发表机构
Massachusetts Institute of Technology（麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出AR(1)源KLT的精确蝶形分解，利用DCT-II/IV核心与秩一校正，实现O(N log N)复杂度，并推广至大N的精度可控应用。
AI 中文摘要
我们推导了已知相关系数rho的一阶自回归（AR(1)）源的Karhunen-Loeve变换（KLT）的精确蝶形风格分解。对于偶数阶N，KLT矩阵分解为一个蝶形级B_N、固定半尺寸的DCT-II和DCT-IV核心CII_{N/2}和CIV_{N/2}，以及两个正交校正因子Qs_{N/2}和Qa_{N/2}。这些校正是显式对角加秩一矩阵的特征向量矩阵。在极限情况rho -> 1下，校正变为恒等矩阵，分解简化为经典的Chen-Smith-Fralick DCT-II分解。对于N = 4，它简化为一个蝶形和两个平面旋转，具有闭式角度tan(2 psi_s) = -2/(1 - rho)和tan(2 psi_a) = 2/(1 + rho)。在一般情况下，校正因子是分治特征求解器中中心秩一更新特征问题的实例；将快速多极和分层方法应用于它们，我们推导出总体复杂度为O(N log N)的分解，在较大N下实现精确KLT的精度可控应用。提供了所得算法的信号流图。
英文摘要
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 $ρ\in(0,1)$, 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 $ρ$-dependence. Fast DCT factorizations are reused unchanged; as $ρ\to1$ 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 $\tan2ψ_s=-2/(1-ρ)$, $\tan2ψ_a=2/(1+ρ)$; 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.
