Conditioning of LMS algorithms with fast sampling

The LMS algorithm is very commonly used in signal processing. Its convergence properties depend primarily on the step size chosen and the condition number of an information matrix associated with the system. In most applications today, the LMS uses a regression vector based on the shift operator (including the ubiquitous tapped delay line). We demonstrate that generically, when fast sampling is employed, these regression vectors lead to poorly conditioned LMS. By comparison, delta operator based regression vectors lend with rapid sampling to improved condition numbers, hence, to better performance.<>