Parseval's Theorem
Discover the relation between signal energy in the time and frequency domains.
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The energy of a length- discrete-time signal is defined as:
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:
In other words, the energy of a signal in the time domain is equal to its energy in the transform domain after scaling by .
Derivation
From the definition of the inverse discrete Fourier transform, we have:
Using this in the expression for the energy of a signal:
From here, we can rearrange the terms after conjugation.
Intuition
Intuitively, going into the frequency domain doesn’t alter the energy contents of the observed signal. In mathematics, this is known as a unitary transformation.
As far as the scaling factor of is concerned, recall that the DFT outputs a sum of complex sinusoids. For a constant input, this kind of sum magnifies the input by . This is why a reduction by is necessary to balance the energy in the two domains. Due to this reason, many software routines define the DFT and inverse DFT with the same scaling factor of .
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