# 2.2 Dsp00108-averaging time series  (Page 9/14)

 Page 9 / 14

## 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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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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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