A new recursive algorithm for linear equation and its application to adaptive signal processing

Abstract

AbstractThe solution of a system of linear equations is very important in various fields of engineering, and various methods have been presented for the numerical solution. Especially, in the adaptive signal processing and adaptive control, the system of equations is often considered where the coefficient matrix is (pseudo‐) positive‐definite and symmetrical. the conjugate graduate method is known as a means to solve such a system of equations. Letting the rank of the coefficient matrix be R, it is a method where the solution for the system of equatins is determined by iterating a certain procedure R times. In this metho, even if the iterative process is terminated on the way due to the constraint for the computation time, for example, an approximate solution which is somewhat accurate is obtained. This method, however, still has room for improvement with respect to the accuracy of the approximate solution.From such a viewpoint, this paper examines geometrically the iterative computation process in the conjugate gradient method and, based on that result, proposes a new method of iterative solution for the system of linear equations with symmetrical coefficient matrix. Then the original method is slightly modified and an adaptive algorithm is derived by applying the result to the parameter estimation of the adaptive filter. Those methods are based on the orthogonal projection operation from the solution to be determined to the linear space. Consequently, it is expected that, even if the iterative computation is terminated on the way, a more accurate approximate solution is obtained, compared to the case of the conjugate gradient method or the adaptive algorithm based on the conjugate gradient method. By the property of the method proposed in this paper the method is useful when the coefficient matrix is ill‐conditioned. It is expected that the method will effectively be utilized in the signal processing and control.