# Iterative Receiver Design for the Estimation of Gaussian Samples in Impulsive Noise +
## 4 Signal Estimation
### Soft Decisions: EP and TP Algorithms  
Hard decisions on every \(i_{k}\) imply that only one part of the information, provided by the lower FG half (Section 4.2), is being used by the signal estimation, in the upper FG half (Section 4.1), and the rest of the information is being discarded. This is clearly a suboptimal approach.  
A better performing soft information strategy can instead be pursued, based on approximating the mixture (41) by minimizing the KL divergence. This can be implemented by using either the EP algorithm or the TP algorithm. To have a better insight into the similarities and differences between EP and TP, considering the posterior marginal \(p(s_{k}|\mathbf{y})\) obtained, according to the SPA rules, as the product of all incoming messages to the variable node \(s_{k}\):  
\[p(s_{k}|\mathbf{y})=p_{f}(s_{k})p_{b}(s_{k})p_{u}(s_{k})=p_{d}(s_{k})p_{u}(s_{k}) \quad, \tag{42}\]  
where \(p_{d}(s_{k})\) in (34) is Gaussian and \(p_{u}(s_{k})\) in (41) is a Gaussian mixture; thus, \(p(s_{k}|\mathbf{y})\) is a Gaussian mixture as well. An approximation for it can be computed by the projection operation (23), once an approximating exponential family \(\mathcal{F}\) is selected. In the problem that we analyze in this manuscript, the Gaussian family seems a natural choice, so that:  
\[\widetilde{p}^{EP}(s_{k}|\mathbf{y})=\text{proj}[p_{d}(s_{k})p_{u}(s_{k})]=g(s_{k} ;s_{k},\phi_{k}^{2})\quad. \tag{43}\]  
This is exactly the approach of the EP algorithm: the posterior marginal is approximated by matching the moments of the approximating Gaussian to those of the mixture, as discussed in Section 3.1, and the upward message \(p_{u}(s_{k})\), sent to the variable node \(s_{k}\) for the next iteration, is replaced by the following, obtained by Gaussian division:  
\[p_{u}^{EP}(s_{k})=\frac{\widetilde{p}^{EP}(s_{k}|\mathbf{y})}{p_{d}(s_{k})}=g(s_{ k};s_{u,k}^{EP},(\phi_{u,k}^{EP})^{2})\quad. \tag{44}\]\[\widetilde{p}^{TP}(s_{k}|\mathbf{y})=p_{d}(s_{k})p_{u}^{TP}(s_{k})\quad, \tag{45}\]  
to finally get the TP symbol estimate from the posterior marginal (45).  
This division of pdfs, which is inherent to the EP algorithm, is not painless and can give rise to improper distributions, which in turn introduce instabilities into the algorithm [21, 26, 27]. This is easily understood by considering a Gaussian division in (44) where the variance of the denominator is larger than the variance of the numerator, so that an absurd negative variance results. Several techniques have been proposed to avoid these instabilities; we adopt the simple improper message rejection discussed in [27].  
An alternative approach, which spontaneously avoids instabilities, is that of the TP algorithm [22], where it is the individual message \(p_{u}(s_{k})\), i.e., the mixture, that is projected onto an approximating (here Gaussian) pdf, instead of the posterior marginal, as done in EP:  
\[p_{u}^{TP}(s_{k})=\text{proj}[p_{u}(s_{k})]=g(s_{k};\tilde{s}_{u,k^{\prime}}^{ TP}(\sigma_{u,k}^{TP})^{2})\quad. \tag{46}\]  
In TP, the posterior marginal, from which the signal sample is estimated, is as usual obtained by multiplying the incoming messages to \(s_{k}\), i.e., the TP message (46) is multiplied by \(p_{d}(s_{k})\):  
Note that the result in (46) would be obtained from the EP projection in (44) if only the operator \(\text{proj}[\cdot]\) in (43) were transparent to \(p_{d}(s_{k})\), i.e., if it were \(\text{proj}[p_{d}(s_{k})p_{u}(s_{k})]=p_{d}(s_{k})\text{proj}[p_{u}(s_{k})]\) (which amounts to stating \(\widetilde{p}^{EP}(s_{k}|\mathbf{y})=\widetilde{p}^{TP}(s_{k}|\mathbf{y})\)), which is in general not the case. Finally, the TP algorithm is inherently stable and does not require any complementary message rejection procedure.