Parseval's Theorem: effect of windowing, step size, and spectral leakage

Hey Mike,

I was wondering if there’s an effect of windowing and/or spectral leakage on Parseval’s Theorem? Moreover, if you run wavelet convolution or multitaper, is it possible to recover the power estimates computed using the FFT by integrating power across time? If so, how could I prove it and how would I factor in window overlap/step size?



Windowing doesn’t change Parseval’s theorem. You just need to keep in mind that the energy in the time-domain signal is no longer the energy in the original signal; it’s the energy in the windowed and tapered signal.

I’m sure it’s possible to work out exactly how this gets implemented, and maybe that’s been done in some old, obscure paper. But I guess it’s a small headache to work out all the details including the energy loss from the windowing, etc. And the normalization is also tricky here, as you now know from your other post :wink:

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