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Consider a set of $p$ stationary correlated time signals of identical duration $T$. I would like to de-correlate them by projection on an orthogonal basis but preserving stationarity. Without the stationarity constraint, a principal component analysis would produce the desired basis. I figure that a functional principal component analysis may allow to impose the stationarity constraint.

It seems that in some cases stationarity of the principal component will ensure from a “side effect” (see this question Fourier bases for a stationary signal & relation to PCA for natural images).

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The link in your question basically addresses the question you posed. Stationary time series on the same duration, sampled in the same way, will tend to decompose into sines and cosines. Conversely, if you want to ensure the stationarity constraint, doing a Fourier analysis would achieve that result.

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