Exact expectation analysis of the LMS adaptive filter without the independence assumption

In almost all analyses of the LMS adaptive filter, it is assumed that the filter coefficients are statistically independent of the input data currently in filter memory, an assumption which is incorrect for shift input data. A method of generating an exact statistical description of the mean and mean-square convergence of the LMS algorithm with shift input data without using this assumption is described. Given input data that are independent from sample to sample, iterations are developed for expectations of products of statistically dependent filter coefficients and data samples, from which important quantities such as the excess mean-square error can be found. Monte Carlo simulations show that the exact analysis produced by this method predicts the mean-square convergence much more accurately than the analysis with the independence assumption for large step sizes, and this accuracy is maintained throughout the useful step size range of the algorithm. With this analysis, phenomena due to the coupling of filter coefficients and past data are discovered and verified