PERMUTATION PROPERTIES OF SIGNAL PROCESSING TRANSFORMS

Abstract

The research work which lead to the preparation of this thesis was undertaken with the objective of defining some new transforms which could he used for signal (message, picture or data) processing and to study the permutation properties of the proposed signal processing transforms. The work contained in this thesis includes generation of higher order orthonormal transform kernels from lower order orthonormal transform kernels, proposing new two-dimensional transforms and studying their permutation properties, modification of some of the existing transforms for pattern recognition to transforms which could be used for trans mission of message, picture and data, and defining a new class of systems which is invariant to some prescribed permutation. It has been observed that the discrete finite system matrices for the proposed class of permutation invariant system are not necessarily matrices with ranks equal to their orders. Conditions have been stipulated under which the resulting system matrices would have ranks equal to their orders. But this, however, needs further investi gation, Two-dimensional transforms could be frequently thought of as two one-dimensional transforms. By taking various combinations of two one-dimensional orthonormal transform kernels one can define a class of two-dimensional transform kernels. The permutation properties of such transform can be deduced from the permutation properties of the component orthonormal transforms. It is known that Kronecker product of two lower order orthonormal matrices results in an orthonormal matrix of higher order. The algebra for Kronecker product is well developed. But it does not commutative. A new matrix product, Chinese product, has been proposed. This product is defined only when the respective dimensions of the twocomponent matrices are coprimes. The matrix resulting from this product has all the properties of the matrix resulting from Kronecker product of the same component mat rices. In addition this matrix product commutes. In fact the former is a rowwise and columnwise permuted version of the latter. Expressions have been derived for permu tation matrices which can help in getting one from another. The notions of these matrix products and partitioning of matrices have buen exploited to obtain higher ordur orthonormal transform kernels from lower order orthonormal transform kernels. Many of the known transforms which find application in pattern recognition are nonlinear in nature. If these transforms could be inverted by some modification then the iii modified transforms could bo useful for message, picture and data signals. It has been proposed that the additional knowledge about the labels at each functional block in the transmitter could lead to the recovery at receiver of the input signal samples at the transmitter. The class of thus modified transforms has been named as labelled sym metric function transform. The thesis ends, as is customary, with references to some problems which could be taken up in future as an extension of this work.