# Intrinsic Interference Use for FBMC-IOTA Systems
## 3 MPA on Intrinsic Interference  
Although the joint MAP detection problem has been translated into an MPF problem as shown in (12) which is much simpler to resolve, the brute-force solution for (11) requires exponential complexity. In order to reduce the complexity, we propose a novel iterative symbol detector IIU based on the generic MPA [21], where MPA is applied to the weight matrix to iteratively approximate the solution of the MPF problem.  
Let \(l_{a\to P}(a_{k,l},P_{k^{\prime},l^{\prime}})\) and \(l_{a\to P}(a_{k,l},P_{k^{\prime},l^{\prime}})\) be the message in the form of log-likelihood ratio (LLR) delivered from \(a_{k,l}\) and \(P_{k^{\prime},l^{\prime}}\), respectively. We denote \(\xi_{k,l}\) as the set of positions \((k^{\prime},l^{\prime})\) that \(a_{k,l}\) interferes on and \(\xi_{k^{\prime},l^{\prime}}\) as the set of positions \((k,l)\) that interfere on \(P_{k^{\prime},l^{\prime}}\), then  
\[l_{a\to P}(a_{k,l},P_{k^{\prime},l^{\prime}}) \tag{14}\] \[=\log\frac{Pr_{\text{ext}}(a_{k,l}=1)}{Pr_{\text{ext}}(a_{k,l}=-1)}\] \[=\sum_{(n,m)\in\xi_{k,l}\setminus(k^{\prime},l^{\prime})}l_{a \to P}(a_{k,l},P_{n,m})\]  
By using the message formulated in (14), we have  
\[Pr_{\text{ext},k^{\prime},l^{\prime}}(a_{k,l})=\lambda_{k,l}\exp\left(\frac{a _{k,l}}{2}l_{a\to P}(a_{k,l},P_{k^{\prime},l^{\prime}})\right) \tag{15}\]  
where \(\lambda_{k,l}\) is chosen such that \(Pr_{\text{ext},k^{\prime},l^{\prime}}(a_{k,l}=+1)+Pr_{\text{ext},k^{\prime},l ^{\prime}}(a_{k,l}=-1)=1\), and the subscript \"ext\" denotes that only the extrinsic information is used.  
The inference of parity symbol \(P_{k^{\prime},l^{\prime}}\) to data symbol \(a_{k,l}\) is updated as follows  
\[l_{a\to P}(a_{k,l},P_{k^{\prime},l^{\prime}}) \tag{16}\] \[=\log\frac{p_{\text{ext},k^{\prime},l^{\prime}}(a_{k,l}=+1|P_{k^ {\prime},l^{\prime}},\mathbf{a}^{[k^{\prime},l^{\prime}]}\backslash a_{k,l})}{p_{ \text{ext},k^{\prime},l^{\prime}}(a_{k,l}=-1|P_{k^{\prime},l^{\prime}},\mathbf{a}^ {[k^{\prime},l^{\prime}]}\backslash a_{k,l})}\] \[=\log\frac{p_{\text{ext},k^{\prime},l^{\prime}}(P_{k,l}|\mathbf{a}^{[ k^{\prime},l^{\prime}]},a_{k,l}=+1)Pr(\mathbf{a}^{[k^{\prime},l^{\prime}]} \backslash a_{k,l})}{p_{\text{ext},k^{\prime},l^{\prime}}(P_{k,l}|\mathbf{a}^{[ k^{\prime},l^{\prime}]},a_{k,l}=-1)Pr(\mathbf{a}^{[k^{\prime},l^{\prime}]} \backslash a_{k,l})}\]  
where the second equality holds following the Bayes rule of (8). Based on MPA, the a priori PMF of \(a_{k,l}\) is not included in the computation of a posteriori PMF of \(a_{k,l}\).  
Substituting (13)-(15) into (16), we can derive \(l_{a\to P}(a_{k,l},P_{k^{\prime},l^{\prime}})\) in the expression of (17), where \(\setminus\) in (16) represents exception.  
\[l_{a\to P}(a_{k,l},P_{k^{\prime},l^{\prime}})=\log\frac{\sum_{\begin{subarray}{ c}\mathbf{a}^{[k^{\prime},l^{\prime}]}\in\mathbb{X}^{\text{exp}}\left(\sum_{(n,m)\in \xi_{k^{\prime},l^{\prime}}\setminus\{k\}}\frac{n_When LLR is converged or the maximum number of iterations is reached, the final estimated inference of \(a_{k,l}\) will be calculated as  
\[l_{k,l}(a_{k,l})=\sum_{(n,m)\in\mathcal{\hat{b}}_{k,l}}l_{a\gets P}(a_{k,l},P _{n,m}) \tag{18}\]  
Consequently, the hard decision of \(a_{k,l}\) is  
\[\hat{a}_{k,l}=\arg\max_{a_{k^{\prime},l^{\prime}}\in\mathcal{X}}l_{k^{\prime},l ^{\prime}}(a_{k,l}) \tag{19}\]