On the joint characteristic function of the complex scalar LMS adaptive weight

In this paper, the joint characteristic function of the single weight complex scalar LMS adaptive algorithm is studied. An integral equation is derived for the joint characteristic function of the real and imaginary parts of the weight. This integral equation is used to obtain the weight moments and is 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 means equal to the Wiener weight. These results are applied to a detection problem using the weight magnitude square as the detection statistic. Assuming the weight is also Gaussian during the transient phase of adaptation, the detection performance is optimized over μ for a fixed number of data samples and a known observation interval. The optimum selection of μ approaches zero so that the adaptation process reduces to a cross-correlation operation. When the observation interval is not known a priori, a μ bounded away from zero is required. This detector is shown to be 3 dB degraded from the optimum narrow-band envelope detector using the same number of data samples.