Over the last decade, a class of equivalent algorithms that accelerate the convergence of the normalized LMS (NLMS) algorithm, especially for colored inputs, has been discovered independently. The affine projection algorithm (APA) is the earliest and most popular algorithm in this class that inherits its name. The usual APA algorithms update weight estimates on the basis of multiple, unit delayed, input signal vectors. We analyze the convergence behavior of the generalized APA class of algorithms (allowing for arbitrary delay between input vectors) using a simple model for the input signal vectors. Conditions for convergence of the APA class are derived. It is shown that the convergence rate is exponential and that it improves as the number of input signal vectors used for adaptation is increased. However, the rate of improvement in performance (time-to-steady-state) diminishes as the number of input signal vectors increases. For a given convergence rate, APA algorithms are shown to exhibit less misadjustment (steady-state error) than NLMS. Simulation results are provided to corroborate the analytical results. I. INTRODUCTION A DAPTIVE filtering techniques are used in a wide range of applications, including adaptive equalization, adaptive noise cancellation, echo cancellation, and adaptive beamforming. The normalized least mean square (NLMS) algorithm [1] is a widely used adaptation algorithm due to its computational simplicity and ease of implementation. Furthermore, this algorithm is known to be robust against finite word length effects. One of the major drawbacks of the NLMS algorithm is its slow convergence for colored input signals. Over the last decade, a class of equivalent algorithms such as the affine projection algorithm (APA), the partial rank algorithm (PRA), the generalized optimal block algorithm (GOBA), and NLMS with orthogonal correction factors (NLMS-OCF) has been developed to ameliorate this problem [2], [3]. The distinguishing characteristic of these algorithms, which was developed independently from different perspectives, is that they update the weights on the basis of multiple, delayed input signal vectors, whereas the NLMS algorithm updates the weights on the basis of a single input vector. In the sequel, we will refer to the entire class of algorithms as affine projection algorithms, since APA (with unit delayed input vectors) is the earliest among these algorithms and since the name APA

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