A Journey of Discovery: Orthogonal Matrices and Wireless Communications

Real orthogonal designs were first introduced in the 1970’s, followed shortly by the introduction of complex orthogonal designs. These designs can be described simply as square matrices whose columns are formally orthogonal, and their existence criteria depend on number theoretic results from the turn of the century. In 1999, generalizations of these designs were applied in the development of successful wireless communication systems, renewing interest in the theory of these orthogonal designs. This area of study represents a beautiful marriage of classical mathematics and modern engineering. This paper has two main goals. First, we provide a brief and accessible introduction to orthogonal design theory and related wireless communication systems. We include neither the mathematical proofs of the relevant results nor the technical implementation details, rather hoping that this gentle introduction will whet the reader’s appetite for further study of the relevant mathematics, the relevant engineering implementations, or, in the best case scenario, both. Second, in light of the dedication of this paper to Joe Gallian, who was an extraordinary undergraduate research advisor to so many of us and who inspired so many of us to become undergraduate research advisors ourselves, we describe the involvement of undergraduates in research involving these orthogonal designs and related communications systems.