AI 中文总结
本文针对AR(1)源的KLT，提出了适用于任意阶数N≥2和ρ∈(0,1)的直接递归分解方法，得到O(N log N)复杂度的精确算法，补充了N=2到8的完整模块，突破了需用DCT近似的传统局限。
AI 中文摘要
平稳AR(1)源的Karhunen-Loève变换（KLT）是变换编码的经典最优性基准，其频率是Ray和Driver超越方程的根，因此该变换长期被视为非结构化稠密矩阵。该领域转而研究其快速近似，最著名的是离散余弦变换（DCT），但本文表明这种退缩为时过早。对于任意阶数N≥2和任意相关系数ρ∈(0,1)，精确AR(1) KLT可实现直接递归分解：偶数阶时，该变换可简化为一个蝶形级、两个自身半阶副本，以及由协方差的秩1边界扰动生成的正交校正；奇数阶时，可简化为单侧预测残差的KLT的两个副本，以及吸收中心样本的箭头级，残差KLT也会以半阶递归。所有校正级均为柯西结构，通过快速多极方法应用这些校正可得到复杂度为O(N log N)的精确KLT算法，与FFT和快速正弦变换的阶数相同。该分解通过协方差矩阵的Sherman-Morrison恒等式和经典久期特征值更新得到，当ρ→1时退化为已知的DCT-II的奇偶分裂。作为次要结果，分解常数被证明是ρ的代数函数，对于所有N≤8可表示为根式，同时给出了N=2到8的完整精确KLT模块。
英文摘要
The Karhunen--Loève 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 $ρ\in(0,1)$: 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 $ρ\to1$, to the known parity splittings of the DCT-II. As secondary results, the factorization constants are shown to be algebraic functions of $ρ$---in radicals for all $N\le8$---and complete exact KLT modules for $N=2,\dots,8$ are given.
