Efficient approximate implementations of the fast affine projection algorithm using orthogonal transforms

The recently-introduced fast aane projection (FAP) adaptive lter has performance rivalling that of the RLS adaptive lter in many situations. However , the sliding-window RLS algorithm embedded within the FAP adaptive lter can be too complex to implement in some cases. In this paper, we describe an alternate implementation of the FAP adaptive lter that uses orthogonal transforms to approximately calculate the aane projection. Simulations show that the performance of this sim-pliied algorithm is nearly as good as the original algorithm when the transform is chosen properly. In addition, we provide a gradient step size (GSS) method for enhancing the performance of the FAP adaptive lter in nonstationary signal environments. 1. INTRODUCTION The performance advantages of the recursive-least-squares (RLS) adaptive lter over the least-mean-square (LMS) adaptive lter are well known. However, the fast transver-sal lter (FTF) version of the RLS adaptive lter has a complexity that makes its use impractical for applications in which the lter lengths L are extremely long, such as in acoustic echo cancellation 1]. Recently, the fast aane projection (FAP) adaptive lter has been applied to both acoustic echo cancellation 2, 3] and active noise control 4]. In many situations, the FAP algorithm's adaptation performance rivals that of the FTF algorithm without the latter algorithm's signiicant computational complexity. The FAP algorithm is summarized in Table 1 for an arbitrary step size sequence k; for a complete description of the algorithm , the reader is referred to 2, 3]. Embedded within the FAP algorithm is the sliding-window fast RLS algorithm of order N, where 2 N L. The value of the projection order N within the FAP algorithm determines both the computational complexity of the system as well as its performance in a given situation. In acoustic echo cancellation with speech signals, the projection order N required for good performance is typically much smaller than the lter length L, and thus the complexity of the FAP algorithm is similar to that of the LMS algorithm. However, in other situations, the lter lengths are shorter, and thus the overhead needed to compute the aane projection is a signiicant portion of the overall complexity of the system. Moreover, the software code for the sliding-window fast RLS algorithm is diicult to implement, A T 0 = 1 0 T ], B T 0 = 0 T 1] 1 Use Nth order sliding window fast RLS