A New Theoretical Model for the Pseudo Affine Projection Algorithm for Unity Step Size and Autoregressive Inputs

The behavior of the Pseudo Affine Projection (PAP) adaptive algorithm is based on the projection of the present input vector onto a subspace defined by a collection of previous input vectors. Existing analytic models for PAP behavior have considered such a projection using expressions that are in fact valid only after a short initialization period. As a consequence, such models are capable of accurately predict the algorithm convergence behavior whenever the parameters of the problem render the effects of this initialization period unimportant. When this is not the case, the PAP behavior predicted by these models can deviate significantly from reality. This work studies the effect of the initialization on the convergence behavior of the PAP algorithm. The analysis is performed for real-valued signals and for unity step-size (fastest convergence). A new analytical model is derived that incorporates a deterministic initial transient phase at the very beginning of the adaptation process. This phase is due both to the arbitrary initialization of the coefficient vector and to the projection subspace, and is responsible for the modified algorithm behavior. Recursive deterministic equations are derived for the mean weight and mean-square error behaviors for a large number of adaptive filter taps, when compared to the algorithm order. Steady-state theoretical equations are also derived. Monte Carlo simulations show significant modeling improvements for specific parameter sets, both during transient and in steady-state, when compared to the most accurate existing PAP model.