# Intrinsic Interference Use for FBMC-IOTA Systems
## 2 MAP Detection on Weight Matrix  
Consider a FBMC-IOTA system with \(N\) subcarriers and \(L\) symbols, the transmitted signal can be written as  
\[s(t)=\sum_{n=0}^{L-1}\sum_{m=0}^{N-1}a_{m,n}g_{m,n}(t) \tag{1}\]  
where \(a_{m,n}\) and \(g_{m,n}(t)\) represent the symbol modulated by the \(m\)th subcarrier at the \(n\)th symbol time and the synthesis basis obtained by the IOTA function \(g(t)\), respectively. After passing by a doubly dispersive channel, the demodulator output can be expressed as [15]  
\[\hat{a}_{k,l} = n_{k,l}+a_{k,l}H_{k,l}\] \[+ \underbrace{\sum_{p,q\
eq 0}a_{k+p,l+q}j^{p+q+p(q+2l)}A_{g}^{*}(q \tau_{0},p\upsilon_{0})}_{h_{k,l}}H_{k,l}\]  
where \(H_{k,l}\) is the channel coefficient at the \(l\)th symbol and the \(k\)th subcarrier frequency, \(A_{g}(\tau,\
u)\) is the auto-ambiguity function of \(g(t)\), \(n_{k,l}\) is the channel noise, \(I_{k,l}\) is the intrinsic interference and can be written in the form of \(l_{k,l}=ja_{k,l}^{(i)}\). Such intrinsic interference contains rich inherent information that has never been used. In fact, the intrinsic interference can be determined by multiplying the neighboring symbols with a weighting matrix, the elements of which can be calculated using the ambiguity function of the employed pulse shape filter [16]. The matrix that presents the weights corresponding to each neighboring symbol is shown below  
\[W_{l_{\text{even}}}= \begin{bmatrix}0&0&-\eta&0&\eta&0&0\\ -\lambda&-\zeta&-\gamma&-\delta&-\gamma&-\zeta&-\lambda\\ -\theta&0&-\beta&1&\beta&0&\theta\\ -\lambda&\zeta&-\gamma&\delta&-\gamma&\zeta&-\lambda\\ 0&0&-\eta&0&\eta&0&0\end{bmatrix}  \tag{3}\]  
\[W_{l_{\text{odd}}}= \begin{bmatrix}0&0&-\eta&0&\eta&0&0\\ \lambda&\zeta&\gamma&\delta&\gamma&\zeta&\lambda\\ -\theta&0&-\beta&1&\beta&0&\theta\\ \lambda&-\zeta&\gamma&-\delta&\gamma&-\zeta&\lambda\\ 0&0&-\eta&0&\eta&0&0\end{bmatrix}  \tag{4}\]  
where \(l_{\text{even}}\) and \(l_{\text{odd}}\) respectively corresponds to the weight for real and imaginary symbols, and \(\delta=0.2486\), \(\beta=0.5756\), \(\gamma=0.1898\), \(\eta=0.0021\), \(\theta=0.0956\), \(\lambda=0.0473\), \(\zeta=0.0991\). For each transmitted information symbol \(a_{k,l}\), its associated intrinsic interference \(a_{k,l}^{(i)}\) is determined by neighboring symbols and can be calculated as in (2) using the above weighting matrices. We find that such principle is essentially the same as block coding where each symbol \(a_{k,l}\) is associated with some non-binary parity symbols \(P_{k,l}=a_{k,l}^{(i)}\). In the following we will focus on the real branch of FBMC-IOTA receiver, and the analysis can be applied to the imaginary branch straightforwardly. Given \(\mathbf{p}=[P_{k,l}:k\in[0,\ldots,N-1]\,\ l\in[0,\ldots,L-1]]\) as the vector of observed parity symbols at real branch, the MAP detection will estimate \(\mathbf{\hat{a}}\) that maximizes the joint a posteriori probability mass function (PMF) of the transmitted symbol vector \(\mathbf{a}=[a_{m,n}:m\in[0,\ldots,N-1]\,\ n\in[0,\ldots,L-1]\), i.e.,  
\[\mathbf{\hat{a}}=\arg\max_{\mathbf{a}}p(\mathbf{a}|\mathbf{p}). \tag{5}\]  
The MAP detector can be implemented by two approaches: individual and joint optimum detection [18; 19]. The joint approach maximizes the joint a posteriori PMF of the transmitted symbol vector as shown above, while the individual optimum MAP detection maximizes the a posteriori PMF, \(p(a_{k,l}|\mathbf{p})\), of each individual symbol. Let \(\mathbb{X}\) be the constellation alphabet, from which \(a_{m,n}\) will take its value, the estimation of \(a_{k,l}\) with individual MAP detection can be written as  
\[\hat{a}_{k,l}=\arg\max_{a_{m,n}\in\mathbb{X}}p(a_{m,n}|\mathbf{p}) \tag{6}\]  
A posteriori PMF for \(a_{k,l}\) can be found by calculating the marginal of the joint a posteriori PMF, thus (6) can be written as  
\[\hat{a}_{k,l}=\arg\max_{a_{m,n}\in\mathbb{X}}\sum_{\mathbf{a}\in\mathbb{X}^{N,L}}p (\mathbf{a}|\mathbf{p}) \tag{7}\]  
Let \(Pr(a)\) be the priori probability of symbol \(a\).