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It has been shown by some researchers that in a problem of weighted least-square (WLS) design of finite-impulse response (FIR) filters, bulk of the design computation is concerned with the evaluation of the inverse of a matrix in order to solve a system of equations. In this paper, a new algorithm for the WLS design of FIR filters is presented, in which an iterative procedure is developed for the inversion of the matrix involved in the design. By imposing a mild constraint on the updation factor of the weighting function, the inverse of a matrix is expanded as a convergent power series. By investigating the properties of some of the matrices from the design formulation, a modified version of the series that converges rapidly is then proposed to evaluate the inverse in each iteration. It is shown that due to the fast convergence of the power series, one needs to evaluate only the first two or three terms of the series except during the initial stages of the iterations, implying that the conventional operation for matrix inversion is simplified significantly.