Parseval's Theorem

Idea

Parseval's theorem usually refers to the result that the Fourier Transform is unitary. That is, the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform.

General Form

Suppose that $A(x)$ and $B(x)$ are two complex-valued functions on $\mathbb{R}$ of period $2\pi$ that are square integrable over intervals of period length, with Fourier Series

$$A(x)=\sum _{n=-\infty }^{\infty }{a}_n\exp (jnx)$$

and

$$B(x)=\sum _{n=-\infty }^{\infty }{b}_n\exp (jnx)$$

respectively. Then

$$\sum _{n=-\infty }^{\infty }{a}_n{\bar{b}}_n=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }A(x)\overline{B(x)}\mathrm{d}$$

In Signal Processing

In Signals and Systems, the theorem is often written as

$${\int }_{-\infty }^{\infty }{\left|x(t)\right|}^2\mathrm{d}t=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{\left|X(\omega )\right|}^2\mathrm{d}\omega$$

where $X(\omega )=\mathcal{F}\left\{x(t)\right\}$ represents the CTFT of $x(t)$.

For discrete time signals, the theorem becomes:

$$\sum _{n=-\infty }^{\infty }{\left|x[n]\right|}^2=\frac{1}{2\pi }{\int }_0^{2\pi }{\left|X(\omega )\right|}^2\mathrm{d}\omega$$

where $X(\omega )$ is the DTFT of $x$.

Alternatively, for the DFT, the relation becomes

$$\sum _{n=0}^{N-1}{\left|x[n]\right|}^2=\frac{1}{N}\sum _{k=0}^{N-1}{\left|X[k]\right|}^2$$

where $X[k]$ is the DFT of $x[n]$, both of length $N$

And for DFS

$$\frac{1}{N}\sum _{n=0}^{N-1}{\left|x[n]\right|}^2=\sum _{k=0}^{N-1}{\left|X[k]\right|}^2$$