# Novel Sparse-Coded Ambient Backscatter Communication for Massive IoT Connectivity
## 4 Iterative Detection on Dyadic Channel
\begin{cases}\widetilde{\mathsf{I}}_{ \widetilde{k}\to n}^{-}(m_{n}),\quad\text{if }\mathcal{D}(\widetilde{k},n),\\ \widetilde{\mathsf{I}}_{\widetilde{k}\to n}(m_{n}),\quad\text{otherwise}, \end{cases}  \tag{49}\]  
where \(\mathrm{I}_{\widetilde{k}\to n}(m_{n})\) denotes the expanded information which is also summarized in Table 1.  
After \(N_{I}\) iterations, VN \(n\) estimates logarithm of the probability of the code \(m_{n}\) as  
\[Q_{n}(m_{n})\leftarrow\ln ap_{n}(m_{n})+\sum_{i\in\mathcal{J}_{n}}\mathrm{I}_ {i\to n}(m_{n}). \tag{50}\]  
Finally, the log-likelihood ratio (LLR) for \(m^{\intercal}\)-th most significant bit (MSB) where \(m^{\intercal}\in\{1,\cdots,\log_{2}M\}\) for tag \(n\) can be calculated as [44]  
\[\Theta_{n,m^{\intercal}}\leftarrow\ln\left(\frac{p(d_{n,m^{\intercal}}^{ \intercal})=0}{p(d_{n,m^{\intercal}}^{\intercal})=1}\right)=\mathsf{m}\mathsf{ x}_{m^{\prime}:d_{n^{\intercal}}^{\intercal}(m^{\prime})=0}\left(Q_{n}(m^{ \prime})\right)-\mathsf{m}\mathsf{x}_{m^{\prime}:d_{m^{\intercal}}^{\intercal}(m^ {\prime})=1}\left(Q_{n}(m^{\prime})\right). \tag{51}\]Then, backscatter data can be detected as  
\[d_{n,m^{\mathrm{t}}}^{\mathrm{t}}= \begin{cases}1,&\text{ if }\Theta_{n,m^{ \mathrm{t}}}\leq 0\\ 0,&\text{ if }\Theta_{n,m^{\mathrm{t}}}>0\end{cases} ,\quad\text{for }a_{n}=1. \tag{52}\]  
Based on the iterative detector utilizing MAI and ISI, high-rate backscatter can be achievable by exploiting inherent characteristic of sparsity in ambient energy harvesting as well as DBC providing additional diversity gain. **Lemma 1**. : _The worst-case complexity of the iterative detector is given by_  
\[\mathcal{O}\left(\left(1-\frac{\widetilde{L}}{L}\right)\widetilde{M}^{K-1}+ \frac{\widetilde{L}}{L}M^{2K-3}\right). \tag{53}\]  
Proof. : Information from FN to VN can be decomposed into two cases. In the first case where ISI is not affected, \(K(L-\widetilde{L})\) FNs are connected to \(K-1\) VNs, respectively, and \(M\)-ary codeword information is passed to VNs, resulting \(M^{K-1}\) combination of information. If \(M\)-ary codeword is projected to \(\widetilde{M}\) symbols, the size of information can be reduced to \(\widetilde{M}^{K-1}\) using information projection [35; 36]. On the other hand, \(K\widetilde{L}\) FNs affected by ISI are connected to additional \(K-2\) VNs in dyadic factor graph, resulting \(M^{2K-3}\) combination of information. Since RF tag relies on energy harvesting, the complexity of detector diminishes as density of active tags becomes sparser. As expressed in Equation (53), ISI increases complexity of the iterative detector for achieving additional diversity gain. However, the fraction of the FNs affected by the interferences is very small when \(\widetilde{L}<<L\) for sufficiently long periods of backscatter symbols. Therefore, we can enjoy low-complexity detection even for non-orthogonal and high-order AmBC. Finally, the average sum throughput of the proposed SC-AmBC can be evaluated as  
\[C=\lambda R_{b}p\left(\hat{\mathcal{I}}|\mathcal{E}\right)p\left(\mathcal{E} \right), \tag{54}\]  
Figure 7: Example of AmBC signals in DBC (\(K=4\), \(L=8\), \(L^{+}=3\)). (**a**) transmitted signal from AP; (**b**) received signal at tag with ISI from forward channel; (**c**) superposition signal at AP with ISI from a dyadic channel. where \(R_{b}=R_{s}/K\log_{2}M\) denotes the bit rate of RF tags and the successful information decoding event and energy harvesting event are denoted by \(\hat{\mathcal{I}}\) and \(\mathcal{E}\), respectively. The probability of successful energy harvesting and information decoding can be calculated as  
\[p\left(\hat{\mathcal{I}}|\mathcal{E}\right) =\mathbb{E}_{a_{n}=1}\left[\frac{\left\|\hat{\mathbf{d}}_{n}^{ \dagger}\odot\mathbf{d}_{n}^{\dagger}\right\|^{2}}{\log_{2}M}\right], \tag{55}\] \[p\left(\mathcal{E}\right) =\mathbb{E}\left[a_{n}\right], \tag{56}\]  
where \(\odot\) denotes exclusive-NOR operator. In conclusion, the comparison of the conventional TD-AmBC and the proposed SC-AmBC are summarized in Table 2.