1. Introduction

An adaptive filter is a kind of learning machine that changes its transfer function automatically. It has two key components: a transversal filter to process the input signal, and a mechanism to control the coefficients of the transversal filter so that it produces an output signal that approximates the desired signal. Adaptive filters can be applied to various real-world problems such as system identification, channel equalization, and acoustic noise cancellation. Among these, the system identification problem is most fundamental covering a wide range of problems, e.g., echo cancellation, active noise control, feedback cancellation in hearing aids, and howling suppression. In the system identification problem, an adaptive filter is used to make a replica of the impulse response of an unknown system. The adaptation is performed by adding a correction vector to the current coefficient vector, making the updated coefficient vector. The correction vector is a function of the current and recent input signal, the desired signal, and the current coefficient vector. This function determines the performance and the complexity of the adaptation algorithm. In classical algorithms such as the least-mean-squares (LMS) algorithm and the normalized least-mean-squares (NLMS) algorithm, a simple function is used. On the other hand, a more elaborate function is employed in the affine projection algorithm (APA), the central theme of this book.