Convergence analysis of alias-free subband adaptive filters based on a frequency domain technique

Convergence analysis of alias-free subband adaptive filters (SADFs) is presented based on a frequency-domain technique where instead of analyzing the adaptive algorithms in the time-domain, the averaging method and the ordinary differential equation (ODE) method are applied to the frequency-domain expressions of the adaptive algorithms converted by the discrete Fourier transform. As an alias-free SADF algorithm, the SADF proposed by Pradhan and Reddy is known. In this paper, this technique is first applied to the Pradhan's SADF. The stability of the Pradhan's SADF is verified in the frequency domain, and a simple formula to evaluate the mean square error (MSE) of the algorithm is theoretically derived. By using a slight modification, the technique can be applied to the two-band delayless subband adaptive filter (DLSADF) with the Hadamard transform. The stability condition and the MSE of the DLSADF with the Hadamard transform are also obtained. Simulation results of both algorithms show the validity of the theoretical results.