Dimensionally-Efficient Transmission and Storage of Unitary Matrices

Abstract: Unitary matrices are the basis of a large number of signal processing applications. In many of these applications, finding ways to efficiently store, and even transmit these matrices, can significantly reduce memory and throughput requirements. In this work, we study the problem of efficient transmission and storage of unitary matrices. Specifically, we explicitly derive a dimensionally-efficient parametrization (DEP) for unitary matrices that allows identifying them with sequences of real numbers, where the dimension coincides with the dimension of the unitary group where they lie. We also characterize its inverse map that allows retrieving the original unitary matrices from their DEP. The proposed approach effectively allows halving the dimension with respect to naively considering all the entries of each unitary matrix, thus reducing the resources required to store and transmit these matrices. Furthermore, we show that the sequence of real numbers associated to the proposed DEP is bounded, and we delimit the interval where these numbers are contained, facilitating the implementation of quantization approaches with limited distortion. On the other hand, we outline ways to further reduce the dimension of the DEP when considering more restrictive constraints for matrices that show up in certain applications. The numerical results showcase the potential of the proposed approach in general settings, as well as in three specific applications of current interest for wireless communications research.