Analysis of a normalized LMS adaptive filter with a singular input covariance matrix

The paper analyzes the signal behavior of an adaptive filter whose adaptation is governed by the normalized least mean square (NLMS) algorithm when the covariance matrix of the filter input is singular. The signal behavior is evaluated in terms of the mean square of the excess output error of the filter. The analysis is done in the context of adaptive identification of a time-invariant plant. The plant input and plant noise are assumed stationary and mutually independent. Under these assumptions, it is found that the long-term average of the mean square excess error of the NLMS algorithm is proportional to the algorithm step size. This implies that in spite of the singularity of the input covariance matrix, the steady state signal behavior of the algorithm can be made arbitrarily fine by using a sufficiently small step size. The analytical results of the paper are supported by simulations.