5. Convergence Behavior of APA

The behavior of an adaptive filter depends on the input signal. However, since there are infinitely many variations of signals, it is difficult to draw general, useful conclusions on the behavior of an adaptive filter taking the waveform of each signal into consideration. Therefore, we often resort to statistical approach, where the signals appearing in the update equation are replaced with random variables. Even in this stochastic framework, we need many assumptions on the statistical properties of those random variables to make the analysis tractable. We are concerned with the behavior of the expectations of the error signal and the squared norm of the coefficient error vector. We are also interested in stability condition on the range of the step-size. In the first part of this chapter, the fundamental behavior of the B-APA is discussed. Then, we review two works based on simplifying assumptions on regressors. One of those works assumes that a regressor can take only one of finite number of orientations. Although this assumption is unrealistic, the analysis shows many of important properties of the B-APA. The other work assumes that the input signal is an autoregressive process, and that the step-size equals unity. This work uses the update equation developed for the D-APA. The last work we review in this chapter gives a general treatment for the convergence behavior of the R-APA. Based on the energy conservation relation and the weighted energy conservation relation, it yields useful results without extreme assumptions.