## Optimal Filtering: Volume II: Spatio-Temporal FieldsIn 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 ¢(. |

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### Contents

Models of continuous fields and associated problems | 43 |

Filtering of spatio temporal fields | 161 |

Optimal filtering of discrete homogeneous fields | 221 |

Copyright | |

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acoustic algorithm antenna array arbitrary assumed boundary conditions boundary value problem bounded coefficients considered continuous spectrum convergent correlation function correlation operator corresponding defined denote desired signal determined differential discrete field divergenceless domain eigen-elements eigenfields eigenvalues electrodynamic electromagnetic field element estimates evolutionary equation expansion expressible factorization finite number formula Fourier transform frequency given in Section Green's function Hilbert space homogeneous field implies inequality integral Laplace operator latticed cone latticed set matrix-valued function method multi-index non-negative obtain optimal filter Pade approximation parametric resonance plane wave polynomial proof of Lemma proof of Theorem quasi-polynomial random rational function realizations regressive equation relation resonance rot rot Ry[t satisfying the condition scalar sequence smooth solution spacial spatially white field spatio-temporal field spectral density spectral density Gy stationary process stationary time series stochastic field stochastic processes subset subspace transfer function valid variables vector virtue waveguide weight function zero