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Encyclopedia > Gibbs phenomenon

Approximation of square wave in 5 steps

Approximation of square wave in 25 steps

Approximation of square wave in 125 steps

In mathematics, the Gibbs phenomenon, named after the American physicist J. Willard Gibbs, (also known as ringing artifacts) 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. The overshoot does not die out as the frequency increases, but approaches a finite limit. Image File history File links Gibbs_phenomenon_10. ... Image File history File links Gibbs_phenomenon_10. ... Created by me (Gady Kozma), 28 Aug. ... Created by me (Gady Kozma), 28 Aug. ... Image File history File links Gibbs_phenomenon_250. ... Image File history File links Gibbs_phenomenon_250. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Josiah Willard Gibbs (February 11, 1839 â€“ April 28, 1903) was an American mathematical physicist who contributed much of the theoretical foundation that led to the development of chemical thermodynamics and was one of the founders of vector analysis. ... The Fourier series, named in honor of Joseph Fourier (1768-1830), is an extremely useful mathematical tool. ... In mathematics, a function f(x) of a real number variable x is defined piecewise, if f(x) is given by different expressions on various intervals. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... Continuous functions are of utmost importance in mathematics and applications. ... In mathematics, a series is a sum of a sequence of terms. ... Sine waves of various frequencies; the lower waves have higher frequencies than those above. ...

The three pictures on the right demonstrate this for a square wave whose Fourier expansion is A square wave is a kind of basic waveform. ...

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

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 $π$ . The integers consist of the positive natural numbers (1, 2, 3, â€¦), their negatives (âˆ’1, âˆ’2, âˆ’3, ...) and the number zero. ...

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

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

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 $0.089490... a$ 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

$int_0^pi frac{sin t}{t} dt = {1.851937052dots} = frac{pi}{2} + pi 0.089490...$

is sometimes known as the Wilbraham-Gibbs constant.

The Gibbs phenomenon was first observed by Albert Michelson via a mechanical graphing machine. Michelson developed a device in 1898 that could compute and re-synthesize the Fourier series. When a square wave was input into the machine, the graph would move to and from around the discontinuities. This would occur, and continue to occur, as the number of Fourier coefficients approached infinity. Albert Abraham Michelson. ... 1898 was a common year starting on Saturday (see link for calendar). ... In mathematics, a coefficient is a multiplicative factor of a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ... Infinity is a term with very distinct, separate meanings which arise in theology, philosophy, mathematics and everyday life. ...

The phenomenon was first explained mathematically by J. Willard Gibbs in 1899. Informally, it reflects the difficulty inherent in approximating a discontinuous function by a series of continuous sine and cosine waves. 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 (and thus very rapidly convergent Fourier series), whereas discontinuous functions will have very slowly decaying Fourier coefficients (and thus very badly convergent Fourier series). Note for instance that the Fourier coefficients $1, 1/3, 1/5, ...$ 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. 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. 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. Josiah Willard Gibbs (February 11, 1839 â€“ April 28, 1903) was an American mathematical physicist who contributed much of the theoretical foundation that led to the development of chemical thermodynamics and was one of the founders of vector analysis. ... 1899 was a common year starting on Sunday (see link for calendar). ... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... See harmonic series (music) for the (related) musical concept. ... In mathematics, a series is a sum of a sequence of terms. ... 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 not hold is a null set, i. ... In mathematical analysis, a sequence { fn } of functions converges uniformly to a limiting function f if the speed of convergence of fn(x) to f(x) does not depend on x. ... 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 functions of a real variable. ... In mathematics, the question whether the Fourier series of a periodic function converges to the given function and in what sense is a rich field of research, sometimes called classic harmonic analysis, a branch of pure mathematics. ...

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. If one uses a wavelet transform instead of the Fourier transform then the Gibbs phenomenon no longer occurs. In mathematics, the FejÃ©r kernel is used to express the effect of CesÃ ro summation on Fourier series. ... In mathematics, Ïƒ-approximation adjusts a Fourier summation to eliminate the Gibbs phenomenon which would otherwise occur at discontinuities. ... Wavelets, wavelet analysis, and the wavelet transform refers to the representation of a signal in terms of a finite length or fast decaying oscillating waveform (known as the mother wavelet). ...

1 Formal mathematical description of the phenomenon
2 The square wave example
3 See also
4 Publications

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 $N geq 1$ , let $S N f$ be the $N$ th partial Fourier series

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

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

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

$a_n := frac{2}{L} int_0^L f(x) cosleft(frac{2pi nx}{L}right) dx$

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

Then we have

$lim_{N to infty} S_N fleft(x_0 + frac{L}{2N}right) = f(x_0^+) + 0.089490... a$

and

$lim_{N to infty} S_N fleft(x_0 - frac{L}{2N}right) = f(x_0^-) - 0.089490... a$

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

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

and

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

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 of a sequence can be thought of as limiting bounds on the sequence. ... In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting bounds on the sequence. ...

The square wave example

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

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

If we introduce the sinc function $operatorname{sinc}(x) := sin(x)/x$ , we can rewrite this as The sinc function sinc(x) from x=-8π to 8π. ...

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

But the expression in square brackets is a numerical integration approximation to the integral $int_0^pi operatorname{sinc}(t) dt$ (more precisely, it is a midpoint rule approximation with spacing $2π / N$ ). Since the sinc function is continuous, this approximation converges to the actual integral as $N to infty$ . Thus we have In numerical analysis, the term numerical integration is used to describe a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe numerical algorithms for solving differential equations. ...

$lim_{N to infty} S_N fleft(frac{2pi}{2N}right) = frac{1}{2} int_0^pi operatorname{sinc}(t) dt = frac{pi}{4} + 0.089490... frac{pi}{2}$

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

$lim_{N to infty} S_N fleft(-frac{2pi}{2N}right) = -frac{1}{2} int_0^pi operatorname{sinc}(t) dt = -frac{pi}{4} - 0.089490... frac{pi}{2}$

Publications

Gibbs, J. W., "Fourier Series". Nature 59, 200 and 606, 1899.
Antoni Zygmund, Trigonometrical series, Dover publications, 1955.
Wilbraham, H. On a certain periodic function, Cambridge and Dublin Math. J., 3 (1848), pp. 198-201.

Antoni Zygmund (25 December 1900 _ 30 May 1992) was a Polish mathematician. ...

Results from FactBites:

Willard Gibbs - Wikipedia, the free encyclopedia (1125 words)
	Josiah Willard Gibbs (February 11, 1839 – April 28, 1903) was an American mathematical physicist who contributed much of the theoretical foundation that led to the development of chemical thermodynamics and was one of the founders of vector analysis.
	Between 1876 and 1878 Gibbs wrote a series of papers collectively entitled "On the Equilibrium of Heterogeneous Substances", considered one of the greatest achievements in physical science in the 19th century and the foundation of the science of physical chemistry.
	Gibbs was born in New Haven, Connecticut, where his father was a professor of sacred literature at Yale University's Divinity School, best known today for his involvement in the Amistad trial.

Gibbs phenomenon - definition of Gibbs phenomenon in Encyclopedia (762 words)
	The Gibbs phenomenon was first observed by Albert Michelson via a mechanical graphing machine.
	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 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.

More results at FactBites »

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Formal mathematical description of the phenomenon GA_googleFillSlot("encyclopedia_square");

The square wave example

See also

Publications

Formal mathematical description of the phenomenon