Exponential FS coefficients and existence

Exponential Fourier series coefficients

Let $x(t)$ be periodic with period $T$, $\omega_0 = 2\pi/T$. It is expressed as a sum of harmonically related complex exponentials (synthesis equation):

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

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

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

The right-hand integral:

$$\int_0^T e^{j(k-n)\omega_0 t}dt = \left[\frac{e^{j(k-n)\omega_0 t}}{j(k-n)\omega_0}\right]_0^T = \frac{e^{j(k-n)2\pi} - 1}{j(k-n)\omega_0} = 0 \quad (k \ne n)$$

and equals $T$ when $k = n$. Only the $k = n$ term remains:

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

This is the analysis equation. $a_0 = \frac{1}{T}\int_T x(t)dt$ is the average value, and $|a_k|$, $\angle a_k$ give the magnitude and phase spectra.

Conditions for existence (Dirichlet conditions)

Over one period:

1. Absolutely integrable: $\int_T|x(t)|dt < \infty$.
   • Violated by $x(t) = 1/t$ on $0 < t \le 1$ (period 1): $\int_0^1 dt/t = \infty$.
2. Finite number of maxima and minima in a period.
   • Violated by $x(t) = \sin(2\pi/t)$ on $0 < t \le 1$: it oscillates infinitely often near $t = 0$.
3. Finite number of discontinuities, each of finite size, in a period.
   • Violated by a staircase that steps down by half on $[0, \frac12), [\frac12, \frac34), [\frac34, \frac78), \dots$: infinitely many jumps in one period.

A signal that satisfies them: a square wave has two finite jumps per period, is bounded and has no oscillations, so its Fourier series exists; it converges to the midpoint value at each jump.