Error whitening criterion for adaptive filtering: Theory and algorithms

Mean squared error (MSE) has been the dominant criterion in adaptive filter theory. A major drawback of the MSE criterion in linear filter adaptation is the parameter bias in the Wiener solution when the input data are contaminated with noise. In this paper, we propose and analyze a new augmented MSE criterion called the Error Whitening Criterion (EWC). EWC is able to eliminate this bias when the noise is white. We will determine the analytical solution of the EWC, discuss some interesting properties, and develop stochastic gradient and other fast algorithms to calculate the EWC solution in an online fashion. The stochastic algorithms are locally computable and have structures and complexities similar to their MSE-based counterparts (LMS and NLMS). Convergence of the stochastic gradient algorithm is established with mild assumptions, and upper bounds on the step sizes are deduced for guaranteed convergence. We will briefly discuss an RLS-like Recursive Error Whitening (REW) algorithm and a minor components analysis (MCA) based EWC-total least squares (TLS) algorithm and further draw parallels between the REW algorithm and the Instrumental Variables (IV) method for system identification. Finally, we will demonstrate the noise-rejection capability of the EWC by comparing the performance with MSE criterion and TLS.