Fast O(n) complexity algorithms for diagonal innovation matrices

Abstract

In general, the direct solution of an n dimensional system of linear equations requires O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) arithmetic operations. Frequently in high data rate signal processing, fast algorithms of complexity lower than O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) are required to solve a large system of linear equations. In several interesting applications, the specific structure of the coefficient matrix associated with the linear system may be used to reduce the required number of operations. Fast algorithms have been developed when the coefficient matrix is a Toeplitz matrix, a Hankel matrix, or when it can be represented as a sum of Toeplitz and Hankel matrices. The arithmetic complexity associated with these fast algorithms is O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). In this paper, fast algorithms of O(n) are presented that can be used for a class of matrices called diagonal innovation matrices (DIM). Previous results for this class of matrices require an arithmetic complexity of O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). A number of results are presented regarding linear systems having DIM coefficient matrices and several special cases are examined. The effect of recursively increasing the order of the coefficient matrix on the number of operations is studied and some observations are made.