# Multi-Dimensional Sparse-Coded Ambient Backscatter Communication for Massive IoT Networks
## 4 Two-Stage Detection
### Stage-1: Utilizing OFDM Structure  
At the reader, when the AmBC signals \(\mathbf{x}_{k,n}\) are received from the backward channel \(\mathbf{G}_{n}^{+}\), \(\mathbf{G}_{n}^{-}\), these signals are interfered by the stronger source signals \(\mathbf{s}_{u,k}\) with the direct-link channel \(\mathbf{H}_{0,u}^{+}\), \(\mathbf{H}_{0,u}^{-}\). The superposition signal \(\mathbf{r}_{k}\) can be represented by [25]  
\[\mathbf{r}_{k}=\underbrace{\sum_{u=1}^{U}\left(\mathbf{H}_{0,u}^{+}\mathbf{s}_ {u,k}+\mathbf{H}_{0,u}^{-}\mathbf{s}_{u,k-1}\right)}_{\text{direct-link interferences}}+ \underbrace{\sum_{u=1}^{N}\left(\mathbf{G}_{n}^{+}\mathbf{x}_{k,n}+\mathbf{G}_ {n}^{-}\mathbf{x}_{k-1,n}\right)}_{\text{backscattered signals}}+\mathbf{w}_{k} \in\mathbb{C}^{L\times 1}, \tag{24}\]  
where \(\mathbf{w}_{k}\) is the circularly symmetric additive white Gaussian noise (AWGN) with variance \(\sigma_{n}^{2}\). As the received signals are affected by the direct-link, forward and backward channels, we define their composite channels \(\hat{\mathbf{h}}_{n,u}\) as  
\[\hat{\mathbf{h}}_{n,u}= \begin{cases}\mathbb{I}\hat{\mathbf{r}}_{n,u}\circ \hat{\mathbf{g}}_{n},&\quad\text{if}\ n\
eq 0\\ \sqrt{L}\hat{\mathbf{h}}_{0,u},&\quad\text{if}\ n=0,\end{cases} ,\quad n=0, \cdots,N,\quad u=1,\cdots,U, \tag{25}\]  
Figure 7: Brief overview of MSC-AmBC detection at the reader. where the case of \(n\
eq 0\) represents the composite forward-backward channels while the case of \(n=0\) the direct-link channels. Then, the time-domain signal \(\mathbf{r}_{k}\) is transformed into the frequency-domain signal \(\mathbf{i}_{k}\) after removing the CP of length \(L_{\text{CP}}\) and using discrete Fourier transform (DFT), that is [25]  
\[\mathbf{i}_{k}=\mathbf{\Omega}^{-1}\mathbf{\Xi}^{-}\mathbf{r}_{k}\in\mathbb{C} ^{L\times 1}, \tag{26}\]  
where \(\mathbf{\Xi}^{-}=[\mathbf{0}_{L,L_{\text{CP}}},\mathbf{I}_{L}]\in\mathbb{B}^{ L\times L}\) is the CP removal matrix with the all-zero matrix \(\mathbf{0}_{L,L_{\text{CP}}}\) of size \(L\times L_{\text{CP}}\). It is noteworthy that the ISI from the previous time slot \(k-1\) can easily be cancelled by removing the CP under the condition, given by [25]  
\[\max\left(\max_{n,\mu}(L_{\mathbf{f}_{n,\mu}}+L_{\mathbf{g}_{n}}),\ \max_{u}(L_{\mathbf{h}_{0,\mu}})\right)-1\leq L_{\text{CP}}. \tag{27}\]  
Since the ranges of the AmBC signals are much shorter than those of the OFDM signals, the above condition can easily be satisfied. There are \(L_{\text{PI}}\) pilot subcarriers with index \(l\in\Phi_{\text{PI}}=[\phi_{1},\cdots,\phi_{L_{\text{PI}}}]\), so the band-pass filter (BPF) matrix to extract the \(L_{\text{PI}}\) pilot subcarriers can be expressed by  
\[\mathbf{\Delta}=\{\delta_{l_{\text{PI}},l}\}\in\mathbb{B}^{L_{\text{PI}} \times L},\quad\text{where}\ \delta_{l_{\text{PI}},l}=\mathbb{I}\{l=\phi_{l_{\text{PI}}}\},\quad l_{\text{ PI}}=1,\cdots,L_{\text{PI}},\quad l=1,\cdots,L. \tag{28}\]  
The output of the BPF is therefore expressed by  
\[\bar{\mathbf{r}}_{k}=\mathbf{\Delta}\dot{\mathbf{r}}_{k} =\sum_{n=0}^{N}\sum_{u=1}^{U}\left(\bar{\mathbf{h}}_{n,\mu}\circ \bar{\mathbf{s}}_{u,k}\right)\Gamma_{k,n}+\bar{\mathbf{w}}_{k} \tag{29}\] \[=\sum_{n=0}^{N}\boldsymbol{\psi}_{k,n}\Gamma_{k,n}+\bar{\mathbf{w }}_{k}\in\mathbb{C}^{L_{\text{PI}}\times 1}, \tag{30}\]  
where \(\bar{\mathbf{h}}_{n,\mu}=\mathbf{\Delta}\dot{\mathbf{h}}_{n,\mu}\in\mathbb{C} ^{L_{\text{PI}}\times 1}\), \(\bar{\mathbf{s}}_{u,k}=\mathbf{\Delta}\dot{\mathbf{s}}_{u,k}\in\mathbb{C}^{L_{ \text{PI}}\times 1}\), and \(\Gamma_{k,0}=1\) for notational convinience. We can observe that \(L_{\text{PI}}\) pilot subcarriers are required for detecting the backscattered signals, and these pilot sequences can be known to the reader as the reader receives the preamble signals from \(U\) ambient sources [3]. Besides, by the virtue of sparsity of the backscattered signals, the channel \(\boldsymbol{\psi}_{k,n}\) can be efficiently obtained by compressed sensing (CS) based channel estimations [42].