An algorithm for solving discrete-time Wiener-Hopf equations based upon Euclid's algorithm

Abstract

An algorithm for solving a discrete-time Wiener-Hopf equation is presented based upon Euclid's algorithm. The discrete-time Wiener-Hopf equation is a system of linear inhomogeneous equations with a given Toeplitz matrix M, a given vector b, and an unknown vector <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\lambda</tex> such that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M\lambda = b</tex> . The algorithm is able to find a solution of the discrete-time Wiener-Hopf equation for any type of Toeplitz matrices except for the all-zero matrix, while the Levinson algorithm and the Trench algorithm are not available when at least one of the principal submatrices of the Toeplitz matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> is singular. The algorithm gives a solution, if one exists, even when the Toeplitz matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> is singular, while the Brent-Gustavson-Yun algorithm only states that the Toeplitz matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> is singular. The algorithm requires <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(t^{2})</tex> arithmetic operations for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</tex> unknowns, in the sense that the number of multiplications or divisions is directly proportional to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t^{2}</tex> , like the Levinson and Trench algorithms. Furthermore, a faster algorithm is also presented based upon the half greatest common divisor algorithm, and hence it requires <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(t \log^{2} t)</tex> arithmetic operations, like the Brent-Gustavson-Yun algorithm.