On the probability density functions of the weight for the complex scalar LMS adaptive algorithm

In this paper, a study is made of the joint probability density function of the single weight for the complex scalar LMS adaptive algorithm. Integral equations are derived for the probability density functions and characteristic functions of the real and imaginary parts of the weights. These integral equations are approximately solved for a sufficiently small adaptation parameter µ. It is shown that, for small µ and, in steady-state, the real and imaginary parts of the weights are statistically independent Gaussian random variables with mean equal to the Wiener weight.