RRT: The regularized resolvent transform for high-resolution spectral estimation

A new numerical expression, called the regularized resolvent transform (RRT ), is presented. RRT is a direct transformation of the truncated time-domai n data into a frequency-domain spectrum and is suitable for high-resolution spectral estimation of multidimensional time signals. One of its forms, un der the condition that the signal consists only of a finite sum of damped s inusoids, turns out to be equivalent to the exact infinite time discrete Fo urier transformation. RRT naturally emerges from the filter diagonalization method, although no diagonalization is required. In RRT the spectrum at ea ch frequency s is expressed in terms of the resolvent R(s)(-1) of a small d ata matrix R(s), that is constructed from the time signal. Generally, R is singular, which requires certain regularization. In particular, the Tikhono v regularization, R-1 approximate to [(RR)-R-dagger + q(2)]R--1(dagger) wit h regularization parameter q, appears to be computationally both efficient and very stable. Numerical implementation of RRT is very inexpensive becaus e even for extremely large data sets the matrices involved are small. RRT i s demonstrated using model 1D and experimental 2D NMR signals. (C) 2000 Aca demic Press.