectra of five different sinusoids of different lengths.

I will begin the discussion with the bottom plot in Figure 11 , which is the computed spectrum for the single sinusoid having a length of 400 samples.

A spectral line

Ideally, since the time series was a single sinusoid, the spectrum should consist of a single non-zero value at the frequency of the sinusoid, (often referred to as a spectral line) and every other value in the spectrum should be zero.

However, because the computation of the spectrum involves the computation of average values resulting from the products of sinusoids, the ideal is not alwaysachieved. In order to achieve the ideal, it would be necessary to multiply and average over an infinite number of points. Anything short of that will result insome measurement error, as exhibited by the bottom plot in Figure 11 .

(The bottom plot in Figure 11 has a large peak in the center with every second point to the left and right of center having a zero value. Iwill explain this structure in more detail later.)

Spectra of shorter sinusoids

Moving from the bottom to the top in Figure 11 , each individual plot shows the result of shorter and shorter averaging windows. As a result, themeasurement error increases and the peak broadens for each successive plot going from the bottom to the top in Figure 11 . The plot at the top, with an averaging window of only 80 samples, exhibits the most measurement error and the broadestpeak.

(It should be noted, however, that even the spectra for the shorter averaging windows have some zero-valued points. Once you understand thereason for the zero-valued points, you can correlate the positions of those points to the length of the averaging windows in Figure 11 .)

Two spectral lines

Now I'm going to show you the detrimental impact of such spectral measurement errors. In particular, the failure of the average to converge on zero for shortaveraging windows limits the spectral resolution of the Fourier transform.

Five time series with two sinusoids each

I will create five new time series, each consisting of the sum of two sinusoids with fairly closely-spaced frequencies. One sinusoid has 32 samplesper cycle as in Figures 10 and 11. The other sinusoid has 26 samples per cycle.

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?
what is a peer
What is meant by 'nano scale'?
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
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 .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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Lily
Do somebody tell me a best nano engineering book for beginners?
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NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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Berger describes sociologists as concerned with
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