Signal behavior of adaptive filtering algorithms in a nonstationary environment with singular data covariance matrix

The paper analyzes the signal behavior of adaptive filtering algorithms when the target weights of the adaptive filter are time varying and the covariance matrix of the filter input is singular. The signal behavior is evaluated in terms of moments of the excess output error of the filter. Two algorithms are considered: the LMS algorithm and the sign algorithm. The analysis is done in the context of adaptive plant identification. The plant parameters vary according to a random walk model. The plant input, plant noise, and plant parameters are assumed mutually independent. Under these assumptions, it is found that the signal behavior of the algorithms is the same as the signal behavior in the case with positive definite input covariance matrix.