Parseval identity for Fourier series ($T^{-1}{\int }_0^T|f{|}^2={\sum }_{r\in \mathbb{Z}}|u_r{|}^2$)

For a square-integrable function of period $T$, use Fourier coefficients $u_r=T^{-1}{\int }_0^{T}f(t)e^{-2\pi irt/T}dt$. Fourier orthogonality proves the displayed identity for a finite trigonometric polynomial. The mean-square convergence of the Fourier series then passes it to the limit. It includes the zero-frequency contribution $|u_0{|}^2$; excluding that contribution gives the variance rather than the full mean-square signal power.