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As before, the lengths of the individual time series will be 80, 160, 240, 320, and 400 samples respectively.

Spectral analysis using fourier transform

I will perform a Fourier transform on each of the time series in an attempt to show that the spectrum of each time series consists of two peaks, (two spectral lines) , with one peak corresponding to each of the sinusoids added together to create the time series. The five spectra are shown in Figure 12 .

Figure 12. Spectra of five different time series of different lengths.
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Discuss the longest time series first

Once again, let's begin with the plot at the bottom of Figure 12 . As you can see, this spectrum shows two very distinct spectral peaks. Thus, for this amountof frequency separation and a length of 400 samples, the Fourier transform did a good job of separating the two peaks.

Resolution for shorter averaging windows

Moving upward in Figure 12 , we see that the Fourier transform on the time series with a length of 320 samples (the fourth plot from the top) also did a good job of separating the two peaks.

However, with respect to separation the process began to deteriorate for lengths of 240 samples and 160 samples.

No peak separation for 80-sample average

For a length of 80 samples, the two peaks merged completely.

A horizontally-expanded view of the spectra

Figure 13 shows a horizontally-expanded view of the same spectral data to give you a better idea of the structure of the peaks. The plots in Figure 13 show only the first fifty frequency values.

Figure 13. Spectra of five different time series of different lengths
missing image

You may find it interesting to make a side-by-side comparison of Figures 13 and 11 in separate browser windows.

Zero-valued points in the spectra

Before leaving this topic, there are a few more things that I want to show you. If you go back and look at the bottom plot in Figure 11 , you will note an interesting characteristic of that plot. In particular, starting at the peak andmoving outward in both directions, every second plotted value is zero. I'm going to explain the reason for and the significance of this characteristic.

(As I mentioned earlier, there are also zero-valued points in the spectra of the time series with the shorter averaging windows. Once youunderstand the reason for the zero-valued points, you can correlate the positions of those points to the length of the averaging window.)

400 spectral values were computed

To begin with, the Fourier transform program that was used to compute this spectrum computed 400 values at equally spaced points between zero and thefolding frequency (only the first 50 values are shown in Figure 11 ). Thus, each of the side-by-side rectangles in Figure 11 represents the spectral value computed at one of the 400 frequency points.

Sampling frequency was one sample per second

The sinusoid that was used as the target for this spectral analysis had 32 samples per cycle. Since this sinusoid was generated mathematically instead ofbeing the result of sampling an analog signal, we can consider the sampling frequency to be anything that we want.

Questions & Answers

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
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
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
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor 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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