Gibbs phenomenon
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Gibbs phenomenon, discovered by Available on-line at:
National Chiao Tung University: Open Course Ware: Hewitt & Hewitt, 1979.
and rediscovered by , is the
oscillatory Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
behavior of the
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
of a
piecewise In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Pi ...
continuously differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
periodic function A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to desc ...
around a
jump discontinuity Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there. The set of ...
. The function's Nth partial Fourier series (formed by summing its N lowest constituent
sinusoids A capillary is a small blood vessel from 5 to 10 micrometres (μm) in diameter. Capillaries are composed of only the tunica intima, consisting of a thin wall of simple squamous endothelial cells. They are the smallest blood vessels in the body: ...
) produces large peaks around the jump which overshoot and undershoot the function's actual values. This
approximation error The approximation error in a data value is the discrepancy between an exact value and some ''approximation'' to it. This error can be expressed as an absolute error (the numerical amount of the discrepancy) or as a relative error (the absolute er ...
approaches a
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of about 9% of the jump as more sinusoids are used, though the
infinite Infinite may refer to: Mathematics * Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...
Fourier
series sum Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...
does eventually converge almost everywhere except the point of discontinuity. The Gibbs phenomenon was observed by experimental physicists, but was believed to be due to imperfections in the measuring apparatus, and it is one cause of ringing artifacts in
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
.


Description

The Gibbs phenomenon involves both the fact that Fourier sums overshoot at a
jump discontinuity Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there. The set of ...
, 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 A square wave is a non-sinusoidal periodic waveform in which the amplitude alternates at a steady frequency between fixed minimum and maximum values, with the same duration at minimum and maximum. In an ideal square wave, the transitions b ...
(of height \tfrac) whose Fourier series is \sin(x)+\frac\sin(3x)+\frac\sin(5x)+\dotsb. More precisely, this square wave is the function f(x) which equals \tfrac between 2n\pi and (2n+1)\pi and -\tfrac between (2n+1)\pi and (2n+2)\pi for every
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
n; thus this square wave has a jump discontinuity of height \tfrac at every integer multiple of \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. Deriving the formula of the limit of the error for the square wave reveals that the error exceeds the height of the square wave (\tfrac) by \frac\int_0^\pi \frac\, dt - \frac = \frac\cdot (0.089489872236\dots)() or about 9% of the jump. More generally, at any discontinuity 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 an error approaching 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 In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dimen ...
of the jump (regardless of the actual value of the original function at the discontinuity). The quantity \int_0^\pi \frac\ dt = (1.851937051982\dots) = \frac + \pi \cdot (0.089489872236\dots)() is sometimes known as the '' Wilbraham–Gibbs constant''.


History

The Gibbs phenomenon was first noticed and analyzed by Henry Wilbraham in an 1848 paper. The paper attracted little attention until 1914 when it was mentioned in
Heinrich Burkhardt Heinrich Friedrich Karl Ludwig Burkhardt (15 October 1861 – 2 November 1914) was a German mathematician. He famously was one of the two examiners of Albert Einstein's PhD thesis ''Eine neue Bestimmung der Moleküldimensionen''. Of Einstein' ...
's review of mathematical analysis in
Klein's encyclopedia Felix Klein's ''Encyclopedia of Mathematical Sciences'' is a German mathematical encyclopedia published in six volumes from 1898 to 1933. Klein and Wilhelm Franz Meyer were organizers of the encyclopedia. Its full title in English is ''Encycloped ...
. In 1898,
Albert A. Michelson Albert Abraham Michelson Royal Society of London, FFRS HFRSE (surname pronunciation anglicized as "Michael-son", December 19, 1852 – May 9, 1931) was a German-born American physicist of Polish/Jewish origin, known for his work on measuring the ...
developed a device that could compute and re-synthesize the Fourier series. 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 about his machine or his later letters to ''
Nature Nature, in the broadest sense, is the physics, physical world or universe. "Nature" can refer to the phenomenon, phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. ...
''. Inspired by correspondence in ''Nature'' between Michelson and A. E. H. Love about the convergence of the Fourier series of the square wave function,
J. Willard Gibbs Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in t ...
published a note in 1898 pointing out the important distinction between the limit of the graphs of the partial sums of the Fourier series of a
sawtooth wave The sawtooth wave (or saw wave) is a kind of non-sinusoidal waveform. It is so named based on its resemblance to the teeth of a plain-toothed saw with a zero rake angle. A single sawtooth, or an intermittently triggered sawtooth, is called a ...
and the graph of 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 Maxime Bôcher (August 28, 1867 – September 12, 1918) was an American mathematician who published about 100 papers on differential equations, series, and algebra. He also wrote elementary texts such as ''Trigonometry'' and ''Analytic Geometry''. ...
gave a detailed mathematical analysis of that overshoot, coining the term "Gibbs phenomenon" and bringing the term into widespread use. After the existence of Henry Wilbraham's paper became widely known, in 1925
Horatio Scott Carslaw Dr Horatio Scott Carslaw FRSE LLD (12 February 1870, Helensburgh, Dumbartonshire, Scotland – 11 November 1954, Burradoo, New South Wales, Australia) was a Scottish- Australian mathematician. The book he wrote with his colleague John ...
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."


Explanation

Informally, the Gibbs phenomenon reflects the difficulty inherent in approximating a
discontinuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
by a ''finite'' series of
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
sinusoidal waves. It is important to put emphasis on the word ''finite'', because even though every partial sum of the Fourier series overshoots around each discontinuity it is approximating, the limit of summing an infinite number of sinusoidal waves does not. The overshoot peaks moves closer and closer to the discontinuity as more terms are summed, so convergence is possible. There is no contradiction (between the overshoot error converging to a non-zero height even though the infinite sum has no overshoot), because the overshoot peaks move toward the discontinuity. The Gibbs phenomenon thus exhibits
pointwise convergence In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared. Definition Suppose that X is a set and ...
, but not
uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...
. For a piecewise continuously differentiable (class ''C''1) function, the Fourier series converges to the function at ''every point'' except at jump discontinuities. At jump discontinuities, the infinite sum will converge to the jump discontinuity's midpoint (i.e. the average of the values of the function on either side of the jump), as a consequence of Dirichlet's theorem. The Gibbs phenomenon is closely related to the principle that the
smoothness In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it ...
of a function controls the decay rate of its Fourier coefficients. Fourier coefficients of smoother functions will more rapidly decay (resulting in faster convergence), whereas Fourier coefficients of discontinuous functions will slowly decay (resulting in slower convergence). For example, the discontinuous square wave has Fourier coefficients (\tfrac,,\tfrac,,\tfrac,,\tfrac,,\tfrac,,\dots) that decay only at the rate of \tfrac, while the continuous
triangle wave A triangular wave or triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function. Like a square wave, the triangle wave contains only odd harmonics. However, the ...
has Fourier coefficients (\tfrac,,\tfrac,,\tfrac,,\tfrac,,\tfrac,,\dots) that decay at a much faster rate of \tfrac. This only provides a partial explanation of the Gibbs phenomenon, since Fourier series with absolutely convergent Fourier coefficients would be
uniformly convergent In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...
by the
Weierstrass M-test In mathematics, the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely. It applies to series whose terms are bounded functions with real or complex values, and is analogous t ...
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 .


Solutions

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. Using a continuous
wavelet A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
transform, the wavelet Gibbs phenomenon never exceeds the Fourier Gibbs phenomenon. Also, using the discrete wavelet transform with Haar basis functions, the Gibbs phenomenon does not occur at all in the case of continuous data at jump discontinuities, and is minimal in the discrete case at large change points. In wavelet analysis, this is commonly referred to as the Longo phenomenon. In the polynomial interpolation setting, the Gibbs phenomenon can be mitigated using the S-Gibbs algorithm.


Formal mathematical description of the phenomenon

Let f: \to 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 jump of a: f(x_0^+) - f(x_0^-) = a \neq 0. For each positive integer N ≥ 1, let S_N f(x) be the Nth partial Fourier series S_N f(x) := \sum_ \widehat f(n) e^ = \frac a_0 + \sum_^N \left( a_n \cos\left(\frac\right) + b_n \sin\left(\frac\right) \right), where the Fourier coefficients \widehat f(n), a_n, b_n are given by the usual formulae \widehat f(n) := \frac \int_0^L f(x) e^\, dx a_n := \frac \int_0^L f(x) \cos\left(\frac\right)\, dx b_n := \frac \int_0^L f(x) \sin\left(\frac\right)\, dx. Then we have \lim_ S_N f\left(x_0 + \frac\right) = f(x_0^+) + a\cdot (0.089489872236\dots) and \lim_ S_N f\left(x_0 - \frac\right) = f(x_0^-) - a\cdot (0.089489872236\dots) but \lim_ S_N f(x_0) = \frac. More generally, if x_N is any sequence of real numbers which converges to x_0 as N \to \infty, and if the jump of a is positive then \limsup_ S_N f(x_N) \leq f(x_0^+) + a\cdot (0.089489872236\dots) and \liminf_ S_N f(x_N) \geq f(x_0^-) - a\cdot (0.089489872236\dots). If instead the jump of a is negative, one needs to interchange
limit superior In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a ...
with
limit inferior In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For ...
, and also interchange the \leq and \ge signs, in the above two inequalities.


Signal processing explanation

From a
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
point of view, the Gibbs phenomenon is the
step response The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the out ...
of a
low-pass filter A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter des ...
, and the oscillations are called
ringing Ringing may mean: Vibrations * Ringing (signal), unwanted oscillation of a signal, leading to ringing artifacts * Vibration of a harmonic oscillator ** Bell ringing * Ringing (telephony), the sound of a telephone bell * Ringing (medicine), a ri ...
or ringing artifacts. Truncating the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
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 In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
of the original signal with the
impulse response In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an Dirac delta function, impulse (). More generally, an impulse ...
of the filter (also known as the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
), which is the
sinc function In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized.. In mathematics, the historical unnormalized sinc function is defined for by \operatornamex = \frac. Alternatively, the u ...
. Thus the Gibbs phenomenon can be seen as the result of convolving a
Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
(if periodicity is not required) or a
square wave A square wave is a non-sinusoidal periodic waveform in which the amplitude alternates at a steady frequency between fixed minimum and maximum values, with the same duration at minimum and maximum. In an ideal square wave, the transitions b ...
(if periodic) with a sinc function: the oscillations in the sinc function cause the ripples in the output. In the case of convolving with a Heaviside step function, the resulting function is exactly the integral of the sinc function, the
sine integral In mathematics, trigonometric integrals are a indexed family, family of integrals involving trigonometric functions. Sine integral The different sine integral definitions are \operatorname(x) = \int_0^x\frac\,dt \operatorname(x) = -\int ...
; 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 \int_^ \frac\,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 Gain or GAIN may refer to: Science and technology * Gain (electronics), an electronics and signal processing term * Antenna gain * Gain (laser), the amplification involved in laser emission * Gain (projection screens) * Information gain in de ...
. 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 In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form f(x) = \exp (-x^2) and with parametric extension f(x) = a \exp\left( -\frac \right) for arbitrary real constants , and non-zero . It is ...
, then the value of the filtered signal will be a
convex combination In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. In other word ...
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 In mathematics, an affine combination of is a linear combination : \sum_^ = \alpha_ x_ + \alpha_ x_ + \cdots +\alpha_ x_, such that :\sum_^ =1. Here, can be elements ( vectors) of a vector space over a field , and the coefficients \alpha_ a ...
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. Image:Gibbs phenomenon 10.svg, Oscillations can be interpreted as convolution with a sinc. Image:Gibbs phenomenon 50.svg, 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 examine the Nth partial Fourier series S_N f(x) of a square wave with a 2\pi period and a \tfrac vertical discontinuity at x = 0. Because the case of odd N is very similar, let us just deal with the case when N is even: S_N f(x) = \sin(x) + \frac \sin(3x) + \cdots + \frac \sin((N-1)x). Substituting x = 0, we obtain S_N f(0) = 0 = \frac = \frac as claimed above. Next, we compute S_N f\left(\frac\right) = \sin\left(\frac\right) + \frac \sin\left(\frac\right) + \cdots + \frac \sin\left( \frac \right). If we introduce the normalized
sinc function In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized.. In mathematics, the historical unnormalized sinc function is defined for by \operatornamex = \frac. Alternatively, the u ...
, \operatorname(x)\,, we can rewrite this as S_N f\left(\frac\right) = \frac \left \frac \operatorname\left(\frac\right) + \frac \operatorname\left(\frac\right)+ \cdots + \frac \operatorname\left( \frac \right) \right But the expression in square brackets is a
Riemann sum In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lin ...
approximation to the integral \int_0^1 \operatorname(x)\ dx (more precisely, it is a
midpoint rule In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lin ...
approximation with spacing \tfrac). Since the sinc function is continuous, this approximation converges to the actual integral as N \to \infty. Thus we have \begin \lim_ S_N f\left(\frac\right) & = \frac \int_0^1 \operatorname(x)\, dx \\ pt& = \frac \int_^1 \frac\, d(\pi x) \\ pt& = \frac \int_0^\pi \frac\ dt \quad = \quad \frac + \frac \cdot (0.089489872236\dots), \end which was what was claimed in the previous section. A similar computation shows \lim_ S_N f\left(-\frac\right) = -\frac \int_0^1 \operatorname(x)\, dx = -\frac - \frac \cdot (0.089489872236\dots).


Consequences

The Gibbs phenomenon is undesirable because it causes artifacts, namely
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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 Magnetic resonance imaging (MRI) is a medical imaging technique used in radiology to form pictures of the anatomy and the physiological processes of the body. MRI scanners use strong magnetic fields, magnetic field gradients, and radio waves ...
, 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 Syringomyelia is a generic term referring to a disorder in which a cyst or cavity forms within the spinal cord. Often, syringomyelia is used as a generic term before an etiology is determined. This cyst, called a syrinx, can expand and elongate ...
. The Gibbs phenomenon manifests as a cross pattern artifact in the
discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex- ...
of an image, 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 In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a par ...
filters have discontinuities in the
frequency domain In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a signa ...
, their exact representation in the
time domain Time domain refers to the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the cas ...
necessarily requires an infinitely-long sinc filter impulse response, since a
finite impulse response In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of ''finite'' duration, because it settles to zero in finite time. This is in contrast to infinite impulse ...
will result in Gibbs rippling in the
frequency response In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of sy ...
near cut-off frequencies, though this rippling can be reduced by windowing finite impulse response filters (at the expense of wider transition bands).


See also

*
Mach bands Mach bands is an optical illusion named after the physicist Ernst Mach. It exaggerates the contrast between edges of the slightly differing shades of gray, as soon as they contact one another, by triggering edge-detection in the human visual s ...
* Pinsky phenomenon *
Runge's phenomenon In the mathematical field of numerical analysis, Runge's phenomenon () is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree over a set of equispaced interpolation ...
(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 In mathematics, trigonometric integrals are a indexed family, family of integrals involving trigonometric functions. Sine integral The different sine integral definitions are \operatorname(x) = \int_0^x\frac\,dt \operatorname(x) = -\int ...


Notes


References

* * * *
Volume 1Volume 2
* *
Paul J. Nahin Paul J. Nahin (born November 26, 1940 in Orange County, California) is an American electrical engineer and author who has written 20 books on topics in physics and mathematics, including biographies of Oliver Heaviside, George Boole, and Claude Sh ...
, ''Dr. Euler's Fabulous Formula,'' Princeton University Press, 2006. Ch. 4, Sect. 4. *


External links

* * * Weisstein, Eric W., "
Gibbs Phenomenon
'". From MathWorld—A Wolfram Web Resource. * Prandoni, Paolo, "

'". * 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 The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...
* A
Python Python may refer to: Snakes * Pythonidae, a family of nonvenomous snakes found in Africa, Asia, and Australia ** ''Python'' (genus), a genus of Pythonidae found in Africa and Asia * Python (mythology), a mythical serpent Computing * Python (pro ...
implementation of the S-Gibbs algorithm mitigating the Gibbs Phenomenon https://github.com/pog87/FakeNodes. {{DEFAULTSORT:Gibbs Phenomenon Real analysis Fourier series Numerical artefacts