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Applications of Fourier Series and Spectrum
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Frequency Interested in signal representation in
frequency domain Use Fourier analysis
Many applications Noise cancellation (hearing aid) Image compression Broadcast TV signals http://www.intmath.com/fourier-series/fourier-
graph-applet.php
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Why Sinusoids? Interested in sinusoidal signals
Signals that have their magnitude changed according to a sine function
In 1807, a famous mathematician, Joseph Fourier, showed that
Almost all signals can be constructed by the summation of sinusoids of different frequencies, amplitudes and phase shifts
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Fourier Series Any continuous-time periodic
waveform can be represented by the Fourier Series as
Summation operator: taking the sum…
0 01
( ) cos 2k kk
x t A A kf t
0 1 0 1 2 0 2
0
( ) cos 2 cos 2 2
cos 2
x t A A f t A f t
A f t
dc of x (dc here stands for signals that will not change in time )
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Fourier Series with negative frequencies
02( ) j kf tk
kx t C e
002 /
00
1 ( )T j kt T
kC x t e dtT
Expressing the signal in terms of a number of sinusoids
Fourier series coefficient: defines the amplitude of the corresponding sinusoids at certain frequencies
02( ) j kf tk
kx t C e
002 /
00
1 ( )T j kt T
kC x t e dtT
Simple Demo: Music
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Matlab: play a piano note scale, DO, RE, MI, ….,
Simple Demo : Music
test1.wav test2.wav
Simple Demo : Music for n = [1,3,5,6,8,10,12,13]
sine_tone(440*2^(n/12));end
function sin_tone(freq)fs=8192;t=[0:1/fs:1];y = sin(2*pi*freq*t);soundsc(y)
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Self-study
demonstration https://soundphysics.ius.edu/?page_id=949 https://www.youtube.com/watch?v=EWzt1bI8AZ0 http://www.mathdemos.org/mathdemos/trigsound
demo/trigsounddemo.html http://www.fearofphysics.com/Sound/sounds.html
http://users.rowan.edu/~shreek/networks1/music.html
Application of frequency idea Signal approximation
http://www.intmath.com/fourier-series/fourier-graph-applet.php
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02( ) j kf tk
kx t C e
002 /
00
1 ( )T j kt T
kC x t e dtT
Application of frequency idea
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Application of frequency idea 20 terms
Online – applet: http://www.intmath.com/fourier-series/fourier-graph-applet.php
Application of frequency idea Progressive Image Transmission
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Fourier domainImage quality: blurring Defocus, motion blur Blurring: Gaussian function
Image quality: blurring
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Frequency domain image steganography
FL: low frequency Noticeable to human
eyes
FM: mid frequency FH: high frequency
Noises, will be removed by compression
References Applications:
https://www.youtube.com/watch?v=aqa6vyGSdos Remove repeating patterns
https://www.youtube.com/watch?v=yyox358zIRw Speech enhancement
https://www.youtube.com/watch?v=itEs9XNI8x8 https://www.youtube.com/watch?v=zIjKoBkF5LU
Noise removal https://www.youtube.com/watch?v=XClduWGEdq
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