A General Method for Generating Discrete Orthogonal Matrices

Discrete orthogonal matrices have several applications, such as in coding and cryptography. It is often challenging to generate discrete orthogonal matrices. A common approach widely in use is to discretize continuous orthogonal functions that have been discovered. The need of certain continuous functions is restrictive. To simplify the process while improving the flexibility, we present a general method to generate orthogonal matrices directly through the construction of certain even and odd polynomials from a set of distinct positive values, bypassing the need of continuous orthogonal functions. We provide a constructive proof by induction that not only asserts the existence of such polynomials, but also tells how to iteratively construct them. Besides the derivation of the method as simple as a few nested loops, we discuss two well-known discrete transforms, the Discrete Cosine Transform and the Discrete Tchebichef Transform, and how they can be achieved using our method with the specific values. We also show some examples of how to generate new orthogonal matrices from arbitrarily chosen values.