Fourier Series and Gibbs Phenomenon

Concept Overview

The Gibbs Phenomenon describes the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity. Specifically, the partial sums of the Fourier series exhibit large oscillations near the jump, which overshoot the function's values. Even as the number of terms approaches infinity, this overshoot does not disappear; instead, the peak of the overshoot moves closer to the discontinuity but its height approaches a non-zero limit of approximately 8.949% of the jump.

Mathematical Model

Consider a 2π-periodic square wave defined by:

f(θ) = A · sign(sin(θ))

The jump at θ = 0 is 2A (from -A to A). The Fourier series partial sum S_N(θ) with N non-zero odd terms is given by:

S_N(θ) = (4A/π) · Σ _j=0^N-1 sin((2j+1)θ) / (2j+1)

The first positive peak of S_N(θ) occurs exactly at θ = π / (2N). The percentage overshoot is the peak's excess above A, divided by the jump size 2A.

Worked Example

Let's calculate the approximation for 1 term and then 3 terms, given A = 1 and Probe θ/π = 0.5.

Learn more about the Gibbs Phenomenon below:

For N = 1: The sum is (4/π)sin(θ). At θ = 0.5π, sin(0.5π) = 1. S₁(0.5π) = 4/π ≈ 1.2732. The peak is at π / (2·1) = 0.5π. The excess is (4/π) - 1, and as a percentage of the jump (2): ((4/π - 1) / 2) × 100% ≈ 13.66%.
For N = 3: S₃(θ) = (4/π) · [sin(θ) + sin(3θ)/3 + sin(5θ)/5]. At θ = 0.5π, we have sin(0.5π) = 1, sin(1.5π) = -1, and sin(2.5π) = 1. S₃(0.5π) = (4/π) · (1 - 1/3 + 1/5) = 52 / (15π) ≈ 1.1035. The first peak occurs at π / (2·3) = π/6. Evaluating S₃(π/6) = (4/π) · [1/2 + 1/3 + 1/10] = 56 / (15π) ≈ 1.1884. The percentage overshoot is ((56/(15π) - 1) / 2) × 100% ≈ 9.42%.

Experiment

You can verify this in the Fourier Series and Gibbs Phenomenon module.

Set the Amplitude A to 1.
Set the Odd-term count N to 1.
Set the Probe theta / pi to 0.5.
Observe the partial sum is approximately 1.2732 and the peak overshoot is 13.66%.
Now set Odd-term count N to 3. The peak moves to 0.167π, but the overshoot remains around 9.42%.
Set Odd-term count N to 2 while Amplitude A = 1 and Probe theta / pi = 0.5. The sum is 8/(3π), about 0.848826363157, while the first peak is 8√2/(3π), about 1.20042175488.
Set Odd-term count N to 20. The near-jump plot marks the first peak at theta / pi = 0.025 and computes its excess rather than forcing the limiting percentage.
Set Probe theta / pi to 0, then -1 and 1. Target midpoint and sum are 0 at each jump. This convention does not mean the continuous finite sum reproduces the one-sided limits there.
Set Amplitude A to 0. Target, sum, peak value and error are 0, while normalized overshoot is unavailable. Restore A = 1, clear Odd-term count N, observe the error and absent outputs, then restore N = 1.

Limitations

The simulation uses a finite sum up to N = 20. The asymptotic 8.95% overshoot is the theoretical limit as N → ∞ and is not artificially forced on finite calculations here. Displayed decimals have eleven significant digits and calculations use floating-point arithmetic. The input theta / pi is dimensionless; sine receives the corresponding angle in radians. Error means absolute difference from the declared target. The two plots have independent labelled scales; target jump segments are drawn separately, with midpoint markers. At amplitude 0 the wave is flat, so there is no isolated first peak. The N used here counts terms; the cited source indexes its terms from 0 through N, so its N + 1 equals this module’s term count.

References

Learn more about the Gibbs Phenomenon below:

Fourier Series & Gibbs Phenomenon (HMC Physics)

Fourier Series and Gibbs Phenomenon