The constrained generalized data windowing conjugate gradient algorithm

Abstract

where F is as given above and P is the projection matrix, which can also be written as P == B (BTB) -1 BT. This paper introduces a constrained version of the GDWCG algorithm, namely the CGDWCG algorithm, and is organized as follows: After Section 2 presents the equations of the GDWCG algorithm, Section 3 presents a step-by-step derivation of the new algorithm, and Section 4 presents the simulation results. Finally, Section 5 concludes this work. algorithm from [3] was introduced in [8]. The constrained version was equivalent, in infinite precision environment, to the MCG algorithm used within the so-called Generalized Sidelobe Canceler (GSC) structure [9,10]. This structure is well-known to be able to transform the linearly constrained minimization problem into an unconstrained minimization problem.In order to nlinimize the Mean Squared-Error (MSE) with respect to w, subject to CTw == f, the GSC structure decomposes the coefficient vector using a transformation matrix that can be represented by [C -B], where C is the constraint matrix, w is the coefficient vector, f is the gain vector, and B is the blocking matrix which spans the null space of the constraint matrix C, i.e., B T C == o. The transformed coefficient vector is intrinsically partitioned yielding and overall filter w(k) == F -BWGsc(k), where F == C(C T C)-lf and w GSC is the reduced-dimension coefficient vector that operates on the input-signal vector modified by the blocking matrix B.Given a projection matrix P == I -C(C T C)-IC T , then any constrained adaptive filter w (k) can be decomposed in two parts: A projection onto the subspace orthogonal to the space spanned by the constraint matrix C, i.e., w(k) premultiplied by the projection matrix P, and a translation that brings the projected vector back to the hyperplane C T w == f, i.e., ABSTRACT This paper introduces a constrained version of a recently proposed generalized data windowing scheme applied to the Conjugate Gradient algorithm. This scheme combines two types of data windowing, the finite-data sliding window and the exponentially weighted data window, in an attempt to attain the best of both methods in a linearly constrained scenario. The proposed algorithm was tested in a simple adaptive beamforming application, where the expected better performance was demonstrated.