Someone argues: "since the Fourier partial sums of a square wave converge to the…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Fourier analysis and integral transforms · Fourier analysis and integral transforms

Mathematical Methods for Physics, Ch Fourier series and periodic decomposition

Problem

Someone argues: "since the Fourier partial sums of a square wave converge to the square wave at every point except the jump, taking enough terms will eventually shrink the bump beside the jump to nothing." For a square wave of amplitude

A = 5.0\,\mathrm{V}

, the peak of the partial sum beside the jump has height

1.188A

for

N = 3

terms,

1.180A

for

N = 9

terms and

1.179A

for

N = 49

terms. (a) Compute the peak voltage for

N=9

and

N=49

, and predict it for

N=999

from the trend, giving your reason. (b) Say, using these numbers, whether the claim is correct, and give the overshoot above

A

as a percentage of the jump

2A

. (c) What value does every partial sum, and the series itself, take exactly at the jump,

x=0

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