A roundoff error analysis of the LMS adaptive algorithm

The steady state output error of the least mean square (LMS) adaptive algorithm due to the finite precision arithmetic of a digital processor is analyzed. It is found to consist of three terms: 1) the error due to the input data quantization, 2) the error due to the rounding of the arithmetic operations in calculating the filter´s output, and 3) the error due to the deviation of the filter´s coefficients from the values they take when infinite precision arithmetic is used. The last term is of paricular interest because its mean squared value is inversely proportional to the adaptation step size μ. Both fixed and floating point arithmetics are examined and the expressions for the final mean square error are found to be similar. The relation between the quantization error and the error that occurs when adaptation possibly ceases due to quantization is also investigated.