Parseval's Theorem

The energy of a length-$N$ discrete-time signal is defined as:

$$E_x = \sum_{n=0}^{N-1} |x[n]|^2$$

Parseval’s relation links the energy of a signal in the time domain to its energy in the frequency domain. In the equation form, it is expressed as:

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

In other words, the energy of a signal in the time domain is equal to its energy in the transform domain after scaling by $1/N$.

Derivation

From the definition of the inverse discrete Fourier transform, we have:

$$x[n]=\frac{1}{N}\sum_{k=0}^{N-1}X[k]e^{+j2\pi\frac{k}{N}n}$$ ...