FS equations and existence conditions

A periodic signal $x(t)$ with period $T$ and fundamental frequency $\omega_0 = 2\pi/T$ can be written as a sum of harmonically related complex exponentials (the exponential Fourier series).

Synthesis equation

The functions $e^{jk\omega_0 t}$, $k = 0, \pm1, \pm2, \dots$, all repeat every $T$, so

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

is periodic with period $T$. This is the synthesis equation; $a_k$ are the Fourier series coefficients.

Analysis equation

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

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

Orthogonality of the exponentials:

$$\int_0^T e^{j(k-n)\omega_0 t}dt = \begin{cases} T, & k = n \\ 0, & k \ne n \end{cases}$$

(for $k \ne n$, the cosine and sine parts complete a whole number of cycles and integrate to zero). Hence

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

This is the analysis equation; $a_0 = \frac{1}{T}\int_T x(t)dt$ is the dc value.

Conditions for existence (Dirichlet conditions)

Over any one period:

1. Absolutely integrable: $\int_T |x(t)|\,dt < \infty$, which guarantees every $|a_k|$ is finite.
2. Finite number of maxima and minima (bounded variation) in one period.
3. Finite number of discontinuities in one period, and each discontinuity is finite.

When these hold, the series converges to $x(t)$ at every point where $x(t)$ is continuous, and to the average $\frac{1}{2}[x(t^-) + x(t^+)]$ at a discontinuity. (A weaker condition, finite energy over one period, $\int_T|x(t)|^2dt < \infty$, guarantees convergence in the mean-square sense.)