Power spectrum with pwelch & question about Parseval's

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AR 2022-8-26

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评论： AR 2022-8-26

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I have xt, a complex sequence of size length N. I use the pwelch function to compute a PSD, using

[yf, ys] = pwelch(xt, w, 0, fsize, fs, 'centered')

fs is the sampling rate. fsize is the fftsize, which is set such that I have some 300 averages or so (say N/300). w is some window function with the same number of points as fsize. Freq resolution on the spectrum is fs/fsize = fstep, say.

I estimate power in the spectrum as sum(ys)*fstep. I want to estimate the power in xt, and expect to find equivalence (by Parseval's relationship). I estimate this as: sum(abs(xt).^2). The two powers are not equal, and bear no sensible relationship to each other either. I have tried 7 different sequences for xt. Power(xt) > Power(ys) by factors ranging from 216 to 1670.

Further, when I upsample xt by a factor of 4, I find that sum(abs(xt_upsampled).^2) is 4 times greater than sum(abs(xt).^2). This is using the resample command. I thought resample was supposed to normalize the power. I am now very uncertain how to estimate power in the time-domain. Any suggestions? Does pwelch do something that invalidates Parseval's or am I wrongly applying it?

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AR 2022-8-26

I just realized that the power estimate from xt was actually energy, and that I haven't accounted for time. So let me rework that and come back if there's still some problem.

AR 2022-8-26

Following up, once the power estimate is corrected as sum(abs(xt).^2)/(fs*total_time_length), it is correct and matches the spectral estimate. One could also just use the bandpower function. Apologize for not having thought about this properly before asking - please scratch the question if that makes sense.

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