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The Fourier transform

In order to perform the spectral analysis, I will perform a Fourier transform on the time series to transform that data into the frequency domain. Then I willplot the data in the frequency domain.

(This module will not provide technical details on the Fourier transform. That information will be forthcoming in a future module.)

Keeping it simple

To keep this explanation as simple as possible, I will stipulate that all of the sinusoids contained in the time series are cosine functions. There are nosine functions in the time series.

(If the time series did contain sine functions, the process would still work, but the explanation would be more complicated.)

A brief description of the fourier transform

Before I get into the results, I will provide a very brief description of how I performed the Fourier transform for these experiments.

The following steps were performed at each frequency in a set of 400 uniformly spaced frequencies across the frequency range from zero to the foldingfrequency.

The steps were:

  • If the time series was shorter than 400 points, extend it to 400 points by appending zero-valued points at the end.
  • Select the next frequency of interest.
  • Generate a cosine function, 400 samples in length, at that frequency.
  • Multiply the cosine function by the time series.
  • Compute the average value of the time series produced by multiplying the cosine function by the time series.
  • Save the average value. Call it the real value for later reference.
  • Generate a sine function, 400 samples in length, at the same frequency.
  • Multiply the sine function by the time series.
  • Compute the average value of the time series produced by multiplying the sine function by the time series.
  • Save the average value. Call it the imaginary value for later reference.
  • Compute the square root of the sum of the squares of the real and imaginary values. This is the value of interest. Plot it.

Why does this work?

No matter how many sinusoidal components are contained in the time series, only one (if any) of those sinusoidal components will match the selected frequency.

Multiply by the cosine and average the product

When that matching component is multiplied by the cosine function having the selected frequency, the new time series created by the multiplication willconsist of a constant value plus a sinusoid whose frequency is twice the selected frequency.

The computed average value of this time series will converge on the value of the constant with the quality of the estimate depending on the number of pointsincluded in the average.

Multiply by the sine and average the product

Since the sinusoids in the time series are stipulated to be cosine functions, when the sinusoid with the matching frequency is multiplied by the sinefunction, the new time series will consist of a constant value of zero plus a sinusoid whose frequency is twice the frequency of the sine function.

The computed average of this time series will converge on zero with the quality of the estimate depending on the number of points in the average.

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
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what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
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what is nanomaterials​ and their applications of sensors.
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what is system testing?
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how to synthesize TiO2 nanoparticles by chemical methods
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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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