Applications
In physics and engineering, Parseval's theorem is often written as:
where represents the continuous Fourier transform (in normalized, unitary form) of x(t) and f represents the frequency component (not angular frequency) of x.
The interpretation of this form of the theorem is that the total energy contained in a waveform x(t) summed across all of time t is equal to the total energy of the waveform's Fourier Transform X(f) summed across all of its frequency components f.
For discrete time signals, the theorem becomes:
where X is the discrete-time Fourier transform (DTFT) of x and Φ represents the angular frequency (in radians per sample) of x.
Alternatively, for the discrete Fourier transform (DFT), the relation becomes:
where X is the DFT of x, both of length N.then:0
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