GIBBS effect at discontinuities for different functions

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Jeff 2017-12-7

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https://ww2.mathworks.cn/matlabcentral/answers/371542-gibbs-effect-at-discontinuities-for-different-functions

编辑： John D'Errico 2017-12-28

The Fourier series of waveforms with discontinuties experiences an overshoot near the discontinuity known as the "Gibbs phenomenon". There is quite a bit of literature showing that the overshoot for a rectangle function is ~ 1.089. What about other functions such as (1-x) or a decaying exponential for x positive? Is there any reason to expect the overshoot ratio to be exactly identical to the rectangle function? I do know for a fact that the behavior of the overshoot is different for the triangle function (1-x) than for the rectangle function. For low harmonics there is an undershoot for the triangle function case, but this is not the case for the rectangle function. The overshoot occurs for the triangle function after a sufficient number of terms are included in the Fourier series. The same is true for the decaying exponential. This is illustrated in the plots.pdf file attached. The m-files that generated the data for the plots are also included. The actual function is represented by variable P & the Fourier series is represented by Pfit.

Does anyone know of MATLAB code that computes the theoretical overshoot if there is an infinite number of terms in the series for different waveforms or functions?

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Jeff 2017-12-27

To David & others knowledgeable on this subject:

From what I have gathered in the literature is that the overshoot from the GIBBS effect is equivalent to the integral of the sinc function over the interval from 0 to pi. However, this has primarily been demonstrated through the STEP input function. This is fine since the jump discontinuity occurs once and the amplitude of the function remains constant after the discontinuity has been encountered.

But what happens if the amplitude changes even after the discontinuity such as the case of the triangle function? I will post another question about this because I have since demonstrated that the rate at which the overshoot approaches the theoretical value of 1.08949 as a function of k terms in the series is quite different depending on the function behavior after the discontinuity. This is why I question the theoretical value of 1.08949 for all functions in general. I have yet to find a good explanation on this matter.

John D'Errico 2017-12-28

编辑：John D'Errico 2017-12-28

You have not gotten a response, because you have not asked a question about MATLAB. And you have now asked multiple questions about the same thing. Answers is about MATLAB questions. When your question is off-topic, then most of the people on this forum will have no interest in responding.

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