An efficient method to derive explicit KLT kernel for first-order autoregressive discrete process

Signal dependent Karhunen-Loève transform (KLT), also called factor analysis or principal component analysis (PCA), has been of great interest in applied mathematics and various engineering disciplines due to optimal performance. However, implementation of KLT has always been the main concern. Therefore, fixed transforms like discrete Fourier (DFT) and discrete cosine (DCT) with efficient algorithms have been successfully used as good approximations to KLT for popular applications spanning from source coding to digital communications. In this paper, we propose a simple method to derive explicit KLT kernel, or to perform PCA, in closed-form for first-order autoregressive, AR (1), discrete process. It is a widely used approximation to many real world signals. The merit of the proposed technique is shown. The novel method introduced in this paper is expected to make real-time and data-intensive applications of KLT, and PCA, more feasible.