Mean field message passing for cooperative simultaneous ranging and synchronization

Distributed clock synchronization and ranging in time-variant wireless networks

“…1. Inserting (9) and (11) into (13) allows each vertex p(yn|xn) to be represented by a subgraph as given in [14], and each vertex p(xn|xn−1) by node-independent transitions. For each state xn, we can thus compute the marginal function m n|n (x k,n ) ∝ Nx k,n (µ k,n , Σ k,n ) locally for each state entry x k,n [xn] k , which corresponds to the MMSE estimate of either clock parameters ϑi,n or a distance dij,n.…”

“…For each state xn, we can thus compute the marginal function m n|n (x k,n ) ∝ Nx k,n (µ k,n , Σ k,n ) locally for each state entry x k,n [xn] k , which corresponds to the MMSE estimate of either clock parameters ϑi,n or a distance dij,n. The resulting FG for BP can be represented graphically by stacking the FG for the posterior function of [14] in discrete-time n and additionally introducing transition functions p(x k,n |x k,n−1 ) for each vertex x k,n . Similar to the discussion in [14], disjoint subgraphs of this FG can be related to physical network nodes, which leads to a distributed algorithm.…”

“…The resulting FG for BP can be represented graphically by stacking the FG for the posterior function of [14] in discrete-time n and additionally introducing transition functions p(x k,n |x k,n−1 ) for each vertex x k,n . Similar to the discussion in [14], disjoint subgraphs of this FG can be related to physical network nodes, which leads to a distributed algorithm.…”

“…. , yn up to n. We use BP to obtain an efficient computation of (14), which means that we apply message passing over the edges of an FG [19][20][21]. In general, an FG is a graphical representation of the factorization of a function; for example, the factorization of (13) is illustrated in Fig.…”

“…In order to avoid frequent resynchronization procedures for drift compensation, an extension of the FG approach in [9] that offers combined estimation of both clock skew and offset was proposed in [12,13]. On the basis of [13], a distributed synchronization and ranging algorithm has recently been introduced in [14], where a time-invariant WN has been assumed.…”