pwelch info: equation involved and problem on the integral

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I would like to reduce the noise in my psd. Given a signal in time u, that has a length of 80001, and decided the length of my window W_len and the fs=1000. I performed this pwelch
[Swelch, fwelch] = pwelch(u, hanning(W_len), W_len*0.5, 4000, 1000,'onesided' )
The problem is that the integral of this psd is different from the original. Is it possible to obtain the same integral with the pwelch?
Finally, I am struggling to find the equation involved in the pwelch. Does it use the correlation theorem, doing |fft(u)|.^2 ???

Accepted Answer

Chunru on 24 Jul 2021
Edited: Chunru on 24 Jul 2021
From "doc pwelch"
Welch’s Overlapped Segment Averaging Spectral Estimation
The periodogram is not a consistent estimator of the true power spectral density of a wide-sense stationary process. Welch’s technique to reduce the variance of the periodogram breaks the time series into segments, usually overlapping.
Welch’s method computes a modified periodogram for each segment and then averages these estimates to produce the estimate of the power spectral density. Because the process is wide-sense stationary and Welch’s method uses PSD estimates of different segments of the time series, the modified periodograms represent approximately uncorrelated estimates of the true PSD and averaging reduces the variability.
The segments are typically multiplied by a window function, such as a Hamming window, so that Welch’s method amounts to averaging modified periodograms. Because the segments usually overlap, data values at the beginning and end of the segment tapered by the window in one segment, occur away from the ends of adjacent segments. This guards against the loss of information caused by windowing.
For the original paper:

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