1d fourier analysis dr. rolf lakaemper. sound let’s have a look at sound: sound: 1 dimensional...
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1D Fourier Analysis
Dr. Rolf Lakaemper
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Sound
Let’s have a look at SOUND:
SOUND: 1 dimensional function of changing (air-)pressure in time
Pre
ssur
e
Time t
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Sound
SOUND: if the function is periodic, we perceive it as sound with a certain frequency (else it’s noise). The frequency defines the pitch.
Pre
ssur
e
Time t
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Sound
The AMPLITUDE of the curve defines the VOLUME
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Sound
The SHAPE of the curve defines the sound character
Flute String
Brass
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Sound
How can the
SHAPE
of the curve be defined ?
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Sound
Listening to an orchestra, you can distinguish between different instruments,
although the sound is aSINGLE FUNCTION !
Flute
String
Brass
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Sound
If the sound produced by an orchestra is the sum of different instruments, could it be possible that there are BASIC SOUNDS,
that can be combined to produce every single sound ?
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Sound
The answer (Jean Baptiste Fourier, 1822):
Any function that periodically repeats itself can be expressed as
the sum of sines/cosines of different frequencies, each
multiplied by a different coefficient
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Sound
Or differently:
Since a flute produces a sine-curve- like sound, a (huge) group of
(outstandingly) talented flautists could replace a classical
orchestra.
( )
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1D Functions
A look at SINE / COSINEThe sine-curve is defined by:
• Frequency (the number of oscillations between 0 and 2*PI)
• Amplitude (the height)
• Phase (the starting angle value)
• The constant y-offset, or DC (direct current)
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1D Functions
The general sine-shaped function:
f(t) = A * sin(t + ) + c
Amplitude
Frequency
Phase
Constant offset(usually set to 0)
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1D Functions
Remember Fourier:
…A function…can be expressed as the sum of sines/cosines…
What happens if we add sine and cosine ?
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1D Functions
a * sin(t) + b * cos(t)
= A * sin(t + )
(with A=sqrt(a^2+b^2) and tan = b/a)
Adding sine and cosine of the same frequency
yields just another sine function with different phase and amplitude, but same frequency.
Or: adding a cosine simply shifts the sine function left/right and stretches it in y-direction. It does NOT change the
sine-character and frequency of the curve.
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1D Functions
Remember Fourier, part II:
Any function that periodically repeats itself…
=> To change the shape of the function, we must add sine-like
functions with different frequencies.
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1D Functions
This applet shows the result:
Applet: Fourier Synthesis
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1D Functions
What did we do ?
• Choose a sine curve having a certain frequency, called the base-frequency
• Choose sine curves having an integer multiple frequency of the base-frequency
• Shift each single one horizontally using the cosine-factor
• Choose the amplitude-ratio of each single frequency
• Sum them up
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1D Functions
This technique is called the
FOURIER SYNTHESIS,
the parameters needed are the sine/cosine ratios of each frequency.
The parameters are called theFOURIER COEFFICIENTS
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1D Functions
As a formula:
f(x)= a0/2 + k=1..n akcos(kx) + bksin(kx)
Fourier Coefficients
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Note:
The set of ak, bk ENTIRELY defines the CURVE synthesized !
We can therefore describe the SHAPE of the curve or the
CHARACTER of the sound by the (finite ?) set of FOURIER
COEFFICIENTS !
1D Functions
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Examples for curves, expressed by the sum of sines/cosines (the
FOURIER SERIES):
1D Functions
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SAWTOOTH Function
1D Functions
f(x) = ½ - 1/pi * n 1/n *sin (n*pi*x)
Freq. sin cos
1 1 0
2 1/2 0
3 1/3 0
4 1/4 0
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SQUARE WAVE Function
1D Functions
f(x) = 4/pi * n=1,3,5 1/n *sin (n*pi*x)
Freq. sin cos
1 1 0
3 1/3 0
5 1/5 0
7 1/7 0
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What does the set of FOURIER COEFFICIENTS tell about the
character of the shape ?
(MATLAB Demo)
1D Functions
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Result:
Steep slopes introduce HIGH FREQUENCIES.
1D Functions
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Motivation for Sound Filtering:
...remove steep slopes to let only lower frequencies pass
it would be nice to be able to get the set of FOURIER COEFFICIENTS if an arbitrary (periodically) function
is given! (why?)
1D Functions
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The Problem now:
Given an arbitrary but periodically 1D function (e.g. a sound), can you tell the FOURIER COEFFICIENTS
to construct it ?
1D Functions
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The answer (Fourier):
YES.
1D Functions
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We don’t want to explain the mathematics behind the answer
here, but simply use the MATLAB Fourier Transformation Function.
1D Functions
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MATLAB - function fft:
Input: A vector, representing the discrete function
Output: The Fourier Coefficients as vector of imaginary numbers,
scaled for some reasons
1D Functions
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Example:
1D Functions
x=0:2*pi/(2047):2*pi;s=sin(x)+cos(x) + sin(2*x) + 0.3*cos(2*x);f=fft(s);
1.3 1026.2 - 1022.8i
310.1 - 1022.1i
-0.4 +
1.6i
Freq. 0 Freq. 1 Freq. 2 Freq. 3
cos
sin
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1D Functions
Fr Re Im
0 1.3 0
1 1026.2 1022.8
2 310.1 1022.1
Fr Re Im
0 ~0 0
1 ~1 ~1
2 ~0.3 ~1
1.3 1026.2 - 1022.8i
310.1 - 1022.1i
-0.4 + 1.6i
Transformation: t(a) = 2*a / length(result-vector)
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1D Functions
The fourier coefficients are given by:
F=fft(function)L=length(F); %this is always = length(function)
Coefficient for cosine, frequency k-times the base frequency:
real(F(k+1)) * 2 / L
Coefficient for sine, frequency k-times the base frequency:
imag(F(k+1)) * 2 / L
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1D Functions
An application using the Fourier Transform:
Create an autofocus system for a digital camera
We did this already, but differently !
(MATLAB DEMO)
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1D Functions
Second application:
Describe and compare 2-dimensional shapes using the Fourier Transform !