On the convergence rate performance of the normalized least-mean-square adaptation

This paper compares the convergence rate performance of the normalized least-mean-square (NLMS) algorithm to that of the standard least-mean-square (LMS) algorithm, which is based on a well-known interpretation of the NLMS algorithm as a form of the LMS via input normalization. With this interpretation, the analysis is considerably simplified and the difference in rate of parameter convergence can be compared directly by evaluating both the condition number of the normalized and unnormalized input correlation matrix. This paper derives the condition number expressions for the normalized input correlation matrix of which the arbitrary-length filter model is linear with respect to its adaptable parameters and contain only two distinct unnormalized eigenvalues. These expressions, which require that the input samples be statistically stationary and zero-mean Gaussian distributed, provide an important insight into the relative convergence performance of the NLMS algorithm to that of the LMS as a function of filter length. This paper also provides a conjecture which set bounds on the NLMS condition number for any arbitrary number of distinct unnormalized eigenvalues.