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In Welch's method a time-series is segmented and then the subsequent FFTs of the segments are averaged. Is there an equivalent time-domain operation?

I guess doing the the Welch estimate and producing an impulse response using an FIR filter might do what I want, but I'm looking for a way to do it in the time-domain directly (getting rid of noise, keeping signal properties)...

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Yes, you can average in time lag (autocorrelation) as well as in frequency. You can also combine the two domains, as long as it is reasonable to treat the data as being stationary.

A. H. Nuttall and G. C. Carter, "Spectral estimation using combined time and lag weighting," in Proceedings of the IEEE, vol. 70, no. 9, pp. 1115-1125, Sept. 1982.

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You can estimate the power density spectrum of a (WSS) random process by first estimating its ACS (autocorrelation sequence), $\phi_{xx}[m]$, and then taking the Fourier transform of the windowed ACS estimate.

Note that one uses a window to capture a block of ACS, just like in the conventional (modified) periodogram based spectral estimation where a block of $x[n]$ is captured by the window. However there is an important difference here.

The spectrum estimate should posses the general properties of a valid power spectrum: being nonnegative, being even symmetric etc. Now in the frequency domain techniques (periodogram, averaged periodogram (Welch)) this is always the case with any real and even window $w[n]$ selection. However in the autocorrelation estimation based method, it's not guaranteed that the resulting spectrum estimate will be nonnegative. Therefore In order to guarantee that the estimated spectrum is nonnegative (valid), not every window but those with a specific charather should be used.

The characteristic (of the window) which yields an always nonnegative (valid) spectrum estimate is that window's Fourier transform be nonnegative; $W(e^{j\omega}) \geq 0$ for all $\omega$. And this condition will be satisfied whenever the window $w[n]$ is obtained by $w[n] = v[n] \star v[-n]$ for an arbitrary $v[n]$. When this is the case, the obtained window will have a nonnegative Fourier transform as can be seen: $$W(e^{j\omega}) = V(e^{j\omega}) V(e^{j\omega})^* = |V(e^{j\omega})|^2 \geq 0$$

which will yield a nonnegative spectrum estimate. Of course $v[n]$ should posses other properties so that $w[n]$ will posses desirable features to improve the estimate. Note that a Bartlett (triangular) window satisfies this condition, but rectengular, Hamming, Hanning, Kaiser etc., do not.

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