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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.

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

1 Answer 1

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My intuition about the equation is how flat is the auto correlation of the window. Assume we have a a flat auto correlation, then each sample is like the peak which is the sum of square of the window as in the numerator. But auto correlation falls, so this gets bigger value than 1. The less flat it is the bigger the ratio.

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