Plenary Lecture
General Problems of the Sampling-Reconstruction Procedure of Random Field Realizations
Professor Vladimir A. Kazakov
National Polytechnic Institute of Mexico
ESIME-Zacatenco, SEPI, Department of Telecommunications
Mexico
E-mail: vkaz41@hotmail.com
Abstract: There are a lot of publications devoted to a statistical description of the Sampling-Reconstruction Procedure (SRP) of random process and field realizations. Unfortunately, this problem is completely not solved until present time. This statement is especially related to the SRP of random fields because their statistical description is more difficult with comparison of random processes. Usually papers devoted to the SRP of random field realizations do not use information about the probability density function (PDF) of sampled fields. In order to overcome this principal drawback we apply the conditional mean rule (CMR) for our investigations.
We consider two different mathematical models of random fields. The first model is the Gaussian field with various types of space covariance functions. The realizations of such fields are continuous. The second model is related with fields having random jumps from one state to another. Such fields can have different number of states and their points of jumps can be described by various flows. Here the simplest case is characterized by Poison’s flow. The SRP of Gaussian field realizations are shortly described by the following manner. The field is described by the multidimensional GaussianPDF. Generally this PDF describes both stationary and non-stationary fields. The non-stationary field can be characterized by changed mathematical expectation, variance and space covariance function. Besides this, we need to know the location of samples. There are some variants of location: triangular, square, pentagonal, and arbitrary. Applying the CMR we obtain the optimal reconstruction surface and the error reconstructionsurface. The SRP of Gaussian field realizationsare completely described by these two surfaces.
The SRP of fields with jumps is investigated by another methodology. In fact, such fields are characterized not only by their covariance functions. Here it is necessary to have a description of locations of random jump pointsin the field. The simple variant of a mathematical model of such field if related with Markov binary processes determined along of both axes. In this case we have a model like chess board with random rectangles. If we fix some functions of the Markov binary process realizations along the both axes then we can have the field with four possible states. Besides this we need to know the location of samples. Once again, using CMR for the estimation of jump points one can describe the surface reconstruction and the error reconstruction surface. Results of proposed investigations have a practical application in two variants: 1) if we know the description of field and the location of samples, then one can obtain reconstruction surface and the error reconstruction surface; 2) if we have a required value of the error reconstruction surface, one can find acceptable intervals between samples along both axes; besides this we can find reconstruction surface and the error reconstruction surface.
Brief Biography of the Speaker: Vladimir Kazakov was born in Moscow region in Russia in 1941. He received the Ph. D. degree in 1967 and the Full Doctor of Science degree in 1990 from Moscow Power Engineering Institute (Technical University). During 1966 – 1996 he worked in Ryazan Radio Engineering University. Since 1996 until the present time he has worked in the National Polytechnic Institute of México. His principal research interests lie in the statistical communication theory. He is the author of more than 200 scientific publications, among of them 3 books, 4 chapters of books, more than 50 papers in the International Journals,17 patents of Russia and 2 patents of Mexico.