# Iterative Receiver Design for the Estimation of Gaussian Samples in Impulsive Noise +
## 3 Factor Graph and Sum-Product Algorithm
In a similar way, the forward and backward messages exchanged along the bottom line of the FG are:  
\[P_{f}(i_{k}) = \sum_{i_{k-1}}\!P(i_{k}\mid i_{k-1})P_{f}(i_{k-1})P_{d}(i_{k-1}) \quad(k=1,\cdots,K-1) \tag{19}\] \[P_{b}(i_{k}) = \sum_{i_{k+1}}\!P(i_{k+1}\mid i_{k})P_{b}(i_{k+1})P_{d}(i_{k+1}) \quad(k=K-2,\cdots,0)\enspace. \tag{20}\]  
where the initial conditions clearly depend on the adopted noise model. In the Markov-Gaussian case, the pdf of the initial impulsive noise state is \(p(i_{0})=P_{G}\delta(i_{0})+P_{B}\delta(i_{0}-1)\), while it is expressed as \(p(i_{0})=\sum_{n=0}^{M-1}P_{n}^{\prime}\delta(i_{0}-n)\), in the case of Markov-Middleton noise, according to the descriptions in Sections 2.1 and 2.2. In both cases, a uniform PMF (obtained by setting a possibly unnormalized equal value, as \(P(i_{K-1})=1\), for any of its realizations) models the absence of prior assumptions on the last impulsive noise state. The upward and downward messages, on the vertical branches of the FG, are directed, respectively, towards the variable nodes \(s_{k}\) and \(i_{k}\) of our interest, and their expressions are:  
\[=\left[\prod_{k=1}^{K-1}p(y_{k}\mid s_{k},i_{k})p(s_{k}\mid s_{k -1})P(i_{k}\mid i_{k-1})\right]p(y_{0}\mid s_{0},i_{0})p(s_{0})P(i_{0})\quad. \tag{13}\]0 \[=\left[\prod_{k=1}^{K-1}p(y_{k}\mid s_{k},i_{k})p(s_{k}\mid s_{k -1})P(i_{k}\mid i_{k-1})\right]p(y_{0}\mid s_{0},i_{0})p(s_{0})P(i_{0})\quad. \tag{13}\]1  
where, given the Gaussian expression (16), the integral in (22) takes the form of a convolution. The described FG, as is evident from Figure 3, includes loops; hence, the SPA does not terminate spontaneously. As is well known, approximate variational inference techniques are required, in these cases [18]. We shall describe some of them in Section 4--including both traditional approaches and novel ones--and the performance of the resulting algorithms will be compared in Section 5. Nevertheless, all of them share a few fundamental features: (i) first, the resulting algorithms are all iterative, passing messages more than once along the same FG edge and direction; (ii) as a consequence, scheduling is a main issue, since it defines the message passing procedure; (iii) during iterations, the complexity of messages spontaneously tends to \"explode\", so that clever message approximation strategies must be devised [18; 19]. More specifically, regarding message approximations (Point iii) above), the usual approach is the following. Cast in qualitative parlance, one should: (1) select an approxi  
\[=\left[\prod_{k=1}^{K-1}p(y_{k}\mid s_{k},i_{k})p(s_{k}\mid s_{k -1})P(i_{k}\mid i_{k-1})\right]p(y_{0}\mid s_{0},i_{0})p(s_{0})P(i_{0})\quad. \tag{13}\]2 Factor graph representation of (13). mating family, i.e., a family of pdfs/PMFs that is flexible and tractable enough, possibly exhibiting closure properties under multiplication, convolution, or other mathematical operations; (2) select an appropriate divergence measure to quantify the accuracy of the approximation, hence to identify the best approximating pdf within the chosen family [20]. While there is no standard method to select the approximating family, the most common choice for the divergence measure is the Kullback-Leibler (KL) divergence, for which we give a few details hereafter.