Gibbs phenomenon

The Gibbs phenomenon is the persistent overshoot and ringing that occurs when a discontinuous function is approximated by a finite Fourier series. No matter how many terms are included in the series, the approximation overshoots the discontinuity by approximately 9% of the jump height. The phenomenon was first observed by Henry Wilbraham in 1848 and later analyzed by J. Willard Gibbs in 1899, and it is a fundamental limit on the convergence of Fourier approximations to discontinuous signals.

The Gibbs phenomenon is the mathematical signature of the fact that no physical filter can have a perfectly sharp transition between passband and stopband. It is the reason why steep filters ring, why sharp digital images show halos, and why any system that tries to separate signal from noise with infinite precision will always pay a price in overshoot.