# Novel Sparse-Coded Ambient Backscatter Communication for Massive IoT Connectivity
## 3 Sparse-Coded Load Modulation
The new graph structure can efficiently exploit the MAI and ISI for backscatter detection, satisfying the second requirement of AmBC for massive connectivity. Based on sparse factor graph structure [39] in UL, the backscatter modulation can be generalized by the pseudo codes in Algorithm 1. Since channel information is not available at RF tags, they encode data using the one-way codebook \(\{\mathbf{B}(m)\}_{m\in\{1,\cdots,M\}}\) without knowing ISI, and the forward and backward codebooks \(\{\mathbf{B}^{+}(m),\mathbf{B}^{-}(m)\}_{m\in\{1,\cdots,M\}}\) containing information on ISI will be used in detection procedure in Section 4. Thus, the reflection coefficients and backscattered signals at RF tags over \(L\) samples can be represented by  
\[\mathbf{x}_{k,n}=\Gamma_{k,n}\mathbf{y}_{k,n}\in\mathbb{C}^{1\times L},\quad \text{where }\Gamma_{k,n}\in \begin{cases}\{0\},&\text{if }a_{n}=0\text{ (idle)},\\ \widetilde{b}_{k,n}(m),&\text{if }a_{n}=1\text{ (active)}.\end{cases}  \tag{15}\]  
Similar to the forward channel, backward channel and interference channel can be defined by  
\[\mathbf{g}_{n} =[g_{n}(1),\cdots,g_{n}(L_{n}^{-})]^{T},\quad\mathbf{h}_{0}=[h_{0 }(1),\cdots,h_{0}(L_{0})]^{T}, \tag{16}\] \[\mathbf{G}^{+}_{n} =\sum_{l^{\prime}=1}^{L_{n}^{-}}g_{n}(l^{\prime})(\mathbf{J}^{+}_ {L})^{l^{\prime}-1}\in\mathbb{C}^{L\times L},\quad\mathbf{G}^{-}_{n}=\sum_{l^ {\prime}=1}^{L_{n}^{-}}g_{n}(l^{\prime})(\mathbf{J}^{-}_{L})^{L-l^{\prime}+1} \in\mathbb{C}^{L\times L},\] \[\mathbf{H}^{+}_{0} =\sum_{l^{\prime}=1}^{L_{0}}h_{0}(l^{\prime})(\mathbf{J}^{+}_{L} )^{l^{\prime}-1}\in\mathbb{C}^{L\times L},\quad\mathbf{H}^{-}_{0}=\sum_{l^{ \prime}=1}^{L_{0}}h_{0}(l^{\prime})(\mathbf{J}^{-}_{L})^{L-l^{\prime}+1}\in \mathbb{C}^{L\times L},\]  
Figure 6: Example of dyadic factor graph composed of forward factor graph \(\widetilde{\mathbf{G}}^{+}\) and backward factor graph \(\widetilde{\mathbf{G}}^{-}\) (\(K=4\), \(N=6\), \(K_{1}=2\), \(L=3\), \(\widetilde{L}=1\)). where \(g_{n}(l^{\prime})\sim\mathcal{CN}(0,\sigma_{g}^{2}/L_{n}^{-})\), \(h_{0}(l^{\prime})\sim\mathcal{CN}(0,\sigma_{h_{0}}^{2}/L_{0})\) and the variance of backward and leakage channels are denoted by \(\sigma_{g}^{2}\) and \(\sigma_{h_{0}}^{2}\), respectively. In Equation (17), \(L_{n}^{-}\) and \(L_{0}\) denote the number of multipath for backward channel and leakage channel, respectively. Then, the signal received at AP at time slot \(k\) can be expressed as  
\[\mathbf{r}_{k} = (\mathbf{H}_{0}^{+}\mathbf{s}_{k}+\mathbf{H}_{0}^{-}\mathbf{s}_{k -1})+\sum_{n=1}^{N}\left(\mathbf{G}_{n}^{+}\mathbf{x}_{k,n}+\mathbf{G}_{n}^{-} \mathbf{x}_{k-1,n}\right)+\mathbf{w}_{k} \tag{17}\] \[= \sum_{n=1}^{N}\underbrace{\left(\mathbf{G}_{n}^{+}\mathbf{F}_{n} ^{+}\mathbf{s}_{k}+\mathbf{G}_{n}^{+}\mathbf{F}_{n}^{-}\mathbf{s}_{k-1} \right)}_{=\widetilde{\mathbf{h}}_{k,n}^{+}}\Gamma_{k,n}+\sum_{n=1}^{N} \underbrace{\left(\mathbf{G}_{n}^{-}\mathbf{F}_{n}^{+}\mathbf{s}_{k-1}+ \mathbf{G}_{n}^{-}\mathbf{F}_{n}^{-}\mathbf{s}_{k-2}\right)}_{=\widetilde{ \mathbf{h}}_{k,n}^{-}}\Gamma_{k-1,n}\] \[+\underbrace{\left(\mathbf{H}_{0}^{+}\mathbf{s}_{k}+\mathbf{H}_{ 0}^{-}\mathbf{s}_{k-1}\right)}_{=\widetilde{\mathbf{h}}_{k,0}}+\mathbf{w}_{k}\] \[= \underbrace{\sum_{n=1}^{N}\widetilde{\mathbf{h}}_{k,n}^{+}\Gamma _{k,n}}_{\text{forward codeword (MAI)}}+\underbrace{\sum_{n=1}^{N}\widetilde{\mathbf{h}}_{k,n}^{-}\Gamma_{k- 1,n}}_{\text{backward codeword (ISI)}}+\underbrace{\widetilde{\mathbf{h}}_{k,0}}_{\text{SI}}+\underbrace{ \mathbf{w}_{k}}_{\text{noise}}\in\mathbb{C}^{L\times 1},\]  
where \(\mathbf{w}_{k}=[w_{1,k},\cdots,w_{L,k}]^{T}\) denotes additive white Gaussian noise with variance \(\sigma_{n}^{2}\). Since the AP knows the ambient signal \(\mathbf{s}_{k}\) for \(k\in\{1,\cdots,K\}\) and the leakage channel \(\mathbf{h}_{0}\), the SI term in Equation (17) can be reconstructed for SIC [2].