Estimating the basis functions of the Karhunen-Loeve transform

A procedure for estimating the basis functions of the Karhunen-Loeve (KL) transform and the associated eigenvalues is presented. This estimator differs from previous results in the literature in that the continuous KL basis functions are estimated, allowing the direct evaluation of the effects of sampling on estimator performance. Theorems describing the convergence of the estimated eigensolution are given. These theorems provide a sound theoretical foundation for the estimation procedure and add insight into the quality of the estimates and the effects of sampling. The previous result of estimating the KL transform using the eigensolution of the sample covariance matrix can be formulated as a special case of the proposed procedure, proving the convergence of this previous result. An example that illustrates the procedure and provides a comparison to previous results is presented