New insights on the transient and steady-state behavior of the quantized LMS algorithm

This correspondence investigates the transient and steady-state behavior of the quantized LMS algorithm for Gaussian inputs. It is shown here that the so-called “stopping” phenomenon is really a “slow-down” phenomenon, which, because of an extremely slow convergence rate, looks as if the algorithm has stopped. The true steady-state MSE is shown to be nearly independent of the number of bits in the digital word-length and very nearly the steady-state MSE of the infinite precision LMS realization. These results assume that the algorithm “misadjustment” effects due to coefficient quantization are negligible in comparison with those due to the “stopping” phenomena. Since the true steady state is rarely achievable with a finite number of iterations, determination of the step size μ that minimizes the residual MSE must be based on a stochastic model for the transient mode of algorithm operation. It is shown that the finite word length and infinite precision design cases differ only in degree and not in kind as far as the selection of μ is concerned