Stability of the quantized LMS algorithm (Q1902507)

The paper deals with the implementation of the least mean square (LMS) algorithm with two truncation quantizations so that numerical stability is achieved. It is shown that a quantized LMS filter can be transformed into a quantized state-space digital filter with variable coefficients. The analysis is restricted to periodically time-varying inputs. Giving a description of the quantized LMS algorithm and its relation to a shift-variant state-space digital filter, two new conditions (and bounds) are established for the input and output stability of the quantized filter. It is shown that it is possible to have both or only one condition satisfied for some problems. One of the conditions is further simplified for the LMS algorithm. Some examples of limit cycles in the quantized LMS algorithm are also presented. The paper will be of interest for the specialists and engineers, working on LMS digital filters design and their VLSI implementations.