Gibbs phenomenon: Difference between revisions

In

This is one cause of ringing artifacts in signal processing.

Description

The Gibbs phenomenon involves both the fact that Fourier sums overshoot at a

The three pictures on the right demonstrate the phenomenon for a square wave (of height ${\displaystyle \pi /4}$) whose Fourier expansion is

${\displaystyle \sin(x)+{\frac {1}{3}}\sin(3x)+{\frac {1}{5}}\sin(5x)+\dotsb .}$

More precisely, this is the function f which equals ${\displaystyle \pi /4}$ between ${\displaystyle 2n\pi }$ and ${\displaystyle (2n+1)\pi }$ and ${\displaystyle -\pi /4}$ between ${\displaystyle (2n+1)\pi }$ and ${\displaystyle (2n+2)\pi }$ for every integer n; thus this square wave has a jump discontinuity of height ${\displaystyle \pi /2}$ at every integer multiple of ${\displaystyle \pi }$.

As can be seen, as the number of terms rises, the error of the approximation is reduced in width and energy, but converges to a fixed height. A calculation for the square wave (see Zygmund, chap. 8.5., or the computations at the end of this article) gives an explicit formula for the limit of the height of the error. It turns out that the Fourier series exceeds the height ${\displaystyle \pi /4}$ of the square wave by

${\displaystyle {\frac {1}{2}}\int _{0}^{\pi }{\frac {\sin t}{t}}\,dt-{\frac {\pi }{4}}={\frac {\pi }{2}}\cdot (0.089489872236\dots )}$(OEIS: A243268)

or about 9 percent of the jump. More generally, at any jump point of a piecewise continuously differentiable function with a jump of a, the nth partial Fourier series will (for n very large) overshoot this jump by approximately ${\displaystyle a\cdot (0.089489872236\dots )}$ at one end and undershoot it by the same amount at the other end; thus the "jump" in the partial Fourier series will be about 18% larger than the jump in the original function. At the location of the discontinuity itself, the partial Fourier series will converge to the midpoint of the jump (regardless of what the actual value of the original function is at this point). The quantity

${\displaystyle \int _{0}^{\pi }{\frac {\sin t}{t}}\ dt=(1.851937051982\dots )={\frac {\pi }{2}}+\pi \cdot (0.089489872236\dots )}$(OEIS: A036792)

is sometimes known as the Wilbraham–Gibbs constant.

History

The Gibbs phenomenon was first noticed and analyzed by

After the existence of Henry Wilbraham's paper became widely known, in 1925 Horatio Scott Carslaw remarked "We may still call this property of Fourier's series (and certain other series) Gibbs's phenomenon; but we must no longer claim that the property was first discovered by Gibbs."^[10]

Explanation

Informally, the Gibbs phenomenon reflects the difficulty inherent in approximating a

Solutions

In practice, the difficulties associated with the Gibbs phenomenon can be ameliorated by using a smoother method of Fourier series summation, such as

From a

(if periodic) with a sinc function: the oscillations in the sinc function cause the ripples in the output.

sine integral

, exhibiting the Gibbs phenomenon for a step function on the real line.

In the case of convolving with a Heaviside step function, the resulting function is exactly the integral of the sinc function, the

sine integral

; for a square wave the description is not as simply stated. For the step function, the magnitude of the undershoot is thus exactly the integral of the (left) tail, integrating to the first negative zero: for the normalized sinc of unit sampling period, this is ${\displaystyle \int _{-\infty }^{-1}{\frac {\sin(\pi x)}{\pi x}}\,dx.}$ The overshoot is accordingly of the same magnitude: the integral of the right tail, or, which amounts to the same thing, the difference between the integral from negative infinity to the first positive zero, minus 1 (the non-overshooting value).

The overshoot and undershoot can be understood thus: kernels are generally normalized to have integral 1, so they result in a mapping of constant functions to constant functions – otherwise they have gain. The value of a convolution at a point is a linear combination of the input signal, with coefficients (weights) the values of the kernel. If a kernel is non-negative, such as for a

• 
  Oscillations can be interpreted as convolution with a sinc.
• 
  Higher cutoff makes the sinc narrower but taller, with the same magnitude tail integrals, yielding higher frequency oscillations, but whose magnitude does not vanish.

Thus the features of the Gibbs phenomenon are interpreted as follows:

• the undershoot is due to the impulse response having a negative tail integral, which is possible because the function takes negative values;
• the overshoot offsets this, by symmetry (the overall integral does not change under filtering);
• the persistence of the oscillations is because increasing the cutoff narrows the impulse response, but does not reduce its integral – the oscillations thus move towards the discontinuity, but do not decrease in magnitude.

The square wave example

Without loss of generality, we may assume the square wave case in which the period L is ${\displaystyle 2\pi }$, the discontinuity ${\displaystyle x_{0}}$ is at zero, and the jump is equal to ${\displaystyle \pi /2}$. For simplicity let us just deal with the case when N is even (the case of odd N is very similar). Then we have

${\displaystyle S_{N}f(x)=\sin(x)+{\frac {1}{3}}\sin(3x)+\cdots +{\frac {1}{N-1}}\sin((N-1)x).}$

Substituting ${\displaystyle x=0}$, we obtain

${\displaystyle S_{N}f(0)=0={\frac {-{\frac {\pi }{4}}+{\frac {\pi }{4}}}{2}}={\frac {f(0^{-})+f(0^{+})}{2}}}$

as claimed above. Next, we compute

${\displaystyle S_{N}f\left({\frac {2\pi }{2N}}\right)=\sin \left({\frac {\pi }{N}}\right)+{\frac {1}{3}}\sin \left({\frac {3\pi }{N}}\right)+\cdots +{\frac {1}{N-1}}\sin \left({\frac {(N-1)\pi }{N}}\right).}$

If we introduce the normalized sinc function, ${\displaystyle \operatorname {sinc} (x)\,}$, we can rewrite this as

${\displaystyle S_{N}f\left({\frac {2\pi }{2N}}\right)={\frac {\pi }{2}}\left[{\frac {2}{N}}\operatorname {sinc} \left({\frac {1}{N}}\right)+{\frac {2}{N}}\operatorname {sinc} \left({\frac {3}{N}}\right)+\cdots +{\frac {2}{N}}\operatorname {sinc} \left({\frac {(N-1)}{N}}\right)\right].}$

But the expression in square brackets is a Riemann sum approximation to the integral ${\displaystyle \int _{0}^{1}\operatorname {sinc} (x)\ dx}$ (more precisely, it is a

${\displaystyle 2/N}$

${\displaystyle N\to \infty }$

${\displaystyle {\begin{aligned}\lim _{N\to \infty }S_{N}f\left({\frac {2\pi }{2N}}\right)&={\frac {\pi }{2}}\int _{0}^{1}\operatorname {sinc} (x)\,dx\\[8pt]&={\frac {1}{2}}\int _{x=0}^{1}{\frac {\sin(\pi x)}{\pi x}}\,d(\pi x)\\[8pt]&={\frac {1}{2}}\int _{0}^{\pi }{\frac {\sin(t)}{t}}\ dt\quad =\quad {\frac {\pi }{4}}+{\frac {\pi }{2}}\cdot (0.089489872236\dots ),\end{aligned}}}$

which was what was claimed in the previous section. A similar computation shows

${\displaystyle \lim _{N\to \infty }S_{N}f\left(-{\frac {2\pi }{2N}}\right)=-{\frac {\pi }{2}}\int _{0}^{1}\operatorname {sinc} (x)\ dx=-{\frac {\pi }{4}}-{\frac {\pi }{2}}\cdot (0.089489872236\dots ).}$

Consequences

In signal processing, the Gibbs phenomenon is undesirable because it causes artifacts, namely clipping from the overshoot and undershoot, and ringing artifacts from the oscillations. In the case of low-pass filtering, these can be reduced or eliminated by using different low-pass filters.

In

• σ-approximation which adjusts a Fourier summation to eliminate the Gibbs phenomenon which would otherwise occur at discontinuities
• Pinsky phenomenon
• Runge's phenomenon (a similar phenomenon in polynomial approximations)
• Sine integral
• Mach bands

Notes