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Kylie Sullivan

on 13 January 2014

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What is Fourier Transformation?
cos(θ) = Adjacent /Hypotenuse
Amplitude = 1
Period = 2Pie
Equation of the Axis = Y=0
Domain is any real number
Range is any real number and must be greater or equal to -1 and smaller or equal to 1.
Cell Phones
Image by Tom Mooring

d=horizontal shift
c=equation of the axis
Real Life Applications
Differential Equations
Radiation from Surface Currents
Signal Processing
Properties of Sinusoidal Functions
Everything that surrounds us can be modeled by a wavelength. For Example: Sound Waves, Radiation, Electromagnetic Fields, Elevation vs Location, Stock Exchange vs Time, etc.
Most Important Fundamental Fact

All waveforms are the sum of simple sinusoidal functions of different frequencies.
sin(θ) = Opposite /Hypotenuse
Amplitude =1
Period= 2Pie
Equation of the Axis = Y=0
Domain is any real number
Range is any real number and must be greater or equal to -1 and smaller or equal to 1.
BUT what makes them different?
COSINE will be at it's maximum height at 0 and repeats that every 2Pie. SINE will be at the Equation of the Axis at 0 and repeats every 2Pie.
Vertical Stretch by a factor of 3 --> a
Horizontal compression by a factor of 1/2 --> 1/k >2
Horizontal translation pie/6 to the left --> d=-pie/6
Vertical translation 1 down -->c=-1
History of Fourier Transform
Learning About Frequencies
Phone Calls and Fourier Transform
Cell phones operate by the principles of electromagnetics.
Every number dialed is based in a protocol called internet protocol. A protocol is a set of rules .
The phone uses your location to determine coordinates to the find the satellite to transmit and receive to the other end.
Our cell phones have to change from an electric or wave system into a voice system that comes from alphabetical words, which then is translated between a numerical system called binaries.
The call will come to a cell tower, where it is transmitted back with special frequencies.
Invented by: Jean Baptiste Joseph Fourier.

Born: 21 March 1768 in Auxerre, Bourgogne, France
Died: 16 May 1830 in Paris, France
Figure 1:This is a simple sine function with 1+2(2pieft), Frequency f= 1/(2pie), Period T=2Pie, Amplitude = 2.

Figure 2:This demonstrates 3 simple cosine functions which have been added together to create g(t) a complicated wave.
* *** The more simple waves added together will allow for a better approximation
What is a Fourier Series?
"The Fourier Series breaks down a periodic function into the sum of sinusoidal functions. It is the Fourier Transform for periodic functions (=)."
Figure 3: This is called the Fourier Series
Figure 4: The Fourier Series is a complicated wave, in this example it is recognized as g(t). It is written from a sum of cosine and sine functions by increasing the fundamental frequency f by n, and by altering the amplitude a^n and b^n. We are able to break down this complicated signal into a more simple form of a wave.

This is an example of the Fourier Transform which has decomposed the frequencies and amplitudes of a complex wave. Within this wave there is A ,B and C which is then created into g(t). The time domain refers to the inverse of Fourier Transform. The frequency domain shows individual frequencies and related amplitudes. The information used is not changed or affected, but rather altered when viewed.
Experiment: I need 4 volunteers with Cell phones. The purpose of this Experiment is to demonstrate how multiple phone calls can occur in the exact same place at the exact same time.
The Cell Tower is the Mathematician.
cell phone will send out an electronic signal which carries a digitized copy of your speech.
ALL cell phones function within the same frequency range (risk of dropped calls and reception failure)
Phone call goes to Cell Tower
Cell Tower breaks down the signal into multiple frequencies which consist of sine and cosine. Every call has it own unique set of frequencies. (Fourier Transform is used here)
Users are a part of a network that holds an infinite amount of calls
Here is an example.
A whistle is blown and it has a frequency of 1,000 hertz.
Thunder is roaring and it has a frequency of 50 hertz.

The low frequencies have done less cycles in the same amount of time making it less loud.
amplitude is volume. When we increase the amplitude it makes the sound louder and if you decrease the amplitude it will sound quieter.
The amplitude of a wave is expressed through the amount of energy it carries. High energy levels = high amplitude, Low energy levels = low amplitude.
Intensity of the wave is the amount of energy that is traveling through a unit area per unit of time in a specific direction.
As the sound reaches a higher amplitude, the intensity of the sound increases.
Frequency is often referred to as pitch - the shorter the wavelength, the higher the frequency therefore the higher the sound of the pitch.

By: Kylie Sullivan
Thank For Watching!

Fourier Transformation
Kylie Sullivan
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