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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 Pressure 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. Pressure 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 a SINGLE 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 / COSINE The 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 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 the FOURIER COEFFICIENTS

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1D Functions As a formula: f(x)= a 0 /2 + k=1..n a k cos(kx) + b k sin(kx) Fourier Coefficients

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Note: The set of a k, b k 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.sincos 110 21/20 31/30 41/40

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SQUARE WAVE Function 1D Functions f(x) = 4/pi * n=1,3,5 1/n *sin (n*pi*x) Freq.sincos 110 31/30 51/50 71/70

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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); 1.31026.2 - 1022.8i 310.1 - 1022.1i -0.4 + 1.6i Freq. 0Freq. 1Freq. 2Freq. 3 cos sin

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1D Functions FrReIm 01.30 11026.21022.8 2310.11022.1 FrReIm 0~00 1~1 2~0.3~1 1.31026.2 - 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 !

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