16.6 : フーリエ級数の収束

The Fourier series of a signal is an infinite sum of complex exponentials. The infinite sum is often truncated to a finite partial sum to make it practical.

Increasing terms in a partial sum should make the approximation converge to the signal. Yet near discontinuities, persistent ripples occur, getting compressed towards the discontinuity - a phenomenon known as the Gibbs phenomenon.

Gibbs observed these high-frequency ripples and overshoots near discontinuities in truncated Fourier series approximations.

To mitigate this, choose a large number of terms so that the ripple's total energy is negligible.

Despite ripples, the energy in the approximation error decreases with more terms, allowing the Fourier series to represent discontinuous signals.

Truncating the Fourier series to a desired number of terms provides the best approximation, minimizing the error. The error reduces with more terms and eventually becomes zero if the signal can be represented by a Fourier series.

For instance, in image processing, reducing the error is crucial when approximating an image signal using a truncated Fourier series to avoid visual artifacts.