Iterative version of the QRD for adaptive RLS filtering

A modiied version of the QR{decomposition (QRD) is presented. It uses approximate Givens rotations instead of exact Givens rotations, i.e., a matrix entry usually annihilated with an exact rotation by an angle is only reduced by using an approximate rotation by an angle ~. The approximation of the rotations is based on the idea of CORDIC. Evaluating a CORDIC{based approximate rotation is to determine the angle ~ = ` = arctan 2 ?` , which is closest to the exact rotation angle. This anglè is applied instead of. Using approximate rotations for computing the QRD results in an iterative version of the original QRD. A recursive version of this QRD using CORDIC{based approximate rotations is applied to adaptive RLS ltering. Only a few angles of the CORDIC sequence, r say (r b, where b is the word length), work as well as using exact rotations (r = b, original CORDIC). The misadjustment error decreases as r increases. The convergence of the QRD{ RLS algorithm, however, is insensitive to the value of r. Adapting the approximation accuracy during the course of the QRD{RLS algorithm is also discussed. Simulations (channel equalization) connrm the results. 1 Introduction In many signal processing applications (e.g. system identiication, channel equalization) an adaptation algorithm is used to adapt the coeecients of the used lter to the instationarity of the underlying processes. The M coeecients w of the lter are adapted such that the diierence between the lter output y(t) and a given reference signal ~ y(t), i.e., the error signal e(t), is minimized (Figure 1). For the minimization of the error signal two diierent criteria are known { the least mean square (LMS) or the least squares (LS) minimization criteria. The algorithms derived from these minimization criteria are the LMS algorithm and the RLS algorithm, respectively 15]. The LMS algorithm is computationally inexpensive (O(M) per sample data vector, where M is the lter length) and easy to implement. The RLS algorithm on the other hand is computationally more expensive (O(M 2)) but its convergence is insensitive to the eigenvalue spread of the data covariance matrix. Due to its computational expense the RLS algorithm does not comply with many real time applications and therefore parallel implementations have been developed. The method of choice for the fast parallel implementation of adaptive RLS ltering is based on the QR decomposition (QRD) X = QR of the data matrix. The respective …