# Novel Sparse-Coded Ambient Backscatter Communication for Massive IoT Connectivity
## 4 Iterative Detection on Dyadic Channel  
For reliable detection of AmBC on DBC, we design the iterative detector with the following characteristics:  
* **Interference Exploitation**: To achieve massive connectivity in AmBC, the interferences (i.e., MAI and ISI) should be exploited in non-orthogonal manners [31; 32], instead of excluding them by using orthogonal signaling as in conventional detectors [2; 23]. In addition, iterative MPA should be modified for DBC to achieve additional gain without loss of backscatter signal power. * **Low Complexity**: The received signals in AP are superposition of multiple backscatter signals with \(M\)-ary modulation. In addition, ISI introduces another superposition of backscatter signals with delay spreads in the dyadic channels, further increasing complexity of iterative detector. Hence, the complexity of detector should be as low as possible for practical applications of SC-AmBC. To achieve these goals, we describe an iterative detector especially on DBC in this section. The SIC output [41] in Equation (18) is rewritten by superposition of \(N\) reflected signals in time slot \(k\) and delayed signals in time slot \(k-1\), which is  
\[ \begin{split}\mathbf{t}_{k}&=[t_{1,k},\cdots,t_{L,k} ]^{\mathrm{T}}=\sum\limits_{n=1}^{N}\widetilde{\mathbf{h}}_{k,n}^{+}\Gamma_{k, n}+\sum\limits_{n=1}^{N}\widetilde{\mathbf{h}}_{k-1,n}^{-}\Gamma_{k-1,n}+ \mathbf{w}_{k}\in\mathbb{C}^{L\times 1},\\ t_{l,k}&=\sum\limits_{n=1}^{N}\widetilde{h}_{k,n,l}^ {+}\Gamma_{k,n}+\sum\limits_{n=1}^{N}\widetilde{h}_{k,n,l}^{-}\Gamma_{k-1,n}+w _{l,k}\in\mathbb{C}.\end{split}  \tag{30}\]  
Let the \(n\)-th codeword \(\gamma_{n}=[\Gamma_{1,n},\cdots,\Gamma_{K,n}]^{\mathrm{T}}\in\mathcal{X}_{n}\), where \(\mathcal{X}_{n}\subset\mathbb{C}^{K\times 1}\) is the codebook of tag \(n\) with cardinality \(|\mathcal{X}_{n}|=M\). Define a codeword matrix \(\mathbf{X}=[\gamma_{1},\cdots,\gamma_{N}]\in\mathbb{C}^{K\times N}\) and a received signal matrix \(\mathbf{Y}=[\mathbf{t}_{1},\cdots,\mathbf{t}_{K}]\in\mathbb{C}^{K\times N}\). Then, the maximum a posteriori (MAP) detection rule for codeword \(\mathbf{X}\) can be expressed as [32]  
\[\mathbf{\hat{X}}=\arg\max_{\mathbf{X}\in\mathcal{X}_{1}\times\cdots\times \mathcal{X}_{N}}p(\mathbf{X}|\mathbf{Y}). \tag{31}\]  
The complexity of the above problem is shown to increase exponentially with the number of tags \(N\) and polynomially with the constellation size \(M\)[32]. Fortunately, the MAP detection problem in Equation (31) can be translated into marginalize product of function (MPF) problem which can be efficiently solved by MPA [44]. In MPA, where codewords are iteratively decoded over the factor graph, we can achieve near-optimal detection performance with low complexity [32; 33; 39; 42]. However, as described in Figure 7, DBC incurs significant ISI for detection of codeword \(\mathbf{X}\). If we simply filter ISI using inaccurate channel estimation as in [2; 23], we lose a large portion of signal power (e.g., maximum 62.5% in Figure 7c), especially for high-rate AmBC applications. Thus, to prevent the SNR loss caused by DBC, we should modify the conventional MPA [44] to exploit the weighted-sum of ISI. In addition, we adopt the log-domain MPA [43; 44], which shows superior decoding performance than MPA for reliable detection of AmBC at low SNR. To achieve this goal, we define the dyadic factor graph for ambient backscatter as follows:  
\[\widetilde{\mathbf{G}}^{\star}=\widetilde{\mathbf{G}}^{+}\oplus\widetilde{ \mathbf{G}}^{-}, \tag{32}\]  
where the operator \(\oplus\) denotes element-wise OR operation. In addition, by the channel reciprocity, \(\widetilde{L}=L^{+}-1\). **Remark 3**.