Statistical analysis of the deficient length affine projection adaptive algorithm

This paper presents a statistical analysis of the Affine Projection (AP) adaptive algorithm for the insufficient order case. Deterministic recursive equations are derived for the mean weight and mean-square error behavior. The analysis assumes a large number of adaptive coefficients when compared to the algorithm´s order, autoregressive input signals and unity step-size. Monte Carlo simulations show excellent agreement with the theoretically predicted behavior. It is shown that the AP coefficients converge in the mean to the initial plant coefficients, producing an unbiased solution even for the correlated input signal case. It is also shown that the steady-state mean square error has a term that is proportional to the power of the unpredictable part of the input signal filtered by the un-modeled part of the unknown impulse response.