Linear Constrained QRD-Based Algorithm

Abstract

The linearly constrained adaptive filtering (LCAF) technique has been extensively used in many engineering applications. In this chapter, we introduce the linearly constrained minimum variance (LCMV) filter, implemented using the linearly constrained recursive least squares (RLS) criterion, with the inverse QR decomposition (IQRD) approach. First, the direct form of recursively updating the constrained weight vector of LS solution based on the IQRD is developed, which is named as the LC-IQRD-RLS algorithm. With the IQRD approach, the parameters related to the Kalman gain are evaluated via Givens rotations and the LS weight vector can be computed without back-substitution. This algorithm is suitable to be implemented using systolic arrays with very large scale integration technology and DSP devices. For the sake of simplification, an alternative indirect approach, referred to as the generalized sidelobe canceler (GSC), is adopted for implementing the LCAF problem. The GSC structure essentially decomposes the adaptive weight vector into constrained and unconstrained components. The unconstrained component can then be freely adjusted to meet any criterion since the constrained component will always ensure that the constraint equations are satisfied. The indirect implementation could attain the same performance as that using the direct constrained approach and possesses better numerical properties. Via computer simulation, the merits of the LC-IQRD-RLS algorithms over the conventional LC-RLS algorithm and its modified version are verified.