Copyright Information: This publication has been posted here to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.

A class of equivalent algorithms that accelerate the convergence of the normalized LMS (NLMS) algorithm, especially for colored inputs, has previously 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.

• Convergence of Numerical Methods
• Convergence Behavior
• Affine Projection Algorithms
• Equivalent Algorithms
• Normalized LMS
• NLMS
• Colored Inputs
• APA
• Weight Estimates
• Multiple Unit Delayed Input Signal Vectors
• Input Signal Vectors
• Convergence Rate
• Performance
• Time-to-Steady-State
• Steady-State Error
• Misadjustment

• Convergence Analyses of Affine Projection Algorithm