Analysis of the hierarchical LMS algorithm

We analyze the hierarchical least mean-square (HLMS) algorithm, providing expressions for its steady-state mean-square error (MSE). We find conditions for the hierarchical structure to be equivalent to the optimal (full-length) Wiener solution. When these conditions are not satisfied, we show that HLMS will compute biased estimates. Our analysis also shows that even when these conditions hold, the MSE obtained using HLMS may be much larger than that obtained using LMS, since the potentially large MSEs at the subfilters in the first hierarchical level directly affect the output MSE.