4. Family of Affine Projection Algorithms

After the birth of the basic affine projection algorithm (B-APA) in the middle of 1980s, several adaptive filtering algorithms that also exploit multiple regressors were put forward independently. They are now recognized as variants of the B-APA, forming a family of affine projection algorithms. In the family, there are such algorithms as the regularized affine projection algorithm (R-APA), the decorrelation affine projection algorithm (D-APA), the partial-rank algorithm (PRA), the normalized least-mean-squares algorithm with orthogonal correction factors (NLMS-OCF), the binormalized data-reusing least-mean-squares algorithm (BNDR-LMS). The R-APA and the D-APA are discussed in the preceding chapter. If we update the coefficient vector every p samples instead of every sample, where p is the projection order, then we obtain the PRA. By applying the Gram–Schmidt orthogonalization to the regressors, the B-APA is transformed into the NLMS-OCF. The BNDR-LMS is just a special case of the B-APA where the projection order equals 2. We can also use sparse regressors in the APAs. These algorithms are formulated by a single update equation with several parameters. By setting those parameters at appropriate values, the update equation for each of the algorithms in the APA family is expressed.