Gibbs phenomenon

Functional approximation of square wave using 5 harmonics

Functional approximation of square wave using 25 harmonics

Functional approximation of square wave using 125 harmonics

The Gibbs phenomenon involves both the fact that Fourier sums overshoot at a jump discontinuity, and that this overshoot does not die out as more sinusoidal terms are added.

The three pictures on the right demonstrate the phenomenon for a square wave (of peak-to-peak amplitude of ${\displaystyle {\tfrac {\pi }{2}}}$) whose Fourier series is

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

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

As more sinusoidal terms are added, the error of the partial Fourier series converges to a fixed height. But because the width of the error continues to narrow, the area of the error – and hence the energy of the error – converges to 0.[3] Deriving the formula of the limit of the error for the square wave reveals that the error exceeds the height (from zero) of the square wave ${\displaystyle ({\tfrac {\pi }{4}})}$ 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% of the jump ${\displaystyle {\tfrac {\pi }{2}}}$. More generally, at any discontinuity of a piecewise continuously differentiable function with a jump of ${\displaystyle a}$, the ${\displaystyle N}$^th partial Fourier series will (for ${\displaystyle N}$ very large) overshoot this jump by an error approaching ${\displaystyle a\cdot (0.089489872236\dots )}$ 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 discontinuity, the partial Fourier series will converge to the midpoint of the jump (regardless of the actual value of the original function at the discontinuity) as a consequence of Dirichlet's theorem.[4] 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.

Let ${\displaystyle f:{\mathbb {R} }\to {\mathbb {R} }}$ be a piecewise continuously differentiable function which is periodic with some period ${\displaystyle L>0}$. Suppose that at some point ${\displaystyle x_{0}}$, the left limit ${\displaystyle f(x_{0}^{-})}$ and right limit ${\displaystyle f(x_{0}^{+})}$ of the function ${\displaystyle f}$ differ by a non-zero jump of ${\displaystyle a}$:

$${\displaystyle f(x_{0}^{+})-f(x_{0}^{-})=a\neq 0.}$$

For each positive integer ${\displaystyle N}$ ≥ 1, let ${\displaystyle S_{N}f(x)}$ be the ${\displaystyle N}$^th partial Fourier series (${\displaystyle S_{N}}$ can be treated as a mathematical operator on functions.)

$${\displaystyle S_{N}f(x):=\sum _{-N\leq n\leq N}{\widehat {f}}(n)e^{\frac {i2\pi nx}{L}}={\frac {1}{2}}a_{0}+\sum _{n=1}^{N}\left(a_{n}\cos \left({\frac {2\pi nx}{L}}\right)+b_{n}\sin \left({\frac {2\pi nx}{L}}\right)\right),}$$

where the Fourier coefficients ${\displaystyle {\widehat {f}}(n),a_{n},b_{n}}$ are given by the usual formulae

$${\displaystyle {\widehat {f}}(n):={\frac {1}{L}}\int _{0}^{L}f(x)e^{-{\frac {i2\pi nx}{L}}}\,dx}$$

$${\displaystyle a_{n}:={\frac {2}{L}}\int _{0}^{L}f(x)\cos \left({\frac {2\pi nx}{L}}\right)\,dx}$$

$${\displaystyle b_{n}:={\frac {2}{L}}\int _{0}^{L}f(x)\sin \left({\frac {2\pi nx}{L}}\right)\,dx.}$$

Then we have

$${\displaystyle \lim _{N\to \infty }S_{N}f\left(x_{0}+{\frac {L}{2N}}\right)=f(x_{0}^{+})+a\cdot (0.089489872236\dots )}$$

and

$${\displaystyle \lim _{N\to \infty }S_{N}f\left(x_{0}-{\frac {L}{2N}}\right)=f(x_{0}^{-})-a\cdot (0.089489872236\dots )}$$

but

$${\displaystyle \lim _{N\to \infty }S_{N}f(x_{0})={\frac {f(x_{0}^{-})+f(x_{0}^{+})}{2}}.}$$

More generally, if ${\displaystyle x_{N}}$ is any sequence of real numbers which converges to ${\displaystyle x_{0}}$ as ${\displaystyle N\to \infty }$, and if the jump of ${\displaystyle a}$ is positive then

$${\displaystyle \limsup _{N\to \infty }S_{N}f(x_{N})\leq f(x_{0}^{+})+a\cdot (0.089489872236\dots )}$$

and

$${\displaystyle \liminf _{N\to \infty }S_{N}f(x_{N})\geq f(x_{0}^{-})-a\cdot (0.089489872236\dots ).}$$

If instead the jump of ${\displaystyle a}$ is negative, one needs to interchange limit superior (${\displaystyle \limsup }$) with limit inferior (${\displaystyle \liminf }$), and also interchange the ${\displaystyle \leq }$ and ${\displaystyle \geq }$ signs, in the above two inequalities.

The sinc function, the impulse response of an ideal low-pass filter. Scaling narrows the function, and correspondingly increases magnitude (which is not shown here), but does not reduce the magnitude of the undershoot, which is the integral of the tail.

From a signal processing point of view, the Gibbs phenomenon is the step response of a low-pass filter, and the oscillations are called ringing or ringing artifacts. Truncating the Fourier transform of a signal on the real line, or the Fourier series of a periodic signal (equivalently, a signal on the circle), corresponds to filtering out the higher frequencies with an ideal (brick-wall) low-pass filter. This can be represented as convolution of the original signal with the impulse response of the filter (also known as the kernel), which is the sinc function. Thus the Gibbs phenomenon can be seen as the result of convolving a Heaviside step function (if periodicity is not required) or a square wave (if periodic) with a sinc function: the oscillations in the sinc function cause the ripples in the output.

The 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 until the first negative zero: for the normalized sinc of unit sampling period, this is ${\textstyle \int _{-\infty }^{-1}{\frac {\sin(\pi x)}{\pi x}}\,dx.}$ The overshoot is accordingly of the same magnitude: the integral of the right tail or (equivalently) 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 Gaussian kernel, then the value of the filtered signal will be a convex combination of the input values (the coefficients (the kernel) integrate to 1, and are non-negative), and will thus fall between the minimum and maximum of the input signal – it will not undershoot or overshoot. If, on the other hand, the kernel assumes negative values, such as the sinc function, then the value of the filtered signal will instead be an affine combination of the input values, and may fall outside of the minimum and maximum of the input signal, resulting in undershoot and overshoot, as in the Gibbs phenomenon.

Taking a longer expansion – cutting at a higher frequency – corresponds in the frequency domain to widening the brick-wall, which in the time domain corresponds to narrowing the sinc function and increasing its height by the same factor, leaving the integrals between corresponding points unchanged. This is a general feature of the Fourier transform: widening in one domain corresponds to narrowing and increasing height in the other. This results in the oscillations in sinc being narrower and taller, and (in the filtered function after convolution) yields oscillations that are narrower (and thus with smaller area) but which do not have reduced magnitude: cutting off at any finite frequency results in a sinc function, however narrow, with the same tail integrals. This explains the persistence of the overshoot and undershoot.

• 
  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.

Animation of the additive synthesis of a square wave with an increasing number of harmonics. The Gibbs phenomenon is visible especially when the number of harmonics is large.

Without loss of generality, we may examine the ${\displaystyle N}$th partial Fourier series ${\displaystyle S_{N}f(x)}$ of a square wave with a ${\displaystyle 2\pi }$ period and a ${\displaystyle {\tfrac {\pi }{2}}}$ vertical discontinuity at ${\displaystyle x=0}$. Because the case of odd ${\displaystyle N}$ is very similar, let us just deal with the case when ${\displaystyle N}$ is even:

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

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 ${\textstyle \int _{0}^{1}\operatorname {sinc} (x)\ dx}$ (more precisely, it is a midpoint rule approximation with spacing ${\displaystyle {\tfrac {2}{N}}}$). Since the sinc function is continuous, this approximation converges to the actual integral as ${\displaystyle N\to \infty }$. Thus we have

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

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

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

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 MRI, the Gibbs phenomenon causes artifacts in the presence of adjacent regions of markedly differing signal intensity. This is most commonly encountered in spinal MRIs where the Gibbs phenomenon may simulate the appearance of syringomyelia.

The Gibbs phenomenon manifests as a cross pattern artifact in the discrete Fourier transform of an image,[14] where most images (e.g. micrographs or photographs) have a sharp discontinuity between boundaries at the top / bottom and left / right of an image. When periodic boundary conditions are imposed in the Fourier transform, this jump discontinuity is represented by continuum of frequencies along the axes in reciprocal space (i.e. a cross pattern of intensity in the Fourier transform).

And although this article mainly focused on the difficulty with trying to construct discontinuities without artifacts in the time domain with only a partial Fourier series, it is also important to consider that because the inverse Fourier transform is extremely similar to the Fourier transform, there equivalently is difficulty with trying to construct discontinuities in the frequency domain using only a partial Fourier series. Thus for instance because idealized brick-wall and rectangular filters have discontinuities in the frequency domain, their exact representation in the time domain necessarily requires an infinitely-long sinc filter impulse response, since a finite impulse response will result in Gibbs rippling in the frequency response near cut-off frequencies, though this rippling can be reduced by windowing finite impulse response filters (at the expense of wider transition bands).[15]

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

• Gibbs, J. Willard (1898), "Fourier's Series", Nature, 59 (1522): 200, doi:10.1038/059200b0, ISSN 0028-0836, S2CID 4004787
• Gibbs, J. Willard (1899), "Fourier's Series", Nature, 59 (1539): 606, doi:10.1038/059606a0, ISSN 0028-0836, S2CID 13420929
• Michelson, A. A.; Stratton, S. W. (1898), "A new harmonic analyser", Philosophical Magazine, 5 (45): 85–91
• Zygmund, Antoni (1959). Trigonometric Series (2nd ed.). Cambridge University Press. Volume 1, Volume 2.
• Wilbraham, Henry (1848), "On a certain periodic function", The Cambridge and Dublin Mathematical Journal, 3: 198–201
• Paul J. Nahin, Dr. Euler's Fabulous Formula, Princeton University Press, 2006. Ch. 4, Sect. 4.
• Vretblad, Anders (2000), Fourier Analysis and its Applications, Graduate Texts in Mathematics, vol. 223, New York: Springer Publishing, p. 93, ISBN 978-0-387-00836-3

• Media related to Gibbs phenomenon at Wikimedia Commons
• "Gibbs phenomenon", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W., "Gibbs Phenomenon". From MathWorld—A Wolfram Web Resource.
• Prandoni, Paolo, "Gibbs Phenomenon".
• Radaelli-Sanchez, Ricardo, and Richard Baraniuk, "Gibbs Phenomenon". The Connexions Project. (Creative Commons Attribution License)
• Horatio S Carslaw : Introduction to the theory of Fourier's series and integrals.pdf (introductiontot00unkngoog.pdf ) at archive.org
• A Python implementation of the S-Gibbs algorithm mitigating the Gibbs Phenomenon https://github.com/pog87/FakeNodes.