Complex Exponential Pseudomodes of LTI Operators Over Finite Intervals

Abstract

In applications including radar and ultrasonic inspection, an observed signal can often be modeled as the output over a finite interval of a linear, time-invariant (LTI) operator having a short-duration impulse response. For processing such as filtering and compression it would be useful to have approximations to eigenfunctions of the operator. It has been shown that discrete-time complex exponentials are asymptotic pseudomodes (i.e., approximate eigenvectors) of sequences of Toeplitz matrices. In this letter we show that complex exponentials over the observation interval are pseudomodes of LTI operators, corresponding to pseudoeigenvalues that are samples from the operator’s frequency response, and how the level of approximation depends on the ratio of the impulse response duration to observation interval length. This implies that the Fourier Series basis for the observation interval can be used as an orthonormal set of approximate eigenfunctions.