# Confusion understanding parseval theorem?

I have tried to study about parseval theorem multiple times and what i am able to understand is that energy of time domain signal remains same when it is converted to frequency domain

Is my understanding correct??

In the field of DSP, Parseval's theorem merely states that the sum of the magnitudes of $$N$$ $$x[n]$$ time samples squared equals $$1/N$$ times the sum of the magnitudes of $$x[n]$$'s spectral samples squared. That is, $$\sum_{n=0}^{N-1} {|x[n]|^2}= \frac{1}{N}\sum_{m=0}^{N-1} {|X[m]|^2}$$ We can show an example of this in Octave/MATLAB using:

x = [1 2 3 4 5];
Spec = fft(x);
Left_side = sum(x.^2)
Right_side = (1/5) * sum(abs(Spec).^2)

• Great answer! It's worth noting that this assumes a specific normalization of the DFT; different literature and different software definitely use conventions here. For example, the FFTw implementation of the IDFT has no prefactor, whereas the Matlab one most definitely does; I'm sure there's also software that $\sqrt{N}^{-1}$ for both IDFT and DFT. Commented May 28, 2022 at 11:35
• unnecessary nerdage, ignore at will: Parseval himself simply assumed the transform to be unitary, but some DFT implementations aren't. (So, he does have a factor of $1$ or $1/(2\pi)$, depending on whether you use the "physicist's" or the "functional analysists" exponent in the Fourier kernel, $e^{i2\pi ft}$, or $e^{i ft}$, not $1/N$, but also, his result is about the product of two square-integrable continuous-time functions that both have rationally related periodicity and hence Fourier series representation with "shared summand $e^{i f_n t}$", not directly about DFTs being mag-squared.) Commented May 28, 2022 at 11:54
• Sir, isn't the right side of your expression/equation exactly same to formula of energy of signal?? Commented May 29, 2022 at 11:52
• @DSPCS If I'm not mistaken, I believe the left side of the equation is called "the total energy of a finite-duration discrete signal." Commented May 30, 2022 at 9:30