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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