The least squares (LS) minimization problem constitutes the cores of many real-time signal processing problems, such as adaptive filtering, system identification and adaptive beamforming. Recently efficient implementations of the recursive least squares (RLS) algorithm and the constrained recursive least squares (CRLS) algorithm based on the numerically stable QR decomposition (QRD) have been of great interest. Several papers have proposed modifications to the rotation algorithm that circumvent the square root operations and minimize the number of divisions that are involved in the Givens rotation. It has also been shown that all the known square root free algorithms are instances of one parametric algorithm. Recently, a square root free and division free algorithm has been proposed [4].

In this paper, we propose a family of square root and division free algorithms and examine its relationship with the square root free parametric family. We choose a specific instance for each one of the two parametric algorithms and make a comparative study of the systolic structures based on these two instances, as well as the standard Givens rotation. We consider the architectures for both the optimal residual computation and the optimal weight vector extraction.

The dynamic range of the newly proposed algorithm for QRD-RLS optimal residual computation and the wordlength lower bounds that guarantee no overflow are presented. The numberical stability of the algorithm is also considered. A number of obscure points relevant to the realization of the QRD-RLS and QRD-CRLS algorithms are clarified. Some systolic structures that are described in this paper are very promising, since they require less computational complexity ( in various aspects) than the structures known to date and they make the VLSI implementation easier.