A stability problem in sign-sign adaptive algorithms

It has been shown by example that the often used sign-sign variant of the LMS can be unstable and produce unbounded parameter estimates. In this paper a modified sufficient excitation condition on the regressor vector for the sign-sign variant is introduced. Satisfaction of the excitation condition guarantees that there exists a sufficiently small step size to produce parameter estimates which converge exponentially to within a region of the correct parameters. The counter example to stability of sign-sign algorithms is reexamined in light of the new excitation condition and it is shown that the excitation condition correctly predicts when instability will occur in this example.