Optimal Filtering, Softcover reprint of the original 1st ed. 1999
Volume II: Spatio-Temporal Fields
Mathematics and Its Applications Series, Vol. 481

In this volume the investigations of filtering problems, a start on which has been made in [55], are being continued and are devoted to theoretical problems of processing stochastic fields. The derivation of the theory of processing stochastic fields is similar to that of the theory extensively developed for stochastic processes ('stochastic fields with a one-dimensional domain'). Nevertheless there exist essential distinctions between these cases making a construction of the theory for the multi-dimensional case in such a way difficult. Among these are the absence of the notion of the 'past-future' in the case of fields, which plays a fundamental role in constructing stochastic processes theory. So attempts to introduce naturally the notion of the causality (non-anticipativity) when synthesising stable filters designed for processing fields have not met with success. Mathematically, principal distinctions between multi-dimensional and one-dimensional cases imply that the set of roots of a multi-variable polyno­ mial does not necessary consist of a finite number of isolated points. From the main theorem of algebra it follows that in the one-dimensional case every poly­ nomial of degree n has just n roots (considering their multiplicity) in the com­ plex plane. As a consequence, in particular, an arbitrary rational function ¢(.

1 Fields and means of describing them.- 1.1 Regular fields.- 1.2 Generalized fields.- 1.3 Spatio-temporal fields and frequency-wave fields.- 1.4 Stochastic discrete fields.- 1.5 Proofs of Lemmas and Theorems.- 1.6 Bibliographical Comments.- 2 Models of continuous fields and associated problems.- 2.1 Fields in electrodynamics.- 2.2 Acoustic fields.- 2.3 Parametric vibrations of distributed systems.- 2.4 Proofs of Lemmas and Theorems.- 2.5 Bibliographical Comments.- 3 Filtering of spatio-temporal fields.- 3.1 Linear filters and antenna arrays.- 3.2 Signal optimal detection.- 3.3 Estimation of angles of arrival of local signals.- 3.4 Proofs of Lemmas and Theorems.- 3.5 Bibliographical Comments.- 4 Optimal filtering of discrete homogeneous fields.- 4.1 Optimal filtering of discrete homogeneous fields.- 4.2 Synthesis of optimal physically realizable stationary filter.- 4.3 Optimal prediction of two-dimensional regressive fields.- 4.4 Multi-dimensional factorization and its attendant problems.- 4.5 Proofs of lemmas and theorems.- 4.6 Bibliographical Comments.- A Appendix: Fields in electrodynamics.- A.1 Self-conjugate Laplace operator.- A.1.1 Laplace operator in invariant subspace.- A.1.2 Invariant subspaces of Laplace operator.- A.1.3 Continuous spectrum of Laplace operator.- A.2 Electrodynamic problem in tube domain.- A.2.1 Eigenfields in tube domain.- A.2.2 Example: Oscillations in rectangular resonator.- A.2.3 Example: Rectangular semi-infinite waveguide.- A.3 Proofs of Lemmas and Theorems.- A.3.1 Proof of Lemma A.1.- A.3.2 Proof of Lemma A.2.- A.3.3 Proof of Lemma A.3.- A.3.4 Proof of Lemma A.4.- A.3.5 Proof of Lemma A.5.- A.3.6 Proof of Lemma A.6.- A.3.7 Proof of Theorem A. 1.- A.4 Bibliographical Comments.- B Appendix: Spectral analysis of time series.- B.1 Reconstruction of spectral densities.- B.1.1 Quasi-stationary signals and their power spectra.- B.1.2 Optimal estimation of power spectrum.- B.2 Padé approximation.- B.2.1 Padé approximation of analytic function.- B.2.2 Padé approximation of spectral density.- B.3 Identification of regressive equation.- B.3.1 Optimal prediction.- B.3.2 Estimation of coefficients of regressive equation.- B.4 Proofs of Lemmas and Theorems.- B.4.1 Proof of Lemma B.l.- B.4.2 Proof of Theorem B.l.- B.4.3 Proof of Theorem B.2.- B.4.4 Proof of Theorem B.3.- B.4.5 Proof of Lemma B.2.- B.4.6 Proof of Theorem B.4.- B.4.7 Proof of Lemma B.3.- B.4.8 Proof of Lemma B.4.- B.4.9 Proof of Lemma B.5.- B.4.10 Proof of Lemma B.6.- B.5 Bibliographical Comments.- C Appendix: Spectral analysis of discrete homogeneous fields.- C.1 Latticed cones and functions.- C.1.1 Latticed cones.- C.1.2 Latticed fields.- C.2 Discrete fields.- C.2.1 Generalized discrete fields.- C.2.2 Stochastic fields.- C.3 Latticed cone filters.- C.3.1 Stable linear filters.- C.3.2 Multi-variate analog of Padé approximation.- C.4 Proofs of Lemmas and Theorems.- C.4.1 Proof of Lemma C.l.- C.4.2 Proof of Theorem C.l.- C.4.3 Proof of Lemma C.2.- C.4.4 Proof of Theorem C.2.- C.5 Bibliographical Comments.- References.- Notation.