18.03 Gibbs’ PhenomenonJeremy OrloffGibbs’ Phenonemon says that the truncated Fourier series near a jump discontinuity over-shoots the jump by about 9% of the size of the jump. Thus, for the standard square wave(which jumps between -1 and 1) the peak value of the truncated Fourier series is about 1.18.The proof is an elaborate and tricky calculus exercise.Start with the square wavef(x) =4πXn oddsin nxnThe trunctated Fourier series isfN(x) =4πsin x +sin 3x3+ . . . +sin(2N − 1)x2N − 1Taking the derivativef0N(x) =4π(cos x + cos 3x + . . . + cos(2N − 1)x)Claim: f0N(x) =2π·sin(2Nx)sin x.Proof: Using complex arithmetic: cos x =eix+ e−ix2, sin x =eix− e−ix2i.⇒ f0N(x) =4π e(−2N +1)ix+ e(−2N +3)ix+ . . . e(2N −3)ix+ e(2N −1)ix2!.This is a geometric series with ratio e2ix⇒f0N(x) =2π·e(−2N +1)ix− e(2N +1)ix1 − e2ix.Multiply top and bottom by e−ixand use the formula for sin x in terms of complex expo-nentials to getf0N(x) =2π·e(−2N )ix− e(2N )ixe−ix− eix=2π·sin(2Nx)sin x. QEDThe spike is at the first positive maximum of fN. The formula for f0N(x) shows this is atπ2N. Since, fN(0) = 0 and all the terms in the sum for fN(π/2N) are positive we concludethat x = π/2N is a local maximum (it is, in fact, the absolute maximum).We now set about estimating fN(π/2N), i.e., the maximum value of fN(x). First wemanipulate the series for fN(π/2N).fNπ2N=4πsin(π/2N)1+sin(3π/2N)3+ . . . +sin((2N − 1)π/2N )2N − 1=2π·πNsin(π/2N)π/2N+sin(3π/2N)3π/2N+ . . . +sin((2N − 1)π/2N )(2N − 1)π/2N118.03 Gibbs’ Phenomenon 2This last is a Riemman sum (using midpoints) for2πZπ0sin xxdx, with ∆x = π/N.Since ∆x → 0 as N → ∞ we getlimN →∞fN(π/2N) =2πZπ0sin xxdx.In words, as N increases the overshoot goes to the value of the integral.All that’s left is to estimate the value of the integral. For this we integrate the power seriesforsin xx. We havesin xx= 1 −x23!+x45!−x67!+ . . .Which gives2πZπ0sin xxdx = 21 −π23 · 3!+π45 · 5!−π67 · 7!+ . . .This series converges very rapidly and after five terms we have the value 1.18 correct to 2decimal places.We have seen that as N gets large the maximum value of fN(x) becomes 1.18. That is itovershoots the correct value by .18, which is 9% of the jump from -1 to
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