Recursive triangular array ladder algorithms

Two recursive-least-squares ladder algorithms for implementation on triangular systolic arrays are presented. The first algorithm computes transversal forward/backward predictor coefficients, ladder reflection coefficients, and forward/backward residual energies. This algorithm has a complexity of three multiplications and additions per rotational (triangular array) element. A second algorithm is presented that facilitates the computation of only the ladder reflection coefficients and the forward/backward residual energies at a cost of two multiplications and additions per rotational element. This way, both algorithms are computationally more efficient than the traditional recursive QR decomposition (Gentleman and Kung array) for any order. The second algorithm is more efficient than Cioffi´s pipelineable linear array fast QR adaptive filter for an order of less than 22 in the prewindowed case, and more efficient than the fast QR for an order of less than 43 in the more general covariance case. A comparison of the presented algorithms and the prominent QR methods is given. The algorithms remain unchanged and the number of arithmetic operations is not increased when finite duration windows are used. The algorithms are based entirely on numerically stable and robust covariance recursions