Fourier series - by Ray Betz

Fourier Series

If

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

then we can write

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

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 ${\alpha }_k$.