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).