# Multi-Dimensional Sparse-Coded Ambient Backscatter Communication for Massive IoT Networks
## 3 Sparse Codes
### Characteristics of Sparse Codes  
Sparse codes [23; 26; 27; 36; 37] are part of a coding scheme based on the sparsity inherent in batteryless RF tags. For example, when there are \(N\) tags in a backscatter network with duty cycle \(D<1\), the expected number of active tags in an arbitrary time slot is \(ND<N\), which is a small portion, so that the MAI among tags is only affected by a fewer number of tags rather than the entire \(N\). By using this sparsity property (i.e., only a small portion of tags are overlapped in time domain) originated from the duty-cycling operation in batteryless RF tags [9], it is possible for multiple tags to achieve long-range and high-rate communications with minimal interference [23]. Furthermore, sparse codes can support massive IoT connectivity by overloading \(\lambda=\frac{N}{K}\) tags per time slot in a non-orthogonal manner. Sparse code multiple access (SCMA) is classified into code-domain NOMA and regarded as one of the prospective radio access techniques in 5G and IoT networks. Especially, when the sparse codes are applied to AmBC, there are several powerful characteristics which can lead to the vision of the massive IoT networks.  
* **Multi-Dimensional Signalling:** In contrast to the TD-AmBC with duty cycle \(D\leq\frac{1}{N}\), sparse-coded AmBC can prolong the duty cycle of RF tags to be \(D\geq\frac{1}{N}\) by allowing non-orthogonal transmissions among tags. When sparse codes are applied to tags, a codeword encoded by each tag spans for \(K\leq N\) compressed time slots by utilizing sparsity and there exist \(2\leq K_{1}\leq K\) non-zero elements and \(K-K_{1}\) zero elements contained in each codeword. By the definition of the duty cycle \(D=\frac{K_{1}}{K}\geq\frac{1}{N}\), sparse codes can prolong the duty cycle of RF tags effectively. Especially, the parameter \(K_{1}\) denotes the dimension of signal constellation which is related to the duty cycle \(D\). Therefore, based on the sparsity of codewords, ambient backscatter signals can be represented in higher dimension for achieving diversity gain [38] by increasing the duty cycle of RF tags. As a result, the multi-dimensional signalling can support long-range AmBC of few tens of meters, enabling the vision of LP-WAN [14].
* **Feasibility of High-Order Modulation:** Instead of manufacturing the customized \(M\)-ary backscatter modulators [3; 20; 21; 22] in the conventional TD-AmBC, sparse codes can implement \(M\)-ary modulation at commercial RF tags with small form-factor by employing binary PSK (BPSK) backscatter modulator. Based on the principle of the low number of projection method [26; 27], \(M\)-ary constellation points can be projected onto \(\tilde{M}=2\) reflection coefficients at the tags, and then decoded at the reader by a low-complexity iterative message passing algorithm (MPA). Theoretically, any high-order modulation (e.g., \(M\geq 32\)) can be implemented in AmBC as long as the dimension \(K_{1}\) is large enough to span \(M\leq 2^{K_{1}}\) constellation points. Thus, sparse codes can also achieve high-rate AmBC with reasonable installation costs for massive IoT.  
Figure 3 illustrates our proposed MSC-AmBC modulation at RF tags. To generate sparse codes, \(M\)-ary data from \(N\) tags are modulated by a \(K_{1}\)-dimensional constellation mapping function, and then spread into a \(K\)-length codeword using a factor graph. Then, sparse codewords are modulated by switching \(\tilde{M}\leq M\) load impedances and backscattered to the reader. In the following subsection, we discuss a design criteria of the constellation mapping function as well as the factor graph.