Gibbs phenomenon: Difference between revisions
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=== History === |
=== History === |
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The Gibbs phenomenon was first noticed and analyzed by [[Henry Wilbraham]] in an 1848 paper.<ref>Wilbraham, Henry (1848) [https://www.google.com/books?id=JrQ4AAAAMAAJ&pg=PA198#v=onepage&q&f=false "On a certain periodic function,"] ''The Cambridge and Dublin Mathematical Journal'', '''3''' : 198–201.</ref> The paper attracted little attention until 1914 when it was mentioned in [[Heinrich Burkhardt]]'s review of mathematical analysis in [[Klein's encyclopedia]].<ref>{{cite book|title=Encyklopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen|volume=Vol II T. 1 H 1|date=1914|publisher=Vieweg+Teubner Verlag|location=Wiesbaden|page=1049|url=http://gdz.sub.uni-goettingen.de/pdfcache/PPN360506208/PPN360506208___LOG_0158.pdf|accessdate=14 September 2016}}</ref> In 1898, [[Albert A. Michelson]] developed a device that could compute and re-synthesize the Fourier series.<ref>{{cite book|last1=Hammack|first1=Bill|last2=Kranz|first2=Steve|last3=Carpenter|first3=Bruce|title=Albert Michelson's Harmonic Analyzer: A Visual Tour of a Nineteenth Century Machine that Performs Fourier Analysis|publisher=Articulate Noise Books|isbn=9780983966173|url=http://www.engineerguy.com/fourier/|accessdate=14 September 2016|language=English|date=2014-10-29}}</ref><ref>{{cite book|last=Wolfram|first=Stephen|title=A New Kind of Science|publisher=Wolfram Media, Inc.|year=2002|page=[https:// |
The Gibbs phenomenon was first noticed and analyzed by [[Henry Wilbraham]] in an 1848 paper.<ref>Wilbraham, Henry (1848) [https://www.google.com/books?id=JrQ4AAAAMAAJ&pg=PA198#v=onepage&q&f=false "On a certain periodic function,"] ''The Cambridge and Dublin Mathematical Journal'', '''3''' : 198–201.</ref> The paper attracted little attention until 1914 when it was mentioned in [[Heinrich Burkhardt]]'s review of mathematical analysis in [[Klein's encyclopedia]].<ref>{{cite book|title=Encyklopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen|volume=Vol II T. 1 H 1|date=1914|publisher=Vieweg+Teubner Verlag|location=Wiesbaden|page=1049|url=http://gdz.sub.uni-goettingen.de/pdfcache/PPN360506208/PPN360506208___LOG_0158.pdf|accessdate=14 September 2016}}</ref> In 1898, [[Albert A. Michelson]] developed a device that could compute and re-synthesize the Fourier series.<ref>{{cite book|last1=Hammack|first1=Bill|last2=Kranz|first2=Steve|last3=Carpenter|first3=Bruce|title=Albert Michelson's Harmonic Analyzer: A Visual Tour of a Nineteenth Century Machine that Performs Fourier Analysis|publisher=Articulate Noise Books|isbn=9780983966173|url=http://www.engineerguy.com/fourier/|accessdate=14 September 2016|language=English|date=2014-10-29}}</ref><ref>{{cite book|last=Wolfram|first=Stephen|title=A New Kind of Science|publisher=Wolfram Media, Inc.|year=2002|page=[https://wolframscience.com/nks/notes-3-12--history-of-experimental-mathematics/ 899]|isbn=978-1-57955-008-0|url=https://www.wolframscience.com/nks/}}</ref> A widespread myth says that when the Fourier coefficients for a square wave were input to the machine, the graph would oscillate at the discontinuities, and that because it was a physical device subject to manufacturing flaws, Michelson was convinced that the overshoot was caused by errors in the machine. In fact the graphs produced by the machine were not good enough to exhibit the Gibbs phenomenon clearly, and Michelson may not have noticed it as he made no mention of this effect in his paper {{harv|Michelson|Stratton|1898}} about his machine or his later letters to ''[[Nature (journal)|Nature]]''.<ref name="Hewitt 1979 129–160"/> Inspired by some correspondence in ''Nature'' between Michelson and Love about the convergence of the Fourier series of the square wave function, in 1898 [[Willard Gibbs|J. Willard Gibbs]] published a short note in which he considered what today would be called a [[sawtooth wave]] and pointed out the important distinction between the limit of the graphs of the partial sums of the Fourier series, and the graph of the function that is the limit of those partial sums. In his first letter Gibbs failed to notice the Gibbs phenomenon, and the limit that he described for the graphs of the partial sums was inaccurate. In 1899 he published a correction in which he described the overshoot at the point of discontinuity (''Nature'': April 27, 1899, p. 606). In 1906, [[Maxime Bôcher]] gave a detailed mathematical analysis of that overshoot, coining the term "Gibbs phenomenon"<ref>Bôcher, Maxime (April 1906) [https://www.jstor.org/stable/1967238?seq=1 "Introduction to the theory of Fourier's series"], ''Annals of Mathethematics'', second series, '''7''' (3) : 81–152. The Gibbs phenomenon is discussed on pages 123–132; Gibbs's role is mentioned on page 129.</ref> and bringing the term into widespread use.<ref name="Hewitt 1979 129–160"/> |
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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."<ref>{{cite journal|last1=Carslaw|first1=H. S.|title=A historical note on Gibbs' phenomenon in Fourier's series and integrals|journal=Bulletin of the American Mathematical Society|date=1 October 1925|volume=31|issue=8|pages=420–424|url=https://projecteuclid.org/euclid.bams/1183486614|accessdate=14 September 2016|language=EN|issn=0002-9904|doi=10.1090/s0002-9904-1925-04081-1|doi-access=free}}</ref> |
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."<ref>{{cite journal|last1=Carslaw|first1=H. S.|title=A historical note on Gibbs' phenomenon in Fourier's series and integrals|journal=Bulletin of the American Mathematical Society|date=1 October 1925|volume=31|issue=8|pages=420–424|url=https://projecteuclid.org/euclid.bams/1183486614|accessdate=14 September 2016|language=EN|issn=0002-9904|doi=10.1090/s0002-9904-1925-04081-1|doi-access=free}}</ref> |
Revision as of 16:35, 13 October 2020
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 ) whose Fourier expansion is
More precisely, this is the function f which equals between and and between and for every integer n; thus this square wave has a jump discontinuity of height at every integer multiple of .
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 of the square wave by
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 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
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
There is no contradiction in the overshoot converging to a non-zero amount, but the limit of the partial sums having no overshoot, because the location of that overshoot moves. We have
The Gibbs 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, ... of the discontinuous square wave described above decay only as fast as the
Solutions
In practice, the difficulties associated with the Gibbs phenomenon can be ameliorated by using a smoother method of Fourier series summation, such as
Formal mathematical description of the phenomenon
Let be a piecewise continuously differentiable function which is periodic with some period . Suppose that at some point , the left limit and right limit of the function differ by a non-zero gap :
For each positive integer N ≥ 1, let SN f be the Nth partial Fourier series
where the Fourier coefficients are given by the usual formulae
Then we have
and
but
More generally, if is any sequence of real numbers which converges to as , and if the gap a is positive then
and
If instead the gap a is negative, one needs to interchange
Signal processing explanation
From a
In the case of convolving with a Heaviside step function, the resulting function is exactly the integral of the sinc function, the
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
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 have less area, but does not reduce the 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.
The square wave example
In the square wave case the period L is , the discontinuity is at zero, and the jump is equal to . For simplicity let us just deal with the case when N is even (the case of odd N is very similar). Then we have
Substituting , we obtain
as claimed above. Next, we compute
If we introduce the normalized sinc function, , we can rewrite this as
But the expression in square brackets is a Riemann sum approximation to the integral (more precisely, it is a
which was what was claimed in the previous section. A similar computation shows
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
The Gibbs phenomenon manifests as a cross pattern artifact in the
See also
- 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
- ^ S2CID 119355426. Available on-line at: National Chiao Tung University: Open Course Ware: Hewitt & Hewitt, 1979. Archived 2016-03-04 at the Wayback Machine
- ISBN 978-0-13-462938-4.
- ^ H. S. Carslaw (1930). "Chapter IX". Introduction to the theory of Fourier's series and integrals (Third ed.). New York: Dover Publications Inc.
- ^ Vretblad 2000 Section 4.7.
- ^ Wilbraham, Henry (1848) "On a certain periodic function," The Cambridge and Dublin Mathematical Journal, 3 : 198–201.
- ^ Encyklopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen (PDF). Vol. Vol II T. 1 H 1. Wiesbaden: Vieweg+Teubner Verlag. 1914. p. 1049. Retrieved 14 September 2016.
{{cite book}}
:|volume=
has extra text (help) - ISBN 9780983966173. Retrieved 14 September 2016.
- ISBN 978-1-57955-008-0.
- ^ Bôcher, Maxime (April 1906) "Introduction to the theory of Fourier's series", Annals of Mathethematics, second series, 7 (3) : 81–152. The Gibbs phenomenon is discussed on pages 123–132; Gibbs's role is mentioned on page 129.
- ISSN 0002-9904. Retrieved 14 September 2016.
- ^ M. Pinsky (2002). Introduction to Fourier Analysis and Wavelets. United states of America: Brooks/Cole. p. 27.
- ^ Rasmussen, Henrik O. "The Wavelet Gibbs Phenomenon." In "Wavelets, Fractals and Fourier Transforms", Eds M. Farge et al., Clarendon Press, Oxford, 1993.
- ^ Kelly, Susan E. "Gibbs Phenomenon for Wavelets." Applied and Computational Harmonic Analysis 3, 1995. "Archived copy" (PDF). Archived from the original (PDF) on 2013-09-09. Retrieved 2012-03-31.
{{cite web}}
: CS1 maint: archived copy as title (link) - ^
De Marchi, Stefano; Marchetti, Francesco; Perracchione, Emma; Poggiali, Davide (2020). "Polynomial interpolation via mapped bases without resampling". J. Comput. Appl. Math. 364. ISSN 0377-0427.
- PMID 25597865.)
{{cite journal}}
: CS1 maint: multiple names: authors list (link
References
- S2CID 4004787
- S2CID 13420929
- Michelson, A. A.; Stratton, S. W. (1898), "A new harmonic analyser", Philosophical Magazine, 5 (45): 85–91
- Antoni Zygmund, Trigonometrical Series, Dover publications, 1955.
- 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: ISBN 978-0-387-00836-3
External links
- "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