A New Sequential Wiener-Type Filter via Invariant Imbedding
Masanori Sugisaka · Transactions of the Society of Instrument and Control Engineers · 1977
Utilizing the theory of invariant imbedding and Karhunen-Loève expansion, an initial value system suitable for real-time calculation is presented for the solutions of the vector-matrix Fredholm integral equations of linear least squares estimation theory when the signal processes are nonstationary. And a new Wiener-type filter using sequential calculations of eigenvalues and eigenfunctions is proposed.In no few systems for control or communication, the information on convariance is more readily available concerning the signal processes than the state space model. Therefore, under the assumption that the kernel matrices of the Fredholm integral equations, which correspond to the covariance (correlation) matrices of the signal processes, are known a priori, the filtering and fixed-point smoothing algorithms are derived for the estimation problem of the signals with semi-degenerate kernels, using the nonminimal covariance factrization technique showed by R. Brockett and the series expansion of the kernel matrix by vector orthonomal eigenfunctions (Karhunen-Loève expansion).In Appendix the initial value systems for a degenerate kernel and a signal process with band-limited spectrum are briefly given for the scalar estimation problem to show the applicability of the derived algorithms for various types of kernels.Some simulation results from the signal processes with semi-degenerate kernels are given.