Fourier series - by Ray Betz

Fourier Series

If

1. $x(t)=x(t+T)$
2. Dirichlet conditions satisfied

then we can write

${\mathbf {x}}(t)=\sum _{k=-\infty }^{\infty }\alpha _{k}e^{\frac {j2\pi kt}{T}}$

The above equation is called the complex fourier series. Given $x(t)$, we may determine $\alpha _{k}$ by taking the inner product of $x(t)$ with $\alpha _{k}$.