# How to understand "independent sample rate" of windows?

A window metric called independent sample rate is given in a book that I recently read. It says that in spectral estimation, the variance of power-level estimates is inversely proportional to the number of independent samples in the analysis window, which is given by

$$M\eta_w/N$$

where $$\eta_w$$ is the independent sample rate, $$M$$ is the data length and $$N$$ is the window length. For rectangular and Hamming window, $$\eta_w=1.5$$ and $$\eta_w=1.91$$, respectively.

If spectrally white noise is assumed in the band and a large analysis time, the independent sample rate is given by, as indicated by the book,

$$\eta_w=N\frac{[\Sigma_{n=0}^{N-1}w^2(n)]^2}{\Sigma_{n=-(N-1)}^{N-1}R_w^2(n)}, (*)$$

where

$$R_w(n)=w(n)*w(-n)$$

is the autocorrelation function of the window $$w(n)$$.

Concerning the above description, my question is

1. why is the independent sample rate not related to the overlap percent of windows between successive data segments. By intuition, I think more independent data samples can be given if proper overlap is applied.
2. How is the equation (*) derived?

P.S.: The equation (*) is derived in Albert IH. Nuttall - SIGNAL-TO-NOISE RATIOS REQUIRED FOR SHORT-TERM NARROWBAND DETECTION OF GAUSSIAN PROCESSES document.

• Sometimes old concepts get new names. Then confusion ensues. Aug 7, 2023 at 5:15