Convergence analysis of the delayless subband adaptive filter based on the frequency domain expression

The upper limit of the step size and the variance of error signal for a two-band delayless subband adaptive filter are theoretically evaluated in the frequency domain. The averaging method and the ordinary differential equation (ODE) method are used for this purpose. It is shown that the calculated result of upper limit of the step size has a similar form of that of the fullband LMS adaptive filter. In the ODE analysis, not only the convergence condition for the adaptive filter but also the variance of the error signal can be derived. The proposed method is also applied to a half-band delayless subband adaptive filter. We demonstrate that these theoretical results give good approximations for simulation results