Kronecker Products, Unitary Matrices and Signal Processing Applications

Discrete unitary transforms are extensively used in many signal processing applications, and in the development of fast algorithms Kronecker products have proved quite useful. In this semitutorial paper, we briefly review properties of Kronecker products and direct sums of matrices, which provide a compact notation in treating patterned matrices. A generalized matrix product, which inherits some useful algebraic properties from the standard Kronecker product and allows a large class of discrete unitary transforms to be generated from a single recursion formula, is then introduced. The notation is intimately related to sparse matrix factorizations, and examples are included illustrating the utility of the new notation in signal processing applications. Finally, some novel characteristics of Hadamard transforms and polyadic permutations are derived in the framework of Kronecker products.