Exact fast factorizations of the AR(1) Karhunen-Loeve transform

Abstract

We derive a fast factorization of the exact Karhunen--Loève transform (KLT) of an AR(1) source by mapping it onto the Discrete Cosine Transform. For even $N$ and every $\rho \in (0,1)$, the KLT factors exactly into Chen's fast DCT-II structure---butterfly plus fixed half-size DCT-II and DCT-IV cores---completed by two orthogonal corrections: eigenvector matrices of diagonal-plus-rank-one matrices carrying the entire $\rho$-dependence. Fast DCT factorizations are reused unchanged; as $\rho \to 1$ the corrections become identities, recovering Chen's algorithm. At short lengths the design is explicit: at $N=4$ the transform is one butterfly and two rotations with $\tan 2{\psi }_s=-2/(1-\rho )$, $\tan 2{\psi }_a=2/(1+\rho )$; at $N=8$ every correction entry is in radicals, via one quartic serving both branches, and factors into six Givens rotations. Complete and scaled realizations run the exact 8-point KLT in 32 and 24 multiplications, about twice the fixed DCT-II; at $N=4$, a single extra multiplication. At large $N$, fast-multipole application of the corrections yields the exact KLT in $O(N\log N)$ operations to prescribed accuracy; this overhead thus peaks at intermediate sizes and vanishes in both limits.