New square-root-free QR-based adaptive filtering algorithms

In this paper we derive some modified QR-RLS adaptive filtering algorithms with zero square-root computation. Reduction of complexity remains a fundamental problem in adaptive filtering due to the time constraints of real applications. The proposed solution to this problem is an enrichment of the GR class of approximate QR based adaptive filtering algorithms. Inside this class, the orthogonal Q transforms (of QR-RLS) can be substituted with a more general partially non-orthogonal G transforms. The new square-root-free algorithms, in this paper, uses an approximate Givens rotation without square-root computation, it is simply based on a taylor expansion and approximation. In order to converge this transform must be followed by an additional compensation transform with O(N) complexity. Unlike previous contributions, this square-root free approach guaranteed the derivation of both standard and fast versions (O(N) complexity). We have tested two square-root free algorithms corresponding to the QR-RLS and its fast QR-based variant. The obtained results are close to those for the original algorithms. Finally, the proposed algorithms constitute a robust approximation with low complexity of the fast converging recursive least-squares.