Gibbs phenomenon - Citizendia

In mathematics, the Gibbs phenomenon (also known as ringing artifacts), named after the American physicist J. Willard Gibbs, is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function f behaves at a jump discontinuity: the nth partial sum of the Fourier series has large oscillations near the jump, which might increase the maximum of the partial sum above that of the function itself. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Josiah Willard Gibbs ( February 11, 1839 &ndash April 28, 1903) was an American theoretical Physicist, Chemist In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions In Mathematics, a piecewise-defined function (also called a piecewise function) is a function whose definition is dependent on the value of the Independent In Mathematics, a periodic function is a function that repeats its values after some definite period has been added to its Independent variable In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with The overshoot does not die out as the frequency increases, but approaches a finite limit. Frequency is a measure of the number of occurrences of a repeating event per unit Time. ^[1]

The overshoot is a consequence of trying to approximate a discontinuous function with a partial (i. e. finite) sum of continuous functions. A finite sum of continuous functions is necessarily continuous, and therefore cannot approximate the discontinuity (and the area "near" it) to within any arbitrarily chosen accuracy. An infinite sum of continuous functions can be discontinuous, and the pointwise limit of the partial sums of a Fourier series does not exhibit an overshoot near a jump discontinuity as do the partial sums themselves. In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions

1 Description
2 Formal mathematical description of the phenomenon
3 The square wave example
4 Notes and references
5 See also
6 Publications
7 External links and references

Description

Approximation of square wave in 5 steps

Approximation of square wave in 25 steps

Approximation of square wave in 125 steps

The three pictures on the right demonstrate the phenomenon for a square wave whose Fourier expansion is

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

More precisely, this is the function f which equals $π / 4$ between $2 n π$ and $(2 n + 1)π$ and $- π / 4$ between $(2 n + 1)π$ and $(2 n + 2)π$ for every integer n; thus this square wave has a jump discontinuity of height $π / 2$ at every integer multiple of $π$ . A square wave is a kind of Non-sinusoidal waveform, most typically encountered in Electronics and Signal processing. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

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 $π / 4$ of the square wave by

$\frac{1}{2}\int_0^\pi \frac{\sin t}{t}\, dt - \frac{\pi}{4} = \frac{\pi}{2}\cdot (0.089490\dots)$

or about 17. 9 percent. 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 $a \cdot (0.089490\dots)$ 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

$\int_0^\pi \frac{\sin t}{t}\ dt = (1.851937052\dots) = \frac{\pi}{2} + \pi \cdot (0.089490\dots)$

is sometimes known as the Wilbraham-Gibbs constant. Henry Wilbraham ( July 25, 1825 – February 13, 1883) was an obscure English mathematician

The Gibbs phenomenon was first noticed and analyzed by the obscure Henry Wilbraham. Henry Wilbraham ( July 25, 1825 – February 13, 1883) was an obscure English mathematician He published a paper on it in 1848 that went unnoticed by the mathematical world. It was not until Albert Michelson observed the phenomenon via a mechanical graphing machine that interest arose. Albert Abraham Michelson ( December 19, 1852 &ndash May 9, 1931) was a Polish - American Physicist known Michelson developed a device in 1898 that could compute and re-synthesize the Fourier series. Year 1898 ( MDCCCXCVIII) was a Common year starting on Saturday (link will display the full calendar of the Gregorian calendar (or a Common When the Fourier coefficients for a square wave were input to the machine, the graph would oscillate at the discontinuities. This would continue to occur even as the number of Fourier coefficients increased. In Mathematics, a coefficient is a Constant multiplicative factor of a certain object

Because it was a physical device subject to manufacturing flaws, Michelson was convinced that the overshoot was caused by errors in the machine. J. Willard Gibbs pointed out in 1899 that the oscillations were a mathematical phenomenon, and would always occur when synthesizing a discontinuous function with a Fourier series. Josiah Willard Gibbs ( February 11, 1839 &ndash April 28, 1903) was an American theoretical Physicist, Chemist Year 1899 ( MDCCCXCIX) was a Common year starting on Sunday (link will display the full calendar of the Gregorian calendar (or a Common Maxime Bôcher gave a detailed mathematical analysis of the phenomenon in 1906 and named it the Gibbs phenomenon. Maxime Bôcher ( August 28 1867 – September 12 1918) was an American Mathematician who published about 100 papers on Year 1906 ( MCMVI) was a Common year starting on Monday (link will display full calendar of the Gregorian calendar (or a Common year starting

Informally, it reflects the difficulty inherent in approximating a discontinuous function by a finite series of continuous sine and cosine waves. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output This phenomenon is also closely related to the principle that the decay of the Fourier coefficients of a function at infinity is controlled by the smoothness of that function; very smooth functions will have very rapidly decaying Fourier coefficients (resulting in the rapid convergence of the Fourier series), whereas discontinuous functions will have very slowly decaying Fourier coefficients (causing the Fourier series to converge very slowly). Note for instance that the Fourier coefficients $1, 1/3, 1/5, \dots$ of the discontinuous square wave described above decay only as fast as the harmonic series, which is not absolutely convergent; indeed, the above Fourier series turns out to be only conditionally convergent for almost every value of x. See Harmonic series (music for the (related musical concept In Mathematics, the harmonic series is the Infinite series In Mathematics, a series (or sometimes also an Integral) is said to converge absolutely if the sum (or integral of the Absolute value of the In Measure theory (a branch of Mathematical analysis) one says that a property holds almost everywhere if the set of elements for which the property does This provides a partial explanation of the Gibbs phenomenon, since Fourier series with absolutely convergent Fourier coefficients would be uniformly convergent by the Weierstrass M-test and would thus be unable to exhibit the above oscillatory behavior. In the mathematical field of analysis, uniform convergence is a type of Convergence stronger than Pointwise convergence. In Mathematics, the Weierstrass M-test is an analogue of the Comparison test for Infinite series, and applies to a series whose terms are themselves By the same token, it is impossible for a discontinuous function to have absolutely convergent Fourier coefficients, since the function would thus be the uniform limit of continuous functions and therefore be continuous, a contradiction. See more about absolute convergence of Fourier series. In Mathematics, the question of whether the Fourier series of a Periodic function converges to the given function is researched by

In practice, the difficulties associated with the Gibbs phenomenon can be ameliorated by using a smoother method of Fourier series summation, such as Fejér summation or Riesz summation, or by using sigma-approximation. In Mathematics, the Fejér kernel is used to express the effect of Cesàro summation on Fourier series. In Mathematics, σ-approximation adjusts a Fourier summation to eliminate the Gibbs phenomenon which would otherwise occur at Discontinuities Using a wavelet transform with Haar basis functions, the Gibbs phenomenon does not occur. A wavelet is a mathematical function used to divide a given function or continuous-time signal into different frequency components and study each component with a resolution

Formal mathematical description of the phenomenon

Let $f: {\Bbb R} \to {\Bbb R}$ be a piecewise continuously differentiable function which is periodic with some period $L > 0$ . Suppose that at some point $x 0$ , the left limit $f(x_0^-)$ and right limit $f(x_0^+)$ of the function $f$ differ by a non-zero gap $a$ :

$f(x_0^+) - f(x_0^-) = a \neq 0.$

For each positive integer N ≥ 1, let S_N f be the Nth partial Fourier series

$S_N f(x) := \sum_{-N \leq n \leq N} \hat f(n) e^{2\pi i n x/L} = \frac{1}{2} a_0 + \sum_{n=1}^N a_n \cos\left(\frac{2\pi nx}{L}\right) + b_n \sin\left(\frac{2\pi nx}{L}\right)$

where the Fourier coefficients $\hat f(n), a_n, b_n$ are given by the usual formulae

$\hat f(n) := \frac{1}{L} \int_0^L f(x) e^{-2\pi i n x/L}\, dx$

$a_n := \frac{2}{L} \int_0^L f(x) \cos\left(\frac{2\pi nx}{L}\right)\, dx$

$b_n := \frac{2}{L} \int_0^L f(x) \sin\left(\frac{2\pi nx}{L}\right)\, dx.$

Then we have

$\lim_{N \to \infty} S_N f\left(x_0 + \frac{L}{2N}\right) = f(x_0^+) + a\cdot (0.089490\dots)$

and

$\lim_{N \to \infty} S_N f\left(x_0 - \frac{L}{2N}\right) = f(x_0^-) - a\cdot (0.089490\dots)$

but

$\lim_{N \to \infty} S_N f(x_0) = \frac{f(x_0^-) + f(x_0^+)}{2}.$

More generally, if $x N$ is any sequence of real numbers which converges to $x 0$ as $N \to \infty$ , and if the gap a is positive then

$\limsup_{N \to \infty} S_N f(x_N) \leq f(x_0^+) + a\cdot (0.089490\dots)$

and

$\liminf_{N \to \infty} S_N f(x_N) \geq f(x_0^-) - a\cdot (0.089490\dots)$

If instead the gap a is negative, one needs to interchange limit superior with limit inferior, and also interchange the ≤ and ≥ signs, in the above two inequalities. In Mathematics, the limit inferior and limit superior (also called infimum limit and supremum limit, or liminf and limsup In Mathematics, the limit inferior and limit superior (also called infimum limit and supremum limit, or liminf and limsup

The square wave example

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.

We now illustrate the above Gibbs phenomenon in the case of the square wave described earlier. In this case the period L is $2π$ , the discontinuity $x 0$ is at zero, and the jump a is equal to $π / 2$ . For simplicity let us just deal with the case when N is even (the case of odd N is very similar). Then we have

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

Substituting $x = 0$ , we obtain

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

as claimed above. Next, we compute

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

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

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

But the expression in square brackets is a numerical integration approximation to the integral $\int_0^1 \operatorname{sinc}(x)\ dx$ (more precisely, it is a midpoint rule approximation with spacing $2 / N$ ). In Mathematics, the sinc function, denoted by \scriptstyle\mathrm{sinc}(x\ and sometimes as \scriptstyle\mathrm{Sa}(x\ has two definitions sometimes In Numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite Integral, and by extension Since the sinc function is continuous, this approximation converges to the actual integral as $N \to \infty$ . Thus we have

$\lim_{N \to \infty} S_N f\left(\frac{2\pi}{2N}\right)$	$= \frac{\pi}{2} \int_0^1 \operatorname{sinc}(x)\ dx$
	$= \frac{1}{2} \int_{x=0}^1 \frac{\sin(\pi x)}{\pi x}\ d(\pi x)$
	$= \frac{1}{2} \int_0^\pi \frac{\sin(t)}{t}\ dt \quad = \quad \frac{\pi}{4} + \frac{\pi}{2} \cdot (0.089490\dots)$

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

$\lim_{N \to \infty} S_N f\left(-\frac{2\pi}{2N}\right) = -\frac{\pi}{2} \int_0^1 \operatorname{sinc}(x)\ dx = -\frac{\pi}{4} - \frac{\pi}{2} \cdot (0.089490\dots).$

Notes and references

^ H S Carslaw (1930). Introduction to the theory of Fourier's series and integrals, Third Edition, New York: Dover Publications Inc. , Chapter IX.

Publications

Gibbs, J. In the mathematical field of Numerical analysis, Runge's phenomenon is a problem that occurs when using Polynomial interpolation with polynomials of In Mathematics, σ-approximation adjusts a Fourier summation to eliminate the Gibbs phenomenon which would otherwise occur at Discontinuities Henry Wilbraham ( July 25, 1825 – February 13, 1883) was an obscure English mathematician W. , "Fourier Series". Nature 59, 200 (1898) and 606 (1899).
Antoni Zygmund, Trigonometrical series, Dover publications, 1955. Antoni Zygmund ( December 25, 1900 – May 30 1992) was a Polish born American mathematician who exerted a major influence
Wilbraham, H. On a certain periodic function, Cambridge and Dublin Math. J. , 3 (1848), pp. 198-201.
Paul J. Nahin, Dr. Euler's Fabulous Formula, Princeton University Press, 2006. Ch. 4, Sect. 4.

External links and references

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)

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