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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