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Prof. Greg Francis1PSY 310: Sensory and Perceptual ProcessesPurdue UniversityFourier AnalysisPSY 310Greg FrancisLecture 08It’s all waves!Purdue UniversityRepresentation of information One of the big issues in perception (andcognition in general) is identifying howthe brain represents information In today’s lecture I want to do two things 1) Discuss a bit about ways of representinginformation 2) Discuss a mathematical property that is acommonly used tool in perception Fourier analysis Don’t freak out!Prof. Greg Francis2PSY 310: Sensory and Perceptual ProcessesPurdue UniversityJoseph Fourier French mathematician(1768-1830) Involved in the Frenchrevolution An administrator forNapolean Bonaparte Discovered a mathematicalproperty that even othermathematicians initiallyfound difficult to acceptPurdue UniversitySines and cosines Take a circle with it’s center at position (0,0)Prof. Greg Francis3PSY 310: Sensory and Perceptual ProcessesPurdue UniversitySines and cosines Take a circle Draw a line from the middle at an angle€ θPurdue UniversitySines and cosines Take a circle Draw a line from the middle at an angle Draw the triangle where the line crosses thecircle€ θ€ θProf. Greg Francis4PSY 310: Sensory and Perceptual ProcessesPurdue UniversitySines and cosines The cosine function cos( ) is the proportion ofthe radius of the circle in the x-direction€ θ€ θ€ θ€ cos(θ)Purdue UniversitySines and cosines The cosine function cos( ) is the proportion ofthe radius of the circle in the x-direction The sine function sin( ) is the proportion of theradius of the circle in the y-direction€ θ€ θ€ θ€ cos(θ)€ θ€ sin(θ)Prof. Greg Francis5PSY 310: Sensory and Perceptual ProcessesPurdue UniversitySines and cosines So€ sin(0) = 0€ cos(0) = 1€ θPurdue UniversitySines and cosines So€ cosπ2      = 0€ θ€ sinπ2      = 1Prof. Greg Francis6PSY 310: Sensory and Perceptual ProcessesPurdue UniversitySines and cosines So€ sinπ( )= 0€ θ€ cosπ( )= −1Purdue UniversitySine function If you plot the sine function as thetavaries you get this nice wave€ θ€ sin(θ)Prof. Greg Francis7PSY 310: Sensory and Perceptual ProcessesPurdue UniversityCos function If you plot the cosine function as theta variesyou get this other nice wave€ θ€ cos(θ)Purdue UniversityProperties There are some interesting properties of thesefunctions€ θ€ cos(θ) = cos(−θ)€ −θProf. Greg Francis8PSY 310: Sensory and Perceptual ProcessesPurdue UniversityProperties There are some interesting properties of thesefunctions€ θ€ sin(−θ) = −sin(θ)€ −θPurdue UniversitySine function You can speed up the wave by multiplyingtheta€ θ€ sin(θ)Prof. Greg Francis9PSY 310: Sensory and Perceptual ProcessesPurdue UniversitySine function You can speed up the wave by multiplyingtheta€ θ€ sin(2θ)Purdue UniversitySine function You can also change the height of the wave Amplitude€ θ€ 0.3sin(θ)Prof. Greg Francis10PSY 310: Sensory and Perceptual ProcessesPurdue UniversityFrequency Suppose you are only interested in thetavalues between -Pi and Pi You can easily generalize to other ranges, but theequations look worse In equations of the form We say the frequency of the wave is n This is how many times the wave cycles(comes back to its starting value)€ sin(nθ)€ cos(nθ)Purdue UniversityOrthonormality This property is harder to prove, but easy to showexamples If you multiple any sine and cosine functions and take theintegral, you get either zero or pi€ sin nθ( )cos mθ( )−ππ∫dθ= 0€ cos nθ( )cos mθ( )−ππ∫dθ=π if n = m0 if n ≠ m   € sin nθ( )sin mθ( )−ππ∫dθ=π if n = m0 if n ≠ m  Prof. Greg Francis11PSY 310: Sensory and Perceptual ProcessesPurdue UniversityOrthonormality Note the positive parts are mirror image of thenegative parts€ x€ sin 2x( )cos x( )Purdue UniversityOrthonormality Note the positive parts are mirror image of thenegative parts€ sin 2x( )sin 4 x( )€ xProf. Greg Francis12PSY 310: Sensory and Perceptual ProcessesPurdue UniversityOrthonormality It’s less obvious here, but each positive partcovers two negative parts€ cos 2x( )cos 6x( )€ xPurdue UniversityOrthonormality If the functions are the same, the integralequals pi€ sin 2x( )sin 2 x( )€ xProf. Greg Francis13PSY 310: Sensory and Perceptual ProcessesPurdue UniversityOrthonormality We can sort of ask how much of one function is made byanother function E.g., sin(2x) has no part of it that is made of sin(3x), orcos(7x) What about other functions? How much of the function ismade up of sine or cosine functions of differentfrequencies? Linear f(x) = mx+b Parabola: f(x) = ax2+b WhateverPurdue UniversityFourier Proved that you can write (almost) any functionas a series of properly weighted (amplified)sine and cosine functions of differentfrequencies€ f (x) = a0+ ancos(nx)n=1∞∑+ bnsin(nx)n=1∞∑ The trick is to find the proper values of a0, an,and bnProf. Greg Francis14PSY 310: Sensory and Perceptual ProcessesPurdue UniversityFinding coefficients The first term is fairly easy to find, it’s just the average€ a0=12f (x)dx−ππ∫ To find the other an terms, just multiply by a cosine function ofthe n-th frequency€ an= f (x)cos(nx)dx−ππ∫ To find the bn terms, just multiply by a sine function of the n-thfrequency€ bn= f (x)sin(nx)dx−ππ∫Purdue UniversityExample Suppose our function is the absolute value function Divided by pi to keep everything between -1 and +1.€ f (x) =xπ€ xProf. Greg Francis15PSY 310: Sensory and Perceptual ProcessesPurdue UniversityExample Suppose our function is the absolute value function Divided by pi to keep everything between -1 and +1.€ f (x) =xπ€ x€ a0=12Purdue UniversityExample Suppose our function is the absolute value function Divided by pi to keep everything between -1 and +1.€ f (x) =xπ€ x€ a0=12€ a1=−4π2€ a0+ a1cos(x)Prof. Greg Francis16PSY 310: Sensory and Perceptual ProcessesPurdue UniversityExample Suppose our function is the absolute value function Divided by pi to keep everything between -1 and +1.€ f (x) =xπ€ x€ a0=12€ a1=−4π2€ a0+ a1cos(x)€ a2= 0Purdue UniversityExample Suppose our function is the


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Purdue PSY 31000 - Fourier Analysis

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