# Novel Sparse-Coded Ambient Backscatter Communication for Massive IoT Connectivity
### System Model
At time slot \(k\in\{1,2,\cdots,K\}\), the ambient signal with average transmit power \(\sigma_{s}^{2}\) can be expressed by  
\[\mathbf{s}_{k}=[s_{1,k},\cdots,s_{L,k}]^{T}\in\mathbb{C}^{L\times 1},\quad s_{ l,k}\sim\mathcal{CN}(0,\sigma_{s}^{2}), \tag{1}\]  
Figure 1: Proposed AmBC network with single AP and multiple RF tags employing ambient RF energy harvesting and backscatter communications using sparse code. Figure 2: Link layer protocol for AmBC with sparse code. where a Gaussian source model with sampling period \(T_{s}\) is assumed to design ambient signal [49]. Next, the signal broadcasted from the AP is attenuated by forward channels. The discrete-time channel impulse response between AP and tag \(n\) can be expressed as [23]  
\[\mathbf{f}_{n}=[f_{n}(1),\cdots,f_{n}(L_{n}^{+})]^{T},\quad\text{ where }f_{n}(l^{\prime})\sim\mathcal{CN}(0,\sigma_{f}^{2}/L_{n}^{+}),\quad\sigma_{f}^ {2}=\frac{G_{s}A_{e}}{4\pi d^{2}},\quad l^{\prime}\in\{1,\cdots,L_{n}^{+}\}, \tag{2}\]  
where \(L_{n}^{+}\) denotes the number of multipath for tag \(n\in\{1,2,\cdots,N\}\). In Equation (2), \(G_{s}\), \(A_{e}\) and \(d\), respectively, denote antenna gain at AP, effective aperture of receive antenna and tag distance from AP [40]. Moreover, we assume independent and identically distributed (i.i.d.) channels among tags [12; 18; 19; 21]. It is noteworthy that the forward channel with wide-band ambient signals (e.g., 20 MHz in Wi-Fi) generally experiences frequency-selective fading in sharp contrast to frequency-flat fading in RFID and bistatic backscatter, causing delay spreads of received signals in the forward channel \(\mathbf{f}_{n}\)[2]. To capture this characteristic, we can use Toeplitz matrix [50], matrix-vector representations of circular-convolution operations. We define Toeplitz lower- and upper-triangular matrices as follows [23]:  
\[\mathbf{F}_{n}^{+}=\sum_{l^{\prime}=1}^{L_{n}^{+}}f_{n}(l^{\prime})(\mathbf{J} _{L}^{+})^{l^{\prime}-1}\in\mathbb{C}^{L\times L},\quad\mathbf{F}_{n}^{-}= \sum_{l^{\prime}=1}^{L_{n}^{+}}f_{n}(l^{\prime})(\mathbf{J}_{L}^{-})^{L-l^{ \prime}+1}\in\mathbb{C}^{L\times L}, \tag{3}\]  
where the matrices \(\mathbf{J}_{L}^{+}\in\mathbb{R}^{L\times L}\) and \(\mathbf{J}_{L}^{-}\in\mathbb{R}^{L\times L}\), respectively, denote the Toeplitz forward shift and backward shift matrices of size \(L\times L\) where the first column of \(\mathbf{J}_{L}^{+}\) and the first row of \(\mathbf{J}_{L}^{-}\) are given by \([0,1,0,\cdots,0]^{T}\) and \([0,1,0,\cdots,0]\), respectively [23; 50]. Due to delay spread, the received signal at tag \(n\) at time slot \(k\) can be expressed by superposition of \(k\)-th signals and \((k-1)\)-th signals such that  
\[\mathbf{y}_{k,n}=\underbrace{\mathbf{F}_{n}^{+}\mathbf{s}_{k}}_{k\text{-th time}}+\underbrace{\mathbf{F}_{n}^{-}\mathbf{s}_{k-1}}_{k-1\text{-th time}}\in\mathbb{C}^{L\times 1}, \tag{4}\]  
where the second term in Equation (4) constitutes ISI for the ambient signal \(\mathbf{s}_{k}\). Since tags rely on RF energy harvesting, it operates in active or idle state depending on the incident power [37] of the received signal in Equation (4), which is expressed as  
\[E_{n}=\mathbb{E}\left[\sum_{k=1}^{K}\left\|\mathbf{y}_{k,n}\right\|^{2}\right]. \tag{5}\]  
As described in Figure 3, the incident power in Equation (5) affects duty-cycling operation and thus sparsity of NOMA. Let \(0<\alpha\leq 1\) be a power ratio between backscattered and incident power, affecting performance of backscatter detection as well as energy harvesting [8; 9; 38]. Let \(K_{1}<K\) be the number of time slots when tag is activated. In a codeword of length \(\widetilde{K}=KL\), each tag is activated for \(K_{1}L\) samples and remains idle for \((K-K_{1})L\). It is remarkable that the delay spread of backscatter channel incurs ISI for \(\widetilde{L}\leq L\) samples. Moreover, backscatter symbols from \(N\geq K\) tags (i.e., NOMA) in active states can be interfered in time-domain causing MAI, further complicating reliable detection of AmBC.