# Novel Sparse-Coded Ambient Backscatter Communication for Massive IoT Connectivity
## 3 Sparse-Coded Load Modulation
The SIC output signal after hybrid digital and analog cancellation [41; 45; 46] can be given by  
\[\mathbf{t}_{k}=\mathbf{r}_{k}-\widetilde{\mathbf{h}}_{k,0}\in\mathbb{C}^{L \times 1}. \tag{18}\]  
However, as discussed in [2; 23], AP estimates the composite forward-backward channel \(\{\mathbf{h}_{n}\}_{n\in\{1,\cdots,N\}}\) rather than the individual estimates of the forward channels \(\{\mathbf{f}_{n}\}_{n\in\{1,\cdots,N\}}\) and the backward channels \(\{\mathbf{g}_{n}\}_{n\in\{1,\cdots,N\}}\). The channels \(\mathbf{h}_{n}\) can be efficiently obtained by compressed sensing based channel estimation techniques [55], which can be defined as  
\[\mathbf{h}_{n} = [h_{n}(1),\cdots,h_{n}(L_{n}^{\star})]^{T},\] \[\mathbf{G}^{+}_{n} =\sum_{l^{\prime}=1}^{L_{n}^{-}}g_{n}(l^{\prime})(\mathbf{J}^{+}_ {L})^{l^{\prime}-1}\in\mathbb{C}^{L\times L},\quad\mathbf{G}^{-}_{n}=\sum_{l^ {\prime}=1}^{L_{n}^{-}}g_{n}(l^{\prime})(\mathbf{J}^{-}_{L})^{L-l^{\prime}+1} \in\mathbb{C}^{L\times L},\]0  
where \(L_{n}^{\star}=L_{n}^{+}+L_{n}^{-}-1\) denotes the number of multipath for the composite channel. Assuming i.i.d. fully-correlated DBC model with channel reciprocity appearing in in-band wireless-powered communication network [56], the backward channel satisfies \(\mathbf{g}_{n}=\mathbf{f}_{n}\) and has \(L^{-}=L_{n}^{-}=L^{+}=L_{n}^{+}\) multipath. Consequently, for AmBC detection, we should estimate the forward channels with length \(L^{+}\) from the dyadic channels with \(L^{\star}=2L^{+}-1\). Based on the circular convolution operation \(\star\), the relation between \(\mathbf{f}_{n}\) and \(\mathbf{h}_{n}\) can be represented by  
\[\mathbf{G}^{+}_{n} =\sum_{l^{\prime}=1}^{L_{n}^{-}}g_{n}(l^{\prime})(\mathbf{J}^{+}_ {L})^{l^{\prime}-1}\in\mathbb{C}^{L\times L},\quad\mathbf{G}^{-}_{n}=\sum_{l^ {\prime}=1}^{L_{n}^{-}}g_{n}(l^{\prime})(\mathbf{J}^{-}_{L})^{L-l^{\prime}+1} \in\mathbb{C}^{L\times L},\]1  
For example, if \(L^{+}=3\), Equation (20) can be expressed by  
\[\mathbf{G}^{+}_{n} =\sum_{l^{\prime}=1}^{L_{n}^{-}}g_{n}(l^{\prime})(\mathbf{J}^{+}_ {L})^{l^{\prime}-1}\in\mathbb{C}^{L\times L},\quad\mathbf{G}^{-}_{n}=\sum_{l^ {\prime}=1}^{L_{n}^{-}}g_{n}(l^{\prime})(\mathbf{J}^{-}_{L})^{L-l^{\prime}+1} \in\mathbb{C}^{L\times L},\]2 (21) \[\mathbf{G}^{+}_{n} =\sum_{l^{\prime}=1}^{L_{n}^{-}}g_{n}(l^{\prime})(\mathbf{J}^{+}_ {L})^{l^{\prime}-1}\in\mathbb{C}^{L\times L},\quad\mathbf{G}^{-}_{n}=\sum_{l^ {\prime}=1}^{L_{n}^{-}}g_{n}(l^{\prime})(\mathbf{J}^{-}_{L})^{L-l^{\prime}+1} \in\mathbb{C}^{L\times L},\]3 \[\mathbf{G}^{+}_{n} =\sum_{l^{\prime}=1}^{L_{n}^{-}}g_{n}(l^{\prime})(\mathbf{J}^{+}_ {L})^{l^{\prime}-1}\in\mathbb{C}^{L\times L},\quad\mathbf{G}^{-}_{n}=\sum_{l^ {\prime}=1}^{L_{n}^{-}}g_{n}(l^{\prime})(\mathbf{J}^{-}_{L})^{L-l^{\prime}+1} \in\mathbb{C}^{L\times L},\]4  
where the second term \(\hat{\mathbf{F}}_{n}^{-}\hat{\mathbf{F}}_{n}^{-}\mathbf{s}_{k-2}=\mathbf{0}_{L}\) is neglected under the condition \(L^{\star}\leq L\)[23]. **Remark 2**. : _The dyadic channels are even such that_  
\[\mathbf{G}^{+}_{n} =\sum_{l^{\prime}=1}^{L_{n}^{-}}g_{n}(l^{\prime})(\mathbf{J}^{+}_ {L})^{l^{\prime}-1}\in\mathbb{C}^{L\times L},\quad\mathbf{G}^{-}_{n}=\sum_{l^ {\prime}=1}^{L_{n}^{-}}g_{n}(l^{\prime})(\mathbf{J}^{-}_{L})^{L-l^{\prime}+1} \in\mathbb{C}^{L\times L},\]5 \[\mathbf{G}^{+}_{n} =\sum_{l^{\prime}=1}^{L_{n}^{-}}g_{n}(l^{\prime})(\mathbf{J}^{+}_ {L})^{l^{\prime}-1}\in\mathbb{C}^{L\times L},\quad\mathbf{G}^{-}_{n}=\sum_{l^ {\prime}=1}^{L_{n}^{-}}g_{n}(l^{\prime})(\mathbf{J}^{-}_{L})^{L-l^{\prime}+1} \in\mathbb{C}^{L\times L},\]6  
Since \(\hat{\mathbf{t}}_{n}=\pm\mathbf{f}_{n}\), we may incorrectly estimate the channels when \(\hat{\mathbf{t}}_{n}=-\mathbf{f}_{n}\). However, the even symmetry of the channels ensures correct estimation of the channels regardless of their directions.