Parseval's Theorem

Discover the relation between signal energy in the time and frequency domains.

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

Ex=n=0N1x[n]2E_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:

n=0N1x[n]2=1Nk=0N1X[k]2\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/N1/N.

Derivation

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

x[n]=1Nk=0N1X[k]e+j2πkNnx[n]=\frac{1}{N}\sum_{k=0}^{N-1}X[k]e^{+j2\pi\frac{k}{N}n} ...