Gibbs phenomenon

The Gibbs phenomenon is the overshoot that appears when you reconstruct a discontinuous signal from finitely many sinusoidal terms of its Fourier series. Near a jump, the partial sums overshoot the true signal by about 9% of the jump size, and this overshoot does not go to zero as you add more terms. It just gets narrower.

Partial Fourier sums of a square wave: the ~9% overshoot near jumps persists as $N\to \infty$.

What happens at a jump

For a signal with a jump from $a$ to $b$ at $t_0$, the full Fourier series converges to the midpoint $(a+b)/2$ at $t_0$ and to the correct values at every other point. So pointwise the series is fine.

The truncated partial sum (sum of the first $N$ harmonics) does something different near the jump: it overshoots above $b$ and undershoots below $a$, by roughly $9\%$ of $|b-a|$, with the overshoot concentrating closer to the jump as $N$ grows but never shrinking in height.

Why it doesn’t go away

The proof is technical, but the picture is that the partial sum is the original signal convolved with a truncated sinc (the Fourier dual of truncating the frequency spectrum). Convolving a jump with a sinc-like kernel produces ringing on both sides of the jump, with peak amplitude proportional to the integral of the kernel up to its first zero, and that integral is a fixed number independent of $N$.

More terms give narrower ringing, not smaller ringing. The peak overshoot stays at about $9\%$ forever.

The phenomenon was first observed numerically by Henry Wilbraham (1848) and rediscovered by J. Willard Gibbs (1899), after whom it is named.

Practical implications

Whenever you reconstruct or filter a signal with sharp edges using finite frequency content, you’ll see ringing artifacts:

• Image processing: edges of objects show ringing if you apply ideal lowpass filtering or aggressive JPEG compression (which throws away high-frequency DCT components).
• Audio: brick-wall lowpass filters at the Nyquist boundary can cause “pre-ringing” before transients.
• Bandlimited reconstruction: forcing a discontinuous signal into a band-limited representation makes Gibbs ringing unavoidable.

The fix in practice is a windowed version of the filter. Multiplying the sinc kernel by a smoother window function (Hamming, Hann, Blackman, etc.) trades some passband flatness and frequency selectivity for reduced overshoot.

Connection to convergence

The Fourier series of a function with bounded jumps still converges (in the $L^2$ sense and pointwise at every continuity point) to the original signal. Gibbs phenomenon is a statement about uniform convergence, which fails near jumps: the partial sums don’t converge uniformly to the function in any neighborhood of a discontinuity. This is a feature of the family of sinusoids, not a defect of the function.