# Instantaneous power estimation by discrete hilbert transform - how far does it smooth?

In my research area, instantaneous power in a specific frequency band is commonly estimated by the following procedure:

1. Apply a bandpass filter on the raw signal (e.g. 80-90Hz bandpass).
2. Estimate the analytical signal by DFT (in Matlab: hilbert())
3. Take the absolute of the result.

I'm wondering how far this procedure smooths. In other words, if I perturb x(t) and then recalculate the instantaneous power, what is the most distant point that will be affected? I understand that the answer will depend on the particular bandpass filter applied. Is there an easy way to approach this?

• I'm curious, what "field of science" is it that you're working in? From your description is also seems that a more efficient approach would be to use a complex bandpass filter that automatically results in an analytic signal (and is trivially causal). Finally, for the local "energy" (better: power) you should definitely take the squared magnitude. With the complex bandbass you can also easily analyze the impulse response to answer your question. An ordinary hilbert transformer is acausal and strongly nonlocal. So you'll have to give a threshold for what you count as "affected". – Jazzmaniac Aug 29 '14 at 9:42
• Thank you for your detailed reply and suggestions. It's electrophysiology (for example see scholarpedia.org/article/Hilbert_transform_for_brain_waves). I understand that the Hilbert transform is strongly non-local, but within the described procedure, isn't this non-locality limited by the preceding band-pass filter? – Trisoloriansunscreen Aug 29 '14 at 10:55
• Tal, I'm afraid the bandpass filter won't reduce the nonlocality. In my opinion there are much better methods to get what you want with much more local results, but that's probably not subject of this question. – Jazzmaniac Aug 29 '14 at 15:42