# Novel Sparse-Coded Ambient Backscatter Communication for Massive IoT Connectivity
## 4 Iterative Detection on Dyadic Channel
: _The dyadic factor graph \(\widetilde{\mathbf{G}}^{\star}\) with \(\widetilde{K}=KL\) FNs and \(N\) VNs satisfies the following properties:_  
* _There are_ \(K(L-\widetilde{L})\) _FNs with degree_ \(K-1\) _and_ \(K\widetilde{L}\) _FNs with degree_ \(2K-3\)_._
* _There are_ \(K\) _VNs with degree_ \(2L+\widetilde{L}\) _and_ \(N-K\) _VNs with degree_ \(2L+2\widetilde{L}\)_._  
If \(\widetilde{L}=0\), the graph is equivalent to the spreading of the one-way factor graph \(\widetilde{\mathbf{G}}\) with spreading factor \(L\). As described in Figure 6, \(\widetilde{L}\) samples are collided by previous backscatter symbols causing ISI and increasing complexity of decoding. The condition \(\widetilde{L}<L\) ensures low-complexity decoding since the collision is only affected by neighboring backscatter symbols [23]. Especially, from Remark 3, information exchanged between FNs and VNs can be decomposed into two cases for reducing computational complexity while exploiting ISI. To perform decoding, we arrange the received signals and channels as follows:  
\[\vec{\mathbf{y}} =[\vec{y}_{1},\cdots,\vec{y}_{\vec{k}}]^{T}=[\mathbf{t}_{1}^{T}, \cdots,\mathbf{t}_{\vec{k}}^{T}]^{T}\in\mathbb{C}^{\vec{K}\times 1}, \tag{33}\] \[\vec{\mathbf{h}}_{n}^{+} =\left[\vec{h}_{1,n^{\prime}}^{+},\cdots,\vec{h}_{\vec{K},n}^{+} \right]^{T}=\left[\left(\vec{\mathbf{h}}_{1,n}^{+}\right)^{T},\cdots,\left(\vec {\mathbf{h}}_{\vec{K},n}^{+}\right)^{T}\right]^{T}\in\mathbb{C}^{\vec{K}\times 1},\] (34) \[\vec{\mathbf{h}}_{n}^{-} =\left[\vec{h}_{1,n^{\prime}}^{-},\cdots,\vec{h}_{\vec{K},n}^{-} \right]^{T}=\left[\left(\vec{\mathbf{h}}_{1,n}^{-}\right)^{T},\cdots,\left( \vec{\mathbf{h}}_{\vec{K},n}^{-}\right)^{T}\right]^{T}\in\mathbb{C}^{\vec{K} \times 1}. \tag{35}\]  
Then, we define symbol vector composed of \(\widetilde{M}=\min(M,3)\leq M\) components as  
\[\vec{\mathbf{s}}= \begin{cases}[\vec{S}(\widetilde{m}_{1}),\vec{S}(\widetilde{ m}_{2})]=\sqrt{\alpha}[1,-1],&\text{for }M=2,\\ [\vec{S}(\widetilde{m}_{1}),\vec{S}(\widetilde{m}_{2}),\vec{S}(\widetilde{m}_{ 3})]=\sqrt{\alpha}[1,0,-1],&\text{for }M\geq 4.\end{cases}  \tag{36}\]  
For notational convenience, we define the ISI condition as follows:  
\[\mathcal{D}(\widetilde{k},n)=\sum_{n=1}^{N}a_{n}\widetilde{g}_{\vec{k},n}^{-}>0. \tag{37}\]  
Given the received signal \(\vec{\mathbf{y}}\), the channels \(\{\vec{\mathbf{h}}_{n}^{+},\vec{\mathbf{h}}_{n}^{-}\}_{\forall n}\), the noise variance \(\sigma_{n}^{2}\), the tag state \(\mathbf{a}\)[15], the factor graphs \(\{\widetilde{\mathbf{G}}^{+},\widetilde{\mathbf{G}}^{-},\widetilde{\mathbf{ G}}^{\star}\}\) and the codebooks \(\{\mathbf{B}^{+}(m),\mathbf{B}^{-}(m)\}_{\forall n}\), the initial information on FN \(\widetilde{k}\) can be formulated as  
\[f_{\widetilde{k}}^{\dagger}(\{m_{i}\}_{i\in\mathcal{I}_{\widetilde{k}}}) \leftarrow \begin{cases}-\frac{1}{2\sigma_{n}^{2}}\left\|\vec{y}_{ \widetilde{k}}-\sum_{i\in\mathcal{I}_{\widetilde{k}}}\left[\vec{h}_{k,i}^{+}B_ {k,i}^{+}(m_{i})+\vec{h}_{k,i}^{-}B_{k,i}^{-}(m_{i})\right]\right\|^{2},&\text {if }\mathcal{D}(\widetilde{k},n),\\ -\frac{1}{2\sigma_{n}^{2}}\left\|\vec{y}_{\widetilde{k}}-\sum_{i\in\mathcal{I}_ {\widetilde{k}}}\vec{h}_{k,i}^{+}B_{k,i}^{+}(m_{i})\right\|^{2},&\text{otherwise },\end{cases}  \tag{38}\]  
where the set \(\mathcal{I}_{\widetilde{k}}=\left\{n\big{|}a_{n}\widetilde{g}_{\vec{k},n}^{ \star}>0\right\}\) denotes the VN indices connected to FN \(\widetilde{k}\) and codeword indices satisfy \(m_{i}\in\{1,\cdots,M\}\). If FN \(\widetilde{k}\) is affected by ISI, the initial information is modified by adding the weighted-sum of ISI represented by the codebook \(\{\mathbf{B}^{-}(m)\}_{\forall m}\) in Equation (38). On the other hand, the dimension of the information in Equation (38) increases exponentially as modulation order \(M\) increases [35; 36].