In mathematics, the Gibbs phenomenon is the oscillatory behavior of the Fourier series of a piecewise continuously differentiable periodic function around a jump discontinuity. The N {\textstyle N} th partial Fourier series of the function (formed by summing the N {\textstyle N} lowest constituent sinusoids of the Fourier series of the function) produces large peaks around the jump which overshoot and undershoot the function values. As more sinusoids are used, this approximation error approaches a limit of about 9% of the jump, though the infinite Fourier series sum does eventually converge almost everywhere. The Gibbs phenomenon was observed by experimental physicists and was believed to be due to imperfections in the measuring apparatus, but it is in fact a mathematical result. It is one cause of ringing artifacts in signal processing. It is named after Josiah Willard Gibbs.
Description
The Gibbs phenomenon is a behavior of the Fourier series of a function with a jump discontinuity and is described as the following:As more Fourier series constituents or components are taken, the Fourier series shows the first overshoot in the oscillatory behavior around the jump point approaching ~ 9% of the (full) jump and this oscillation does not disappear but gets closer to the point so that the integral of the oscillation approaches zero.At the jump point, the Fourier series gives the average of the function's both side limits toward the point.
Square wave example The three pictures on the right demonstrate the Gibbs phenomenon for a square wave (with peak-to-peak amplitude of c {\textstyle c} from − c / 2 {\textstyle -c/2} to c / 2 {\textstyle c/2} and the periodicity L {\textstyle L} ) whose N {\textstyle N} th partial Fourier series is
2 c π ( sin ( ω x ) + 1 3 sin ( 3 ω x ) + ⋯ + 1 2 N − 1 sin ( ( 2 N − 1 ) ω x ) ) {\displaystyle {\frac {2c}{\pi }}\left(\sin(\omega x)+{\frac {1}{3}}\sin(3\omega x)+\cdots +{\frac {1}{2N-1}}\sin((2N-1)\omega x)\right)}
where ω = 2 π / L {\textstyle \omega =2\pi /L} . More precisely, this square wave is the function f ( x ) {\textstyle f(x)} which equals c 2 {\displaystyle {\tfrac {c}{2}}} between 2 n ( L / 2 ) {\textstyle 2n(L/2)} and ( 2 n + 1 ) ( L / 2 ) {\textstyle (2n+1)(L/2)} and − c 2 {\textstyle -{\tfrac {c}{2}}} between ( 2 n + 1 ) ( L / 2 ) {\textstyle (2n+1)(L/2)} and ( 2 n + 2 ) ( L / 2 ) {\textstyle (2n+2)(L/2)} for every integer n {\textstyle n} ; thus, this square wave has a jump discontinuity of peak-to-peak height c {\textstyle c} at every integer multiple of L / 2 {\textstyle L/2} . As more sinusoidal terms are added (i.e., increasing N {\textstyle N} ), 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. The square wave analysis reveals that the error exceeds the height (from zero) c 2 {\displaystyle {\tfrac {c}{2}}} of the square wave by
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