Gibbs phenomenon

The oscillations crowd closer to the discontinuity as more terms are added, but the approximately 9% overshoot remains.

\(u_T(t)=\begin{cases}0,&-T/2<t<0\\1,&0<t<T/2\end{cases},\quad u_T(t+T)=u_T(t)\)

\(x_N^{(\alpha)}(t)=\dfrac12+\dfrac{2}{\pi}\displaystyle\sum_{m=0}^{N-1}w_{m,\alpha}\dfrac{\sin\!\left((2m+1)2\pi t/T\right)}{2m+1}\)

\(w_{m,\alpha}=(1-\alpha)+\alpha\!\left(1-\dfrac{2m+1}{2N}\right)\)

Odd harmonics \(N\) N = 10 · through k = 19 Fejér taper \(\alpha\) 0%

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Periodic step and \(x_{10}(t)\)

Formula preview

\(x_N(t)\) \(t/T\)

periodic steppartial sum

Near the discontinuity

zoom around \(t=0\)

\(x_N(t)\) \(t/T\)

periodic steppartial sum

Maximum 1.091 Overshoot 9.1% Python not run

Variables: \(T\), period; \(N\), number of odd harmonics; \(m\), term index; \(t\), time; \(\alpha\), taper strength. References: H. Wilbraham, “On a Certain Periodic Function,” 1848; J. W. Gibbs, “Fourier’s Series,” Nature 59, 200 (1898), and p. 606 (1899); L. Fejér, “Sur les fonctions bornées et intégrables,” 1900.