Notes on Discrete Karhunen-Loeve Transform (DKLT) and DFT

• In general, the DKLT is obtained from eigenvalue decomposition.
• The correlation matrix of a stationary process is Toeplitz.
• If the autocorrelation sequence of a random process is periodic with fundamental period $M$, its correlation matrix becomes circulant.
• The DFT provides the DKLT of periodic random sequences.
• This can be easily seen because DFT defines the complete set of eigenvectors for all circulant matrices.