IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 1, JANUARY 2010 175 Multichannel Fast QR-Decomposition Algorithms: Weight Extraction Method and Its Applications Mobien Shoaib, Student Member, IEEE, Stefan Werner, Senior Member, IEEE, and José Antonio Apolinário Jr., Senior Member, IEEE Abstract—Multichannel fast QR decomposition RLS (MC- FQRD-RLS) algorithms are well known for their good numerical properties and low computational complexity. The main limi- tation is that they lack an explicit weight vector term, limiting themselves to problems seeking an estimate of the output error signal. This paper presents techniques which allow us to use MC-FQRD-RLS algorithms with applications that previously have required explicit knowledge of the adaptive filter weights. We first consider a multichannel system identification setup and present how to obtain, at any time, the filter weights associated with the MC-FQRD-RLS algorithm. Thereafter, we turn to problems where the filter weights are periodically updated using training data, and then used for fixed filtering of a useful data sequence, e.g., burst-trained equalizers. Finally, we consider a particular control structure, indirect learning, where a copy of the coefficient vector is filtering a different input sequence than that of the adaptive filter. Simulations are carried out for Volterra system identification, decision feedback equalization, and adaptive pre- distortion of high-power amplifiers. The results verify our claims that the proposed techniques achieve the same performance as the inverse QRD-RLS algorithm at a much lower computational cost. Index Terms—Adaptive systems, equalizer, fast algorithms, indi- rect learning, multichannel algorithms, predistortion, QR decom- position, Volterra system identification, weight extraction. I. INTRODUCTION F AST QR-DECOMPOSITION RECURSIVE LEAST- SQUARES (FQRD-RLS) algorithms based on backward prediction errors are a popular class of least-squares based algorithms that are known for their numerical stability and reduced computational complexity, unlike fast versions of the RLS algorithm [1]–[3]. The idea in FQRD-RLS algorithms (single or multichannel versions) is to exploit the underlying time-shift structure of the input data vector in order to replace matrix update equations with vector update equations [4]. The time-shift structure of Manuscript received September 19, 2008; accepted July 13, 2009. First pub- lished August 18, 2009; current version published December 16, 2009. This work was supported in part by the Academy of Finland, Smart and Novel Ra- dios (SMARAD) Center of Excellence, GETA, PSATRI/STC-Chair, CAPES, and by CNPq. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Dennis R. Morgan. M. Shoaib is with the Department of Signal Processing and Acoustics, Helsinki University of Technology, Finland. He is also with the Prince Sultan Advanced Technology Research Institute (PSATRI), King Saud University, Saudi Arabia (e-mail: mobien@ieee.org). S. Werner is with the Department of Signal Processing and Acoustics, Helsinki University of Technology, Finland. J. A. Apolinário, Jr. is with the Military Institute of Engineering, Rio de Janeiro, Brazil. Digital Object Identifier 10.1109/TSP.2009.2030594 a multichannel input data vector is not necessarily trivial, as it may contain a number of single channel vectors of dif- ferent orders. The vector update equations are derived from forward and backward predictions. This paper considers mul- tichannel algorithms [4]–[7] based on the update of backward prediction errors which are numerically robust [8]. Note that single channel FQRD-RLS algorithm is a particular case of multichannel FQRD-RLS (MC-FQRD-RLS) algorithms. In this paper our focus is on MC-FQRD-RLS algorithms and its applications, unless mentioned otherwise. The main limitation of the MC-FQRD-RLS algorithms is the unavailability of an explicit weight vector term. Furthermore, it does not directly provide the variables allowing for a straight- forward computation of the weight vector, as is the case with the conventional QRD-RLS algorithm, where a back-substitution procedure can be used to compute the coefficients. Therefore, the applications are limited to output error based (e.g., noise cancellation), or to those requiring a decision-feedback estimate of the training signal (e.g., equalizers operating in decision-di- rected mode). The goal of this paper is to extend the range of application of the MC-FQRD-RLS algorithm. We will focus on three different application scenarios: 1) System identification, where knowledge of the explicit weights of the adaptive filter is not needed at each algo- rithm iteration. 2) Burst-trained systems, where the coefficient adaptation is performed in a training block. The weight vector obtained at the end of the training block is then kept fixed and used for output filtering, e.g., periodically updated channel equalizers. 3) Indirect-learning, where at each time instant a copy of adaptive filter is used for filtering a different input sequence than that of the adaptive filter, e.g., predistortion of high- power amplifiers (HPAs). For the case of system identification, we propose a mecha- nism in which the coefficients of the transversal weight vector can be obtained in a sequential manner at any chosen iteration at a total computational complexity cost of , i.e., per coefficient, without compromising the accuracy, where is the total number of filter coefficients. Obviously, if the transversal weight vector is not required at every iteration (typically after convergence), then the overall computational complexity using this approach will be much lower than when using a conven- tional QRD-RLS algorithm as in [9]. In addition, the peak-com- plexity (when we have different complexity in distinct itera- tions, corresponds to the maximum value within a time frame) 1053-587X/$26.00 © 2009 IEEE Authorized licensed use limited to: INSTITUTO MILITAR DE ENGENHARIA. Downloaded on February 3, 2010 at 14:21 from IEEE Xplore. Restrictions apply.