Fast design of multichannel FIR least-squares filters with optimum lag

This paper is mainly concerned with the efficient design of optimum lag FIR least-squares filters. These filters result when the minimization of the total mean-square error encompasses a time shift between the input signal and the desired response. The solution of this problem leads to a family of normal systems of equations whose associated matrix is the same, while the right-hand side members are parametrized by the lag ranging within a specified interval. The fast algorithm derived here for the determination of the optimum lag filter provides the solutions of the above family of systems by step-up, step-down, Levinson-like order recursions coupled with time updating Kalman-like relations and offers a recursive error capability as well. This is achieved by exploiting the near-to-Toeplitz structure of the matrix and a certain shift invariance property the right-hand side members possess. Finally motivated by the optimum lag problem, systems of equations with near-to-Toeplitz parameters are considered and a fast algorithm is developed. The proposed schemes find important applications in seismic signal processing, system identification, economic forecasting, etc.