Gibbs Phenomenon

Fourier Analysis · Axiom Academy

Understanding the overshoot at discontinuities in Fourier series Let's approximate a square wave using its Fourier series. Watch what happens at the discontinuity as we add more terms - the overshoot gets narrower but never disappears! 2. The Dirichlet Kernel Explanation The Gibbs phenomenon occurs because sine and cosine functions are smooth, but a discontinuity is fundamentally non-smooth. When smooth functions try to approximate a sharp jump, they "overshoot" and create ripples. The maximum overshoot can be calculated precisely. For a unit jump (from 0 to 1), the overshoot approaches approximately 0.08949 , or about 8.95% of the jump height. The Gibbs phenomenon has important consequences in real-world applications: 5. Methods to Reduce Gibbs Phenomenon While we cannot eliminate the Gibbs phenomenon entirely, several techniques can reduce its effects:

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