Exponential FS coefficients; Gibbs phenomenon

Exponential Fourier series coefficients

A periodic signal $x(t)$ with period $T$ and $\omega_0 = 2\pi/T$ is written as

$$x(t) = \sum_{k=-\infty}^{\infty}a_k e^{jk\omega_0 t} \qquad \text{(synthesis)}$$

Multiply both sides by $e^{-jn\omega_0 t}$ and integrate over one period $T$:

$$\int_0^T x(t)e^{-jn\omega_0 t}dt = \sum_{k=-\infty}^{\infty}a_k\int_0^T e^{j(k-n)\omega_0 t}dt$$

For $k \ne n$:

$$\int_0^T e^{j(k-n)\omega_0 t}dt = \frac{e^{j(k-n)\omega_0 T} - 1}{j(k-n)\omega_0} = \frac{e^{j2\pi(k-n)} - 1}{j(k-n)\omega_0} = 0$$

For $k = n$ the integral is $\int_0^T 1\,dt = T$. Therefore

$$\int_0^T x(t)e^{-jn\omega_0 t}dt = a_n T \quad\Rightarrow\quad a_k = \frac{1}{T}\int_T x(t)e^{-jk\omega_0 t}dt \qquad \text{(analysis)}$$

• $a_0 = \frac{1}{T}\int_T x(t)dt$ is the dc value.
• $a_k$ is complex: $|a_k|$ is the magnitude spectrum and $\angle a_k$ the phase spectrum.
• For real $x(t)$, $a_{-k} = a_k^*$.

Example: a square wave of height 1 for $|t| < T_1$ in period $T$ gives $a_0 = 2T_1/T$ and $a_k = \frac{\sin(k\omega_0 T_1)}{k\pi}$.

Gibbs phenomenon

When a signal with a jump discontinuity (e.g. a square wave) is approximated by a finite number of Fourier terms,

$$x_N(t) = \sum_{k=-N}^{N}a_k e^{jk\omega_0 t},$$

the partial sum shows ripples near the discontinuity, with an overshoot of about 9% of the jump height on each side of it.

Key features:

• As $N$ increases, the ripples become narrower and crowd closer to the discontinuity, but the peak overshoot does not decrease; it stays at about 9% of the jump (about 1.09 times for a unit jump).
• At the discontinuity itself, the series converges to the midpoint $\frac{1}{2}[x(t^-) + x(t^+)]$.
• The energy of the error $\int_T|x(t) - x_N(t)|^2dt \to 0$ as $N \to \infty$ (mean-square convergence), because the area under the ripples shrinks.
• It was explained by J. W. Gibbs (1899) after Michelson noticed the overshoot in a mechanical harmonic analyser.

 x_N(t) near a jump (square wave)
       overshoot ~9%
        _/\_/\/\/\________
       |
       |   finite N ripples
 ______|
  \/\/\
       ^ jump at t = T1

Practical meaning: truncating a Fourier series (or ideal filtering, which cuts off high harmonics) produces ringing near sharp edges; window functions are used to reduce it.