The Two-Dimensional Power Spectral Density: A Connection Between 2-D Rational Functions and Linear Systems

Author

Pulford, Graham W.

Author_Institution

Underwater Syst., Brest, France

Volume

56

Issue

7

fYear

2011

fDate

7/1/2011 12:00:00 AM

Firstpage

1729

Lastpage

1734

Abstract

A derivation of the power spectral density (PSD) matrix of a continuous-time, linear, constant-coefficient system with white Gaussian noise input is given. The assumption of wide-sense stationarity is not made. By transforming the two-dimensional (2-D) autocorrelation function (ACF), a 2-D spectral density is obtained that takes the system´s transient behavior into account. The 2-D PSD is a non-separable rational function. The results are applicable to second-order Volterra series representations of nonlinear systems. Simulations are presented for a fourth order system that compare the 2-D PSD amplitude (both theoretical and empirical) with its 1-D counterpart on 50 realizations of a short data segment containing an initial transient.

Keywords

Gaussian noise; Volterra series; continuous time systems; linear systems; matrix algebra; nonlinear systems; 2D autocorrelation function; 2D rational functions; constant-coefficient system; continuous-time system; fourth order system; linear systems; nonlinear systems; second-order Volterra series representations; system transient behavior; two-dimensional power spectral density matrix; white Gaussian noise; Kernel; Laplace equations; Polynomials; Transfer functions; Transforms; Transient analysis; 2-D Laplace transform; 2-D fast Fourier transform (FFT); Autocorrelation function (ACF); S-transform; Volterra series; nonseparable rational function; nonstationary system; power spectral density (PSD); transient response;

fLanguage

English

Journal_Title

Automatic Control, IEEE Transactions on

Publisher

ieee

ISSN

0018-9286

Type

jour

DOI

10.1109/TAC.2011.2135170

Filename

5740950