On the real and complex least mean square adaptive filter algorithms

The real and complex LMS adaptive algorithms are compared. In general, it is shown that, with respect to the weights, the real algorithm is not equivalent to the real part of the complex algorithm. However, when the algorithm is configured as an ALE with sinusoidal inputs, the mean weight behavior of the real algorithm is identical to the real part of the mean weight of the complex algorithm. This result is useful because there are simplicities in the analysis of the complex model that do not exist in the analysis of the real model Some of these simplicities are discussed.