# 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.

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
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
Do somebody tell me a best nano engineering book for beginners?
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
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
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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
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.
how did you get the value of 2000N.What calculations are needed to arrive at it
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