  This paper analyzes the correlation matrix between the a priori error and
measurement noise vectors for affine projection algorithms (APA). This
correlation stems from the dependence between the filter tap estimates and the
noise samples, and has a strong influence on the mean square behavior of the
algorithm. We show that the correlation matrix is upper triangular, and compute
the diagonal elements in closed form, showing that they are independent of the
input process statistics. Also, for white inputs we show that the matrix is
fully diagonal. These results are valid in the transient and steady states of
the algorithm considering a possibly variable step-size. Our only assumption is
that the filter order is large compared to the projection order of APA and we
make no assumptions on the input signal except for stationarity. Using these
results, we perform a steady-state analysis of the algorithm for small step
size and provide a new simple closed-form expression for mean-square error,
which has comparable or better accuracy to many preexisting expressions, and is
much simpler to compute. Finally, we also obtain expressions for the
steady-state energy of the other components of the error vector.
