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Equation (3) is a practical symmetrical estimation of Equation (2) when the observation time $T$ increases. Equation (2) is a reformulation of Equation (1) using the concept of ergodicity. Here, $X(t)$ is modeled as a continuous random variable. I see it as a variable phenomenon that can takes observed values at time $t$, depending on some probability ...

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The definition of auto-correlation depends on the type of signal. For random processes, the auto-correlation function is defined by the expectation given in Eq. $(1)$ of your question. For deterministic signals, there are two definitions, depending on whether the signal is an energy signal (i.e., has finite energy), or a power signal (i.e., has finite power ...

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I believe the OP's question is simplified to the following (confirming I didn't actually miss the salient question): Given we can compute a power spectral density from the DFT using the conjugate product as: $$S(k) = |X(k)|^2 = X(k)X^*(k)$$ Can we use a more efficient algorithm, such as the Goertzel, to compute a subset of $S(f)$ when only a ...

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The equation looks reasonable to me: The scaling by $\frac{1}{2\pi}$ is to have the result in units of normalized frequency $f$ instead normalized angular frequency $\omega$. What may be confusing is using the index $k$ from the autocorreation instead of $n$ since it would be the time domain variable for $R[k]$ and $w[k]$, while $k$ is often associated with ...

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