Direct Factorization of the Karhunen-Lo\`eve Transform of AR(1) Sources

Abstract

The Karhunen--Lo\`eve transform (KLT) of the stationary AR(1) source is the classical optimality benchmark of transform coding. Its frequencies are roots of the transcendental equations of Ray and Driver, and the transform has therefore long been treated as an unstructured dense matrix. Rather than compute the exact KLT, the field turned to fast approximations of it, most famously the discrete cosine transform (DCT). This paper shows that the retreat was premature. The exact AR(1) KLT admits direct recursive factorizations, at every order $N\ge2$ and every correlation coefficient $\rho\in(0,1)$: at even orders, the transform reduces to a butterfly stage, two half-order copies of itself, and orthogonal corrections generated by rank-one boundary perturbations of the covariance; at odd orders, to two copies of the KLT of the one-sided prediction residual and an arrowhead stage absorbing the center sample. The residual KLT recurses through half order as well. All correction stages are Cauchy-structured, and applying them by the fast multipole method yields exact-KLT algorithms of $O(N\log N)$ complexity---the same order as the FFT and the fast sinusoidal transforms. The factorizations follow from the covariance matrix by the Sherman--Morrison identity and the classical secular eigenvalue updates, and degenerate, as $\rho\to1$, to the known parity splittings of the DCT-II. As secondary results, the factorization constants are shown to be algebraic functions of $\rho$---in radicals for all $N\le8$---and complete exact KLT modules for $N=2,\dots,8$ are given.