Systems of equations with near-to-Toeplitz or near-to-Hankel parameters and applications to signal processing

In various signal processing applications we are confronted with the problem of finding the optimum in the least squares sense linear filter that matches the input signal with a certain lag of the desired response. The system of equations yielding all filters coefficients turns out to have a near-to-Toeplitz or near-to-Hankel structure with displacement rank 1 or 2 generically depending on whether windowing is used. Motivated by these questions the general linear system is considered and recursive procedures are derived for the determination of the solution which when the parameters have low displacement rank are proved very efficient. In particular if the parameters are Toeplitz the resulting algorithm coincides with a scheme introduced recently by the authors which in turn performs better than the well-known Simpson´s sideways recursions.