Gibbs Phenomenon

The Gibbs phenomenon is the persistent overshoot that appears when you approximate a discontinuous signal using a finite number of Fourier series terms. It affects every signal processing system that truncates a spectrum — from audio equalizers to digital filters — and explains why sharp cutoff filters ring in the time domain.

Core Concept

When you truncate a Fourier series at the Nth harmonic, you are multiplying the true (infinite) coefficient sequence by a rectangular window. In frequency terms, a rectangular window corresponds to a sinc function in time. Convolution of the original signal with this sinc function causes ringing near every discontinuity.

As N increases, the overshoot spike gets narrower and moves closer to the discontinuity, but its height does not decrease. It converges to approximately 8.9% of the magnitude of the jump. For a square wave of height A, the overshoot settles at about 0.089A above A and 0.089A below zero. This result was first rigorously explained by J. Willard Gibbs in 1899.

The Gibbs phenomenon cannot be fixed by adding more harmonics. It can only be reduced by changing the window. Replacing the rectangular truncation with a smoother window — such as the Hanning, Hamming, or Blackman window — tapers the high-frequency coefficients gradually. This reduces the overshoot at the cost of a less sharp transition in the approximation.

Key Formula

The N-term partial sum of the square wave Fourier series is S_N(t) = (4A/π) Σ sin(kω0t)/k for odd k from 1 to N. The maximum value of S_N near the discontinuity approaches A · Si(π)/π/2 as N → ∞, where Si(π) = ∫₀^π sinc(u) du ≈ 1.8519. The overshoot percentage = [Si(π)/π - 1/2] · 2 · 100% ≈ 8.9%. Here Si(π) is the sine integral evaluated at π.

Problem: Verify Gibbs overshoot for a square wave (A=1, T=2π, ω0=1).

Step 1: Write partial sum with N terms (odd k only).
         S_N(t) = (4/π) Σ sin(kt)/k  for k=1,3,5,...,N

Step 2: The maximum of S_N occurs near t = π/N.
         As N → ∞, this peak approaches:
         S_peak = (2/π) ∫₀^π sin(u)/u du
                = (2/π) · Si(π)

Step 3: Evaluate Si(π) numerically.
         Si(π) = ∫₀^π sin(u)/u du ≈ 1.8519

Step 4: Compare peak to ideal step height A = 1.
         S_peak = (2/π) · 1.8519 ≈ 1.1789/π · π ≈ 1.0895
         Wait: S_peak = 2 · Si(π) / π = 2 · 1.8519 / π ≈ 1.179 / 1 ≈ actually:
         S_peak = (4/π) · (π/2) · (2/π) · Si(π) ... simplify:
         Peak of (4A/π) ∑ sin(kt)/k → (2A/π)·Si(π)
         For A = 1: peak ≈ (2/π)·1.8519 ≈ 1.1789/π... 
         Correct evaluation: (2/π)·Si(π) = 2·1.8519/3.1416 ≈ 1.0895

Step 5: Overshoot = 1.0895 - 1 = 0.0895, i.e., 8.95% of jump height.

Final Answer:
  Overshoot ≈ 8.9% of step height, independent of N.

Exam Tip: GATE questions on Gibbs phenomenon usually ask the overshoot percentage (8.9% or 9%), what happens as N increases (overshoot narrows but does not decrease), and which window reduces it. The answer to window choice is any smooth non-rectangular window such as Hanning or Hamming. Remember: Gibbs phenomenon occurs only at jump discontinuities, not at smooth peaks.

Properties Summary

• Overshoot magnitude: approximately 8.9% of the jump height at each discontinuity.
• Independence of N: the overshoot percentage does not decrease as more harmonics are included.
• Location: the overshoot spike moves toward the discontinuity as N increases but never vanishes.
• Cause: rectangular spectral truncation is equivalent to sinc convolution in time, causing ringing.
• Reduction: smooth window functions (Hanning, Hamming, Blackman) taper coefficients and reduce overshoot.
• Convergence: S_N(t) converges to [x(t+) + x(t-)]/2 at discontinuities, the midpoint of the jump.

Quick Revision

• Gibbs overshoot = 8.9% of the jump height, always.
• Adding more Fourier terms narrows the spike but does not reduce its height.
• At a discontinuity, the partial sum converges to the average of the left and right limits.
• Rectangular truncation in frequency ↔ sinc ringing in time.
• Smooth windows (Hanning, Hamming) reduce Gibbs overshoot at the cost of wider transition bands.
• Gibbs phenomenon occurs only where x(t) has a jump discontinuity — not at smooth extrema.
• Digital FIR filter design uses windowing to control Gibbs-type ripple in the frequency response.
• Exam trap: stating that increasing N eliminates the Gibbs overshoot — it does not; only the width shrinks, not the height.