Gibbs Phenomenon

gibbs-phenomenon.md

Gibbs Phenomenon

By Andy Pan (Qbane), 2016/3/10. (See Gist for revision history.)

This is a test document on MathJax support on various platforms, mainly on printed material.

For large $N$, the graph of the $N$th partial sum of the Fourier series of $f$ on $[-L, L]$ overshoots the graph of the function at a jump discontinuity by approx. $9%$ of the magnitude of the jump.

(a) $N$th Partial Sum

We write out the $N$th partial sum of the Fourier series of $f(x)$ and expand its coefficients: $$ \begin{align} S_N(x) &= a_0 + \sum_{n=1}^N \left[ a_n \cos \left(\frac{n\pi x}{L}\right) + b_n \sin \left(\frac{n\pi x}{L}\right) \right] \ &= \frac{1}{2L} \int_{-L}^L f(t) dt \ &+ \frac{1}{L} \sum_{n=1}^N \left[ \int_{-L}^L f(t) \cos\left( \frac{n\pi t}{L} \right) dt \right] \cos\left( \frac{n\pi x}{L} \right) \ &+ \frac{1}{L} \sum_{n=1}^N \left[ \int_{-L}^L f(t) \sin\left( \frac{n\pi t}{L} \right) dt \right] \sin\left( \frac{n\pi x}{L} \right) \ &= \frac{1}{L} \int_{-L}^L f(t) \left{ \frac{1}{2} + \sum_{n=1}^N \cos\left[ \frac{n\pi (t-x)}{L} \right] \right} dt. \end{align} $$

(b) Trigonometric Identity

It can be shown that $$\sum_{n=1}^N \cos (n\xi) = \frac{\sin \left[(N+\frac 1 2)\xi \right]}{2 \sin \left( \frac \xi 2 \right) } - \frac 1 2. $$

Here is a trick: multiply it by $\sin \left( \frac \xi 2 \right)$ and try to lead to an identity. It can also be simplified by transforming $\cos x = \frac{e^{ix} + e^{-ix}}{2}$ and transforming back to $\sin$ terms. (It would be tricky if you didn't know RHS, though.)

(c) Approximating (a)

For small $x$, $\sin x \approx x $, and using (b), (a) can be expressed as $$S_N(x) \approx \frac 1 {2L} \int_{-L}^L f(t) \frac{\sin \left[ \frac \pi L (N + \frac 1 2) (t-x) \right] }{ \frac{\pi}{2L}(t-x) } dt. $$

(d) Estimating a Discontinuity

To find out the amount, we use the following function with a jump discontinuity $1$ at $x = L$, $$ f(x) = \begin{cases} 0 & \text{if } -L < x \le x_0, \ 1 & \text{if } x_0 < x < L. \end{cases} $$

Inserting it into $S_N$ yields $$ \begin{align} S_N(x) &\approx \frac 1 {2L} \int_{x_0}^L \frac{\sin \left[ \frac \pi L (N + \frac 1 2) (t-x) \right] }{ \frac{\pi}{2L}(t-x) } dt \ &= \frac 1 \pi \int_{(\pi/L)(N + 1/2)(x_0-x)}^{(\pi/L)(N + 1/2)(L-x)} \frac{\sin s}{s} ds. \end{align}$$

(e) Putting $N \to \infty$

The integral $\int_0^z \frac{\sin t}{t} dt$ is called the sine integral, denoted as $\text{Si}(z)$.

Computing $S_N$ when $N \to \infty$ diverges to three cases depending on $x$, in which we are only interested in the case that $x &gt; x_0$, that is $(\pi/L)(N + 1/2)(x_0-x) \to -\infty$:

$$ \begin{align} \lim_{N \to \infty} S_N(x) &\approx \frac 1 \pi \int_{(\pi/L)(N + 1/2)(x_0-x)}^\infty \frac{\sin s}{s} ds \\ &= \frac 1 \pi \left{ \lim_{z \to \infty} \text{Si}(z) - \text{Si} \left[ \left(\frac \pi L \right) \left(N + \frac 1 2 \right) \left(x_0-x \right) \right] \right}, \end{align} $$

(f) Evaluating the Sine Integral

$\text{Si}(z)$ at infinity can be computed as follows, $$ \begin{align} \lim_{z \to \infty} \text{Si}(z) &= \int_0^\infty \frac{\sin t}{t} dt \ &= \int_0^\infty \int_0^\infty e^{-st} \frac{\sin t}{t}ds,dt \ &= \int_0^\infty \int_0^\infty e^{-st} \frac{\sin t}{t}dt,ds \ &= \int_0^\infty \frac{1}{s^2 + 1}ds \ &= \frac \pi 2. \end{align} $$

$\text{Si}(z)$ is maximized when $\text{Si}'(z^) = 0$, that is $\frac{\sin z^}{z^} = 0$, and the minimal $x^$ satisfying the condition is $z^* = \pi$.

Thus the overshoot is at most $$ \frac 1 \pi \left[\frac \pi 2 - \text{Si}(\pi) \right] = \frac 1 2 - \frac{\text{Si}(\pi)}{\pi} \approx 0.0895, $$

which is approximately $9%$.

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