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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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
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Can you compute that for me. Ty
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what is inorganic
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Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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answer
Magreth
progressive wave
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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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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