Does the PWELCH function provide unbiased estimation of the power spectral density?

I would like to know if I need to multiply the result of PWELCH by "norm(w)^2/sum(w)^2", where w is the window vector?

Also, I would like to know if the old function PSD can be used together with "norm(w)^2/sum(w)^2" to get the unbiased estimation of the power spectral density.

Answer 1

[MathWorks Support Team]

The PWELCH function implements Welch's method of power spectrum estimation.

Welch's method yields a biased estimate, (see for example Hayes, M. Statistical Digital Signal Processing and Modeling, Wiley, 1996, pg. 419) although it is unbiased when the data is white noise, and asymptotically unbiased for any stationary data.

Under certain conditions - namely, if the width of window is narrow with respect to the spacing between a certain spectral peak and other spectral features - the bias can be approximated to (sum(w)^2/norm(w)^2)*unbiased value.

However, this does NOT mean that you can eliminate the bias of a particular estimate by accounting for the scale factor. It means that ON AVERAGE, if you scale appropriately, your estimate will be approximately unbiased. How that would affects a particular estimate depends on the variance of your estimator.

For practical purposes this is only useful if the variance of your estimator is small. The variance of Welch's method depends on both the overlap used and the window chosen for the estimate (see reference above).