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

Abstract

A new numerical expression, called the regularized resolvent transform (RRT), is presented. RRT is a direct transformation of the truncated time-domain data into a frequency-domain spectrum and is suitable for high-resolution spectral estimation of multidimensional time signals. One of its forms, under the condition that the signal consists only of a finite sum of damped sinusoids, turns out to be equivalent to the exact infinite time discrete Fourier transformation. RRT naturally emerges from the filter diagonalization method, although no diagonalization is required. In RRT the spectrum at each frequency s is expressed in terms of the resolvent R(s)(-1) of a small data matrix R(s), that is constructed from the time signal. Generally, R is singular, which requires certain regularization. In particular, the Tikhonov regularization, R(-1) approximately [R(dagger)R + q(2)](-1)R(dagger) with regularization parameter q, appears to be computationally both efficient and very stable. Numerical implementation of RRT is very inexpensive because even for extremely large data sets the matrices involved are small. RRT is demonstrated using model 1D and experimental 2D NMR signals. Copyright 2000 Academic Press.