Expectation Propagation and Transparent Propagation in Iterative Signal Estimation in the Presence of Impulsive Noise

Iterative Receiver Design for the Estimation of Gaussian Samples in Impulsive Noise

“…Exponential families have the pleasant property of being closed under the multiplication operation. Furthermore, it can be shown [20] that the projection operation in (23) simply reduces to equating the expectation of each feature with respect to the true distribution p(x), E p [G i (x)], with that computed with respect to the approximating distribution q(x), E q [G i (x)]. Such a procedure of matching the expectations (of the features) is at the heart of (and gives the name to) the celebrated EP (expectation propagation) algorithm [21], further applied in Section 4 to our problem.…”

“…where ∑ i α i = 1 for normalization. We shall approximate it by the nearest Gaussian q(x) in the sense of the KL divergence, i.e., by applying (23). For that purpose, given that the expectations of the features of a Gaussian distributions are simply the mass, the mean, and the mean-squared value of that Gaussian (i.e., the expectations of one, x and x 2 ), the matching of these moments computed under p(x) with those computed under q(x) simply amounts to equating the mean and variance of the two distributions (assuming that both p(x) and q(x) are normalized).…”

“…where p d (s k ) in ( 34) is Gaussian and p u (s k ) in ( 41) is a Gaussian mixture; thus, p(s k |y) is a Gaussian mixture as well. An approximation for it can be computed by the projection operation (23), once an approximating exponential family F is selected. In the problem that we analyze in this manuscript, the Gaussian family seems a natural choice, so that:…”

“…We analyze different estimation algorithms and provide a performance comparison between several suboptimal and close-to-optimal techniques, in various channel conditions. In particular, this paper is an extension of our recent work [23], where EP and TP were applied for the first time to a channel with Markov-Middleton impulsive noise. Besides describing these algorithms in more detail, here, we critically review the channel model and compare it with the simpler Markov-Gaussian model.…”

“…This is the case of the celebrated expectation propagation (EP) algorithm proposed by Minka [21], which is based on KL divergence. The EP algorithm and a similar algorithm called transparent propagation (TP) were recently applied to signal estimation in impulsive noise channels, showing good performance [22,23].…”