Investigation of Delayless Subband Adaptive Filters within the Unified Framework

Abstract

Conventional subband adaptive filter (SAF) solved a complexity and convergence rate problem from long adaptive filter by doing adaptive filtering in subbands. The complexity is reduced because the subband adaptive filters have a much lower length and run parallel in a decimated rate. Faster convergence is achieved because each subband spectrum is stretch out making it more flatter, such that it resembles a white signal. Essentially we decorrelate the signal. However, conventional SAF is plague with delay in the signal path. The delay is introduced mainly from the convolution of analysis and synthesis filter bank. With the delayless subband adaptive filter (DSAF) introduced in 1995 by Morgan and Thi the delay is eliminated by doing a fullband adaptive filtering, but the (fullband) weight-update is done by the subband adaptive filters’ weights through a weight transformation. We keep the benefit of convergence speed, but have increased the computational complexity. DSAF comes in two varieties. An open loop version which resemble the initial problem with Wiener filter based adaptive filter algorithm such as the (N)LMS algorithm. This version does not converge to the true Wiener solution. The other variety is the closed loop version. This version converges to the true Wiener solution. However, the open loop is redundant because the closed loop requires less computation. In this thesis we have derived an equation set which describe the closed loop version of DSAF. From a unified framework we can derive different adaptive filter algorithms within 3 steps. Therefore a proposition is to reverse these 3 steps to find the underlying equation set for the adaptive filter. Where we can investigate convergence properties from tools available from linear algebra. A central part of DSAF is the weight transform which can be viewed as a reconstruction problem with a synthesis filter bank. We have therefore optimized the synthesis and analysis filters, with the help of the derived equation set, to gain better convergence speed.