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This module builds the tools necessary for the frequency analysis of brain waves as recording by an electroencephalograph. We proceed from the Pythagorean Theorem to sine waves, the trapezoid rule and finally to Fourier decomposition.

Table of contents

0. Background

1. Sine and Cosine Waves

2. Trapezoid Rule for estimating area

3. Fourier Method for decomposing signals

4. Spectrogram application to analyzing brain waves

Background: brain waves and the eeg

Signals are sent through the brain using both chemical and electrical means. The synchronized electrical activity of individual neurons adds up to something big enough to detect on from outside the head. To measure it, we use a set of electrical nodes called an electroencephalogam (EEG). The measured activity reflects different states of the brain which in turn tell us something about the mindset of the person. Our goal in this module is to decompose an EEG signal into its different frequencies, which is intuitively the most meaningful piece of information.

Sine and cosine waves

Brainwaves have complex shapes that are not easily interpreted. In order to study these waves, we need to develop some mathematical tools that will tell us about different waves. To outline, we begin by talking about pure (sine or cosine) waves, then move to the trapezoid rule for estimating area under a curve. Next, we develop Fourier analysis for picking out the frequencies in a jumbled signal, and finally use these tools to create spectrograms, which allow us to track different frequencies over time.

Sine waves

The sine wave is a mathematical function. It describes many physical phenomena, including sound waves and oscillation. It looks just like a wave. MATLAB uses the sin function to make sin waves. For example, to make Figure 1, we use the code:

>>t = 0:.01:1;>>y = sin(2*pi*t);>>plot(t,y);

The sine wave is defined by the lengths and angles of a triangle. Run sincirc.m (copied below) to see how the sine and cosine values relate to the angle ϕ of the triangle. As you can see, if ϕ is the angle of a right triangle with hypotenuse 1 (illustrated by the circle) , sin ( ϕ ) is the height of the triangle and cos ( ϕ ) is the base of it:

A sin wave

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The relation of sine and cosine to a triangle and unit circle

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Two illustrations of the sine function.
% sincirc.m %% sincirc.m illustrates the relation of the sin and cosine waves to the circle. %define parametersNturns = 2; steps_per_turn = 9;step_inc = 2*pi/steps_per_turn; %set up points for circlecirc_x = cos(0:.01:2*pi); circ_y = sin(0:.01:2*pi);axis equal %loop over triangles with different anglesfor n = 1:Nturns * steps_per_turn; phi = n * step_inc + pi/4;%plot circle, then triangle, then text plot(circ_x, circ_y);axis([-1 1 -1 1] * 1.5);line([0 cos(phi)], [0 sin(phi)]); line([1 1]* cos(phi), [0 sin(phi)]);line([0 cos(phi)], [0 0]); text(cos(phi)/2 , -.1*sign(sin(phi)),'cos(\varphi)')text(cos(phi) + .1*(sign(cos(phi))-.5), sin(phi)/2, 'sin(\varphi)') text(cos(phi)*.2, sin(phi)*.1,'\varphi');pause(.5); end

Characteristics of the sine wave

The sin wave has three primary characteristics:

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, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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