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Unit 3 · Lesson 3-2
Convergence, the Gibbs Phenomenon and Parseval
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Convergence, the Gibbs Phenomenon and Parseval

Dirichlet conditions, mean-square convergence, the 9 % overshoot at a jump, Parseval's relation and smoothing the partial sums

Letters to Nature — 1898 and 1899
In the 1890s Albert Michelson, the physicist of precision measurement, built a mechanical harmonic analyzer: a machine of springs and levers that computed Fourier coefficients and drew the partial sums of the series. He and S. W. Stratton described it in 1898 (Albert A. Michelson). That year Michelson and the mathematician A. E. H. Love argued in the pages of Nature about how the Fourier series of a square wave converges near its jumps. J. Willard Gibbs joined in with a note of his own — and missed the effect. In April 1899 he published a correction describing it: near a jump, the partial sums overshoot, and the overshoot does not die away as terms are added. Maxime Bôcher analysed it in 1906 and named it the Gibbs phenomenon. Henry Wilbraham had in fact discovered and analysed it in 1848, in a paper almost nobody read (Gibbs phenomenon). A precision instrument met a mathematical fact that no precision could remove. This lesson is about that fact, and about the sense in which Fourier series do converge.
The partial sum xN(t)=∑∣k∣≤Nakejkω0tx_N(t) = \sum_{|k| \le N} a_k e^{jk\omega_0 t} is what any real computation uses. When does it approach x(t)x(t)? The Dirichlet conditions give a safe answer for engineering signals: if over one period xx is absolutely integrable and has finitely many maxima, minima and jumps, then
  • at every point where xx is continuous, xN(t)→x(t)x_N(t) \to x(t);
  • at a jump, xN(t)x_N(t) converges to the midpoint of the jump.
So the square wave's series gives exactly 00 at its switching instants — halfway between −1-1 and +1+1.
💡Convergence in energy
There is a second, often more useful sense. For any signal with finite energy per period, the energy of the error goes to zero: 1T∫T∣x(t)−xN(t)∣2 dt  ⟶  0(N→∞).\frac1T\int_T |x(t) - x_N(t)|^2\,dt \;\longrightarrow\; 0 \qquad (N \to \infty). This mean-square convergence holds even where pointwise convergence is awkward. It says the error may stay large at isolated points, but it occupies less and less time — which is exactly what happens at a jump.
The Gibbs phenomenon. Near a jump, xNx_N overshoots: for the ±1\pm1 square wave the first peak of the partial sum reaches about 1.1791.179 instead of 11. Add terms and the peak moves closer to the jump and narrows, but its height does not fall — it tends to about 9 % of the jump on each side (0.179 on a jump of 2). The wiggles carry less and less energy, so mean-square convergence still holds; but the maximum error never goes below 9 % of the jump.
Derivation
Where the 9 % comes from
Write the square wave's partial sum with MM odd harmonics and differentiate term by term:
xN(t)=4π∑m=1Msin⁡((2m−1)ω0t)2m−1,dxNdt=4ω0π∑m=1Mcos⁡((2m−1)ω0t)=2ω0π sin⁡(2Mω0t)sin⁡(ω0t).x_N(t) = \frac4\pi\sum_{m=1}^{M}\frac{\sin\big((2m-1)\omega_0 t\big)}{2m-1}, \qquad \frac{dx_N}{dt} = \frac{4\omega_0}{\pi}\sum_{m=1}^{M}\cos\big((2m-1)\omega_0 t\big) = \frac{2\omega_0}{\pi}\,\frac{\sin(2M\omega_0 t)}{\sin(\omega_0 t)} .
The first zero of the derivative after the jump at t=0t = 0 — the first peak — is at 2Mω0t=π2M\omega_0 t = \pi. Integrate from the jump to there, substituting u=2Mω0tu = 2M\omega_0 t and using sin⁡(ω0t)≈ω0t\sin(\omega_0 t) \approx \omega_0 t for large MM:
xN(tpeak)≈2π∫0πsin⁡uu du=2π Si(π)=2π×1.85194=1.17898.x_N(t_{peak}) \approx \frac{2}{\pi}\int_0^{\pi}\frac{\sin u}{u}\,du = \frac2\pi\,\mathrm{Si}(\pi) = \frac{2}{\pi}\times 1.85194 = 1.17898 .
The integral no longer depends on MM: the peak height is fixed while its position tpeak=π/(2Mω0)t_{peak} = \pi/(2M\omega_0) shrinks towards the jump. The overshoot 0.1790.179 is 8.95 %8.95\,\% of the jump of 2.
💡Parseval's relation
The average power of a periodic signal equals the sum of the powers of its harmonics: 1T∫T∣x(t)∣2 dt=∑k=−∞∞∣ak∣2.\frac1T\int_T |x(t)|^2\,dt = \sum_{k=-\infty}^{\infty} |a_k|^2 . It follows from orthogonality: expanding ∣x∣2=x x∗|x|^2 = x\,x^* as a double sum, every cross term between different harmonics integrates to zero. The list of ∣ak∣2|a_k|^2 against frequency is the signal's power spectrum — where its power lives.
📐Where a square wave keeps its power
The ±1\pm1 square wave has average power 11. Its harmonics have ∣ak∣2=4/(π2k2)|a_k|^2 = 4/(\pi^2 k^2) for odd kk, counted twice (±k\pm k):
  • the fundamental alone: 2×4π2=8π2=81.1 %2 \times \dfrac{4}{\pi^2} = \dfrac{8}{\pi^2} = 81.1\,\% of the power;
  • harmonics 1, 3 and 5: 8π2(1+19+125)=93.3 %\dfrac{8}{\pi^2}\left(1 + \tfrac19 + \tfrac1{25}\right) = 93.3\,\%;
  • all odd harmonics: 8π2∑k odd1k2=8π2⋅π28=100 %\dfrac{8}{\pi^2}\sum_{k\ \text{odd}} \dfrac{1}{k^2} = \dfrac{8}{\pi^2}\cdot\dfrac{\pi^2}{8} = 100\,\%.
The last line is a famous sum obtained for free: 1+19+125+⋯=π2/81 + \tfrac19 + \tfrac1{25} + \cdots = \pi^2/8. Most of the power sits in the first few harmonics, even though the jumps need infinitely many to draw.
Taming the overshoot. The overshoot comes from cutting the series off abruptly at NN. Taper the coefficients instead and it shrinks. Fejér's averaging of the partial sums (Cesàro means) removes the overshoot entirely, at the price of a softer jump. Lanczos's σ-factors, multiplying aka_k by sinc⁡(k/N)\operatorname{sinc}(k/N), cut it to about 1 % while keeping the jump fairly sharp. The same trade — sharpness against ripple — reappears in every window of Lesson 6-3 and every FIR filter of Lesson 9-1.
The whole is simpler than its parts.
– J. Willard Gibbs, as quoted by Irving Fisher, Bulletin of the American Mathematical Society 36 (1930), on Wikiquote
🕰Wilbraham, Gibbs, Bôcher
Henry Wilbraham analysed the overshoot in an 1848 paper that attracted little attention until it was mentioned in a 1914 review. Gibbs's 1898 note in Nature missed it; his correction of 27 April 1899 described it. Maxime Bôcher's detailed analysis of 1906 coined the name "Gibbs phenomenon", and the constant ∫0πsin⁡uudu\int_0^\pi \tfrac{\sin u}{u}du is sometimes called the Wilbraham–Gibbs constant.
💡Ringing you can see
Compress a photograph of black text on white paper hard enough and faint ripples appear beside every letter. The compressor kept only the low spatial frequencies of each block, and the edges ring — the Gibbs phenomenon in two dimensions. Audio codecs fight the same effect in time as "pre-echo" before sharp attacks.
💡Where you will use this
Parseval returns for the Fourier transform (Lesson 4-2) and the DFT (Lesson 6-2). Truncation ripples are the spectral leakage of Lesson 6-3 and the passband ripple of window-designed FIR filters in Lesson 9-1. Mean-square convergence is why least-squares fitting with sinusoids works.
📚Further reading
  • Oppenheim and Willsky, Signals and Systems, section 3.4 — convergence of the Fourier series and the Gibbs phenomenon.
  • Edwin Hewitt and Robert E. Hewitt, "The Gibbs–Wilbraham phenomenon: an episode in Fourier analysis", Archive for History of Exact Sciences 21 (1979) — the full story.
  • Gibbs phenomenon on Wikipedia.
↓But there is a problem
We now know what a Fourier series is, how it converges and where its power lives. Lesson 3-1 promised the reward: an LTI system multiplies each harmonic by H(jkω0)H(jk\omega_0), and that is all it does. So what happens when a square wave goes through a filter — or when a power grid full of rectifiers carries currents that are anything but sinusoidal? Harmonic by harmonic, the answer is a list of multiplications.