A new method for low-rank transform domain adaptive filtering

This paper introduces a least squares, matrix-based framework for adaptive filtering that includes normalized least mean squares (NLMS), affine projection (AP) and recursive least squares (RLS) as special cases. We then introduce a method for extracting a low-rank underdetermined solution from an overdetermined or a high-rank underdetermined least squares problem using a part of a unitary transformation. We show how to create optimal, low-rank transformations within this framework. For obtaining computationally competitive versions of our approach, we use the discrete Fourier transform (DFT). We convert the complex-valued DFT-based solution into a real solution. The most significant bottleneck in the optimal version of the algorithm lies in having to calculate the full-length transform domain error vector. We overcome this difficulty by using a statistical approach involving the transform of the signal rather than that of the error to estimate the best low-rank transform at each iteration. We also employ an innovative mixed domain approach, in which we jointly solve time and frequency domain equations. This allows us to achieve very good performance using a transform order that is lower than the length of the filter. Thus, we are able to achieve very fast convergence at low complexity. Using the acoustic echo cancellation problem, we show that our algorithm performs better than NLMS and AP and competes well with FTF-RLS for low SNR conditions. The algorithm lies in between affine projection and FTF-RLS, both in terms of its complexity and its performance.