Exact convergence analysis of affine projection algorithm: The finite alphabet inputs case

The affine projection algorithm (APA) is a very promising algorithm that has good convergence properties when the input signal is correlated. In particular, it's used to perform communications systems: echo cancellation, equalization... However, due to its complexity, there is no available transient and steady state analysis. In this paper, we present an exact analysis approach tailored for digital transmission context. In such context, the input signal remains in a finite alphabet set. With a discrete Markov chain model of the inputs, we can describe accurately the APA's behavior without any unrealistic assumption. In particular we calculate the exact value of critical and optimum step size. Moreover, we provide the exact Mean Square Deviation for all step size and input correlation. The influence of high order statistics can be enhanced.