# Multi-Dimensional Sparse-Coded Ambient Backscatter Communication for Massive IoT Networks
## 2 System Model
### Channel Model  
Once the \(U\) ambient signals are generated by the sources, they can be excitation signals for \(N\) RF tags for AmBC and backscattered to the reader for \(K\) time slots. In AmBC, channels among the sources and the reader can be characterized as follows:  
**Direct-link Channels:** They denote the direct-links from sources to the reader in AmBC and can be harmful interferences for decoding tag data when these channels are strong [2; 3; 11; 12]. Let \(d_{\text{S}_{u}R}\) denote  
Figure 2: General structure of OFDM carrier in frequency domain. the distance from the source \(u=1,\cdots,U\) to the reader and \(L_{\mathbf{h}_{0,u}}\) indicate the number of multipaths [25] (e.g., they include asynchronism, Doppler shift, propagation delay and so on). Then the impulse responses of the direct-link channels for every sampling period \(T_{c}\) are represented by  
\[\mathbf{h}_{0,u}=[h_{0,u}(1),\cdots,h_{0,u}(L_{\mathbf{h}_{0,u}})]^{T}\in \mathbb{C}^{L_{\mathbf{h}_{0,u}}\times 1},\quad h_{0,u}(l)\sim\mathcal{CN} \left(0,\frac{\sigma_{\mathbf{h}_{0,u}}^{2}}{L_{\mathbf{h}_{0,u}}}\right), \quad\sigma_{\mathbf{h}_{0,u}}^{2}=\frac{G_{\mathrm{s}}A_{e}}{4\pi d_{\mathrm{ S_{u}}R}^{2}}, \tag{5}\]  
where \(G_{\mathrm{s}}\) is the antenna gain and \(A_{e}\) the effective aperture (e.g., \(A_{e}\approx 0.0012\) in 2.4 GHz) [35]. Since the ambient signals are wideband (e.g., 20 MHz) and experience frequency-selective fading, the frequency-domain channel representation using discrete Fourier transform (DFT) is necessary for analysis, which is given by  
\[\hat{\mathbf{h}}_{0,u}=\mathbf{\Omega}^{-1}[\mathbf{h}_{0,u}^{T}\ \mathbf{O}_{L-L_{\mathbf{h}_{0,u}}}^{T}]^{T}\in\mathbb{C}^{L\times 1}. \tag{6}\]  
On the other hand, the time-domain direct-link channels considering multipath components can be obtained by [25]  
\[\mathbf{H}_{0,u}^{+}=\sum_{l^{\prime}=1}^{L_{\mathbf{h}_{0,u}}}h_{0,u}(l^{ \prime})\left(\mathbf{J}_{L}^{+}\right)^{l^{\prime}-1}\in\mathbb{C}^{L\times L },\quad\mathbf{H}_{0,u}^{-}=\sum_{l^{\prime}=1}^{L_{\mathbf{h}_{0,u}}}h_{0,u} (l^{\prime})\left(\mathbf{J}_{L}^{-}\right)^{L-l^{\prime}+1}\in\mathbb{C}^{L \times L}, \tag{7}\]  
where the matrix \(\mathbf{J}_{L}^{+}\in\mathbb{B}^{L\times L}\) denotes the Toeplitz forward shift matrix whose first column is given by \([0,1,0,\cdots,0]^{T}\) while the matrix \(\mathbf{J}_{L}^{-}\in\mathbb{B}^{L\times L}\) the Toeplitz backward shift matrix whose first row is given by \([0,1,0,\cdots,0]\)[34]. **Forward Channels:** They denote the links from the ambient sources \(u=1,\cdots,U\) to the tags \(n=1,\cdots,N\). These channels are related to efficiency of ambient energy harvesting and thus determine the number of active RF tags connected to the reader. Similarly, the impulse responses of the forward channels can be formulated as  
\[\mathbf{f}_{n,u}=[f_{n,u}(1),\cdots,f_{n,u}(L_{\mathbf{f}_{n,u}})]^{T}\in \mathbb{C}^{L_{\mathbf{f}_{n,u}}\times 1},\quad f_{n,u}(l)\sim\mathcal{CN} \left(0,\frac{\sigma_{\mathbf{f}_{n,u}}^{2}}{L_{\mathbf{f}_{n,u}}}\right), \quad\sigma_{\mathbf{f}_{n,u}}^{2}=\frac{G_{\mathrm{s}}A_{e}}{4\pi d_{\mathrm{ S_{u}}T_{n}}^{2}}, \tag{8}\]  
where \(d_{\mathrm{S_{u}}T_{n}}\) denotes the distance from the source \(u\) to the tag \(n\).