Block realization of multirate adaptive digital filters

In this paper, we study multirate transversal adaptive digital filters (ADF's) based on the block least-mean-square (BLMS) algorithms. These include an ADF with decimation (ADFD) and an ADF with interpolation (ADFI). We first formulate these filters based on the block mean-squared error (BMSE) criterion and derive BLMS multi-rate weight-adjustment algorithms, and then study efficient realization of those filters. It is shown that the BLMS ADFD can be realized efficiently in the direct form or in the filter-bank structure both using the fast Fourier transform (FFT) and an appropriate sectioning procedure. According to our analysis of computational complexity, the FFT realization of the BLMS ADFD in the filter-bank structure is more efficient than in the direct form, and the two structures using the FFT become more efficient in comparison to the LMS ADFD as the number of weights increases. Unlike the direct form realization, the filter-bank realization of the BLMS ADFD using the FFT becomes more efficient as the decimation ratio increases. Similar results have been obtained for the BLMS ADFI. Finally, we investigate by computer simulation the effects of different weight-adjustment algorithms and several system parameters on the performances of ADFD's. The results of simulation indicate that, as expected, the convergence speed of the BLMS ADFD can be significantly improved by self-orthogonalization of weight adjustment in the frequency domain. Furthermore, the convergence factor of the self-orthogonalizing BLMS ADFD can be chosen such that the steady-state performance of the new ADFD is the same as that of the existing LMS ADFD.