A Simplified Normalized Subband Adaptive Filter (NSAF) with NLMS-like complexity

The Normalized Subband Adaptive Filter (NSAF) is a popular algorithm exhibiting moderate computational complexity and enhanced convergence speed relative to the ubiquitous Normalized Least Mean Square (NLMS) algorithm. Traditionally, the NSAF has made use of sophisticated perfect reconstruction (PR) filter banks and a block updating scheme, in which the adaptive filter vector is updated once every N samples, with N being equal to the number of subbands. Here we argue, first from a theoretical point of view, that an extremely simple two band filter bank with the simplest possible length 2 FIR filters, {1, −1} and {1, 1}, can be successfully used either with a sample by sample adaptive filter update, or with a block update performed for every second input signal sample. We demonstrate that this scheme actually works well through simulations. In short we obtain better convergence performance than the NLMS with a (multiplicative) computationally complexity proportional to 2M, M being the length of the adaptive filter to be identified, with the block update and even better performance if we are willing to accept a computational complexity proportional to 4M.