2.5 Parseval's theorem for the fourier series

Recall that in Chapter 1, we defined the power of a periodic signal as

where $T$ is the period. Using the complex form of the Fourier series, we can write

where we have used the fact that $x{\left(t\right)}^2=x\left(t\right)x{\left(t\right)}^{*}$ , i.e. since $x\left(t\right)$ is real $x\left(t\right)=x{\left(t\right)}^{*}$ . Applying [link] and [link] gives

Substituting this quantity into [link] gives

It is straight-forward to show that

This leads to Parseval's Theorem for the Fourier series:

which states that the power of a periodic signal is the sum of the magnitude of the complex Fourier series coefficients.