A STATE-VARIABLE APPROACH TO THE SOLUTION OF FREDHOLM INTEGRAL EQUATIONS.
Abstract
A method of solving Fredholm integral equations of the second kind by state-variable techniques is presented. The principal advantage of this method is that it leads to efficient computer algorithms for calculating numerical solutions. The only assumptions that are made are (a) the kernel of the integral equation is the covariance function of a random process; (b) this random process is the output of a linear system having a white noise input; (c) this linear system has a finite dimensional state-variable description of its input-output relationship. Both the homogeneous and nonhomogeneous integral equations are reduced to two linear first-order vector differential equations plus an associated set of boundary conditions. The coefficients of these differential equations follow directly from the matrices that describe the linear system. In the case of the homogeneous integral equation, the eigenvalues are found to be the solutions to a transcendental equation. The eigenfunctions also follow directly. In the case of the nonhomogeneous equation, the vector differential equations are identical to those obtained in the state-variable formulation of the optimum linear smoother. In both cases analytical and numerical examples are presented. Finally, the optimum linear smoother (unrealizable filter) structure is derived by using a new approach. In this approach, the filter is required to be linear; then the resulting Wiener-Hopf equation is used in conjunction with techniques developed in the report to find the differential equations and boundary conditions specifying the optimum estimate. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Nov 15, 1967
- Accession Number
- AD0666215
Entities
People
- Arthur Bernard Baggeroer
Organizations
- Massachusetts Institute of Technology