IEEE SIGNAL PROCESSING LETTERS, VOL. 4, NO. 11, NOVEMBER 1997 307 A New Fast QR Algorithm Based on a Priori Errors Jos´ e Antonio Apolin´ ario Jr., Student Member, IEEE, and Paulo S. R. Diniz, Senior Member, IEEE Abstract—This letter presents a new fast QR algorithm based on Givens rotations using a priori errors. The principles behind the triangularization of the weighted input data matrix via QR decomposition and the type of errors used in the updating process are exploited in order to investigate the relationships among different fast algorithms of the QR family. These algorithms are classified according to a general framework and a detailed description of the new algorithm is presented. Index Terms—Adaptive filters, fast QR decomposition, recur- sive least squares, RLS algorithms. I. INTRODUCTION F AST recursive least squares (RLS) algorithms based on QR decomposition (using Givens rotations) are among those adaptive filtering algorithms with desired characteris- tics such as numerical robustness and possibility of efficient implementation. From the conventional QR decomposition method [1], [2], a number of fast algorithms were derived [3]–[6]. These algorithms can be classified in terms of the type of triangularization applied to the input data matrix (upper or lower triangular) and type of errors (a posteriori or a priori) involved in the updating process. As will be clear later, an upper triangularization (in the notation of this work) involves the updating of forward prediction errors, while a lower triangularization involves the updating of backward predic- tions errors. The classification is summarized in Table I. This table also indicates how these algorithms will be designated hereafter. The proposed algorithm, referred as FQR_PRI_F, is a fast QR that updates a priori forward prediction errors. The FQR_PRI_B algorithm was independently developed in [5] and [6] using different approaches. The approach that will be used here derives from concepts used in the inverse QR algorithm [5], [7] (where the inverse Cholesky factor is updated). II. BASIC CONCEPTS OF QR DECOMPOSITION ALGORITHMS This section reviews the basic concepts of the conventional and inverse QR algorithms in order to establish the notation of this letter. The RLS algorithms minimize the following cost Manuscript received April 14, 1997. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. J. J. Shynk. J. A. Apolin´ ario, Jr. is with the Department of Electrical Engineering, Instituto Militar de Engenharia, Rio de Janeiro, RJ-Brazil. P. S. R. Diniz is with the Program of Electrical Engineering, COPPE/EE/Federal University of Rio de Janeiro, Rio de Janeiro, RJ- Brazil (e-mail: diniz@coe.ufrj.br). Publisher Item Identifier S 1070-9908(97)08178-9. TABLE I CLASSIFICATION OF THE FAST QR ALGORITHMS function: (1) where each component of the vector is the a posteriori error at instant weighted by is the forgetting factor). The vector is given by (2) In (2), is the weighted desired signal vector, is the weighted input data matrix, is the order (the number of coefficients is , and is the coefficient vector. The premultiplication of the above equation by the orthonormal matrix triangularizes without affecting the cost function. (3) The weighted-square error in (1) is minimized by choosing such that the term is zero. Equation (3) can be written in a recursive form, as follows, while avoiding ever increasing order for the vectors and matrices involved [1]: (4) where is a sequence of Givens rotations that annihilates the elements of the input vector in the equation (5) The following relation also used in the conventional QR algorithm is obtained by postmultiplying by the pinning vector (6) where is the first element of the first row of 1070–9908/97$10.00  1997 IEEE