Convergence analysis of filtered-X LMS algorithm with secondary path modeling error

A more relaxed sufficient condition for the convergence of filtered-X LMS (FXLMS) algorithm is presented, which shown that if some positive real condition is satisfied within all the frequency range, FXLMS converges whatever the reference signal is. But if above positive real condition is satisfied only within some frequency range, the convergence of FXLMS algorithm is dependent on the spreading of power spectral density of the reference signal.
Applying the conclusion above to the delayed LMS (DLMS) algorithm, it is shown that to guarantee the convergence of DLMS, the power of the reference signal should be dominant in some discrete frequency bands. The width of those bands is determined by the 'time-delay estimation error frequency' which is equal to one fourth of the inverse of the estimated error of the time delay.