Distributed LMS estimation over networks with quantised communications

This article investigates the problem of distributed least mean-square (D-LMS) estimation over a network with quantised communication. Each node in the network has a quantiser consisting of a first-order dynamical encoder-decoder and can only communicate with its neighbours. Then a new D-LMS estimator is proposed by employing a weighted sum of the internal state differences between each node's quantiser and those of its neighbours. Performance analysis of the proposed quantised D-LMS algorithm is studied in terms of mean-square transient and steady-state measurements. We show that for Gaussian data and sufficiently small step sizes, the proposed cooperative D-LMS with quantisation is mean-square stable in all quantisation levels including the 1-bit case, and its performance approaches the cooperative D-LMS without quantisation when the quantisation step size is fairly small. Furthermore, although in unquantised case (infinite precision), the cooperative D-LMS always outperforms the non-cooperative D-LMS scheme (without communications among the neighbours); however, we show that due to the existence of quantisation error, the cooperative D-LMS with quantisation does not always outperform the non-cooperative D-LMS scheme, especially when the quantisation step size is quite large. Finally, numerical simulations also demonstrate that our theoretical performance matches well with experimental performance.