# Iterative Receiver Design for the Estimation of Gaussian Samples in Impulsive Noise +
## 3 Factor Graph and Sum-Product Algorithm  
As a common technique to perform maximum a posteriori (MAP) symbol estimation, we use a factor graph representation for the variables of interest and for the relationships among them that stem from the system model at hand. These variables and relationships are in turn the variable nodes and the factor nodes appearing in the FG [18]. The aim is to compute the marginal distributions (pdf) of the random variables to be estimated, i.e., the transmitted signal samples \(s_{k}\), conditioned on the observation of the received noisy samples \(y_{k}\) in (1), whereas the overall FG represents the joint distribution of all the variable nodes, i.e., of transmitted samples, as well as of channel states \(i_{k}\). Despite that this marginalization task is nontrivial, in the case of many (e.g., thousands of) variable nodes, it is brilliantly accomplished by the sum-product algorithm (SPA), which is a message passing algorithm able to reach the exact solution in the case of FGs without loops [18]. Thanks to [PERSON] rule, the joint posterior distribution of signal samples \(\mathbf{s}=\{s_{k}\}\) and channel states \(\mathbf{i}=\{i_{k}\}\) given the observation samples \(\mathbf{y}=\{y_{k}\}\) can be factorized as follows. \[p(\mathbf{s},\mathbf{i}\mid\mathbf{y}) \propto p(\mathbf{y}\mid\mathbf{s},\mathbf{i})p(\mathbf{s})P(\mathbf{i})\] \[=\left[\prod_{k=1}^{K-1}p(y_{k}\mid s_{k},i_{k})p(s_{k}\mid s_{k -1})P(i_{k}\mid i_{k-1})\right]p(y_{0}\mid s_{0},i_{0})p(s_{0})P(i_{0})\quad. \tag{13}\]  
The Markovianity of both the signal sequence and of the noise sequence is a key factor in the application of the chain rule in (13), where simplified expressions result for its factors. Such a short-term memory (The Markov property of signal and noise sequences could be easily extended to a memory larger than one. This would however complicate the resulting notation without introducing an extra conceptual contribution.) implies that, besides the two types of variable nodes, i.e., \(s_{k}\) and \(i_{k}\), the FG consists of three types of factor nodes, i.e., \(p(y_{k}\mid s_{k},i_{k})\), \(p(s_{k}\mid s_{k-1})\), and \(P(i_{k}\mid i_{k-1})\), the latter involving only pairs of time-adjacent variable nodes. This feature is further evidenced by the FG depicted in Figure 3, which graphically represents the joint pdf in (13). The pdfs \(p(s_{k}\mid s_{k-1})\) and \(p(y_{k}\mid s_{k},i_{k})\) and the conditional probability mass function (PMF) \(P(i_{k}\mid i_{k-1})\) simply arise from the system model, i.e., from Equations (1), (2), and (5) or (11), and from the statistical description of the random variables therein. By denoting \(g(x;\eta,\gamma^{2})\) the standard Gaussian pdf with mean \(\eta\) and variance \(\gamma^{2}\), we have:  
\[p(s_{k}\mid s_{k-1}) = g(s_{k};a_{1}s_{k-1},(1-a_{1}^{2})\sigma_{s}^{2}) \tag{14}\] \[P(i_{k}\mid i_{k-1}) = \mathbf{P}(i_{k-1}+1,i_{k}+1)\] (15) \[p(y_{k}\mid s_{k},i_{k}) = g(y_{k};s_{k},\sigma_{i,k}^{2}) \tag{16}\]  
where \(\sigma_{i,k}^{2}=\sigma_{G}^{2}+i_{k}\sigma_{I}^{2}\) is the variance of the observed sample at time \(k\), which depends on the corresponding impulsive noise state \(i_{k}\), represented by the variable node under the factor node (16), in Figure 3. In (15), the transition probability between successive impulsive noise states is found from the entries \(\mathbf{P}(i,j)\) of the matrix (5) or (11), depending on which of the noise models is adopted. A straightforward application of the SPA [18] yields the forward and backward messages exchanged along the top line of the graph,  
\[p_{f}(s_{k}) = \int p_{f}(s_{k-1})p_{u}(s_{k-1})p(s_{k}\mid s_{k-1})ds_{k-1} (k=1,\cdots,K-1) \tag{17}\] \[p_{b}(s_{k}) = \int p_{b}(s_{k+1})p_{u}(s_{k+1})p(s_{k+1}\mid s_{k})ds_{k+1} (k=K-2,\cdots,0) \tag{18}\]  
where the initial condition for (17) is \(p(s_{0})=g(s_{0};0,\sigma_{s}^{2})\), while a constant value, as \(p(s_{K-1})=1\), is assumed to bootstrap the backward messages, which models the absenceof a posteriori information on the last signal sample.