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I have been reading about the non-parametric estimation of the spectral density, and I'm particularly interested in the Blackman-Tukey method. If we take the rectangular window we will have the truncated periodogram, computed on a reduced covariance sequence length $M$, where $M<N$: $$\hat{P}(\omega)=\sum\limits_{k=0}^{M-1}\hat{r}(k)e^{-jk\omega}$$ where $\hat{r}(k)$ is the estimate of the covariance of the sample and $\omega$ is the frequency of interest. What I'm having trouble is: first, how do we choose our new sample of length $M$? could someone please explain how it does actually work without being too technical, like this explanation on how does the Bartlett method works

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The resolution of the Blackman-Tukey periodogram estimator is $\mathcal{O}(1/M)$, whereas its variance is on the $\mathcal{O}(M/N)$. This compromise between resolution and variance is to be considered when choosing the window’s length.

I have paraphrased Peter Stoica's beautiful explanation from here (check Section 2.5)

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  • $\begingroup$ so we just calculate the periodogram using the truncated estimated covariance? Is there any example that computes the BM periodogram? $\endgroup$ – Toney Shields Oct 20 '17 at 20:29
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    $\begingroup$ Yes, Toney. Sorry, I don't have any examples right now. But a quick google search gives multiple ones. $\endgroup$ – itismeghasyam Oct 20 '17 at 21:13

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