OLRIV: a new fast adaptive algorithm for rectangular-block Toeplitz systems

In this paper, we propose a new algorithm for solving overdetermined systems when they are rectangular-block Toeplitz (the blocks can have more rows than columns). It is based on the expression of the matrix of the system to solve as the intercorrelation between two vectorial processes, namely, the original and the instrumental processes. The instrumental process has generally more components than the original one; therefore, it takes the overdetermined character of the system into consideration. The proposed algorithm known as the overdetermined lattice recursive instrumental variable (OLRIV) belongs to the fast-RLS family and lies on a double lattice structure, where one lattice performs the prediction of the original process and the other one the prediction of the instrumental process. The geometric complete derivation of the proposed algorithm is given in the paper. Afterwards, we show how OLRIV can be applied to perform blind adaptive identification of AR channels using high order statistics, where the systems to solve are often overdetermined to ensure identifiability. Both the scalar and the vectorial cases are investigated. Simulation results are finally given to show the performances of OLRIV.