WSEAS Transactions on Signal Processing
Print ISSN: 1790-5052, E-ISSN: 2224-3488
Volume 18, 2022
A Matrix-Valued Inner Product for Matrix-Valued Signals and Matrix-Valued Lattices
Author:
Abstract: A matrix-valued inner product was proposed before to construct orthonormal matrix-valued wavelets for matrix-valued signals. It introduces a weaker orthogonality for matrix-valued signals than the orthogonality of all components in a matrix that is commonly used in orthogonal multiwavelet constructions. With the weaker orthogonality, it is easier to construct orthonormal matrix-valued wavelets. In this paper, we re-study the matrix-valued inner product more from the inner product viewpoint that is more fundamental and propose a new but equivalent norm for matrix-valued signals. We show that although it is not scalar-valued, it maintains most of the scalarvalued inner product properties. We introduce a new linear independence concept for matrix-valued signals and present some related properties. We then present the Gram-Schmidt orthonormalization procedure for a set of linearly independent matrix-valued signals. Finally we define matrix-valued lattices, where the newly introduced Gram-Schmidt orthogonalization might be applied.
Search Articles
Keywords: Matrix-valued inner product, matrix-valued signal space, nondegenerate matrix-valued signals, linearly independent matrix-valued signals, matrix-valued lattices, matrix-valued wavelets
Pages: 146-152
DOI: 10.37394/232014.2022.18.21