# Novel Sparse-Coded Ambient Backscatter Communication for Massive IoT Connectivity
## 3 Sparse-Coded Load Modulation
Accordingly, the required number of load impedances is \(N_{Z}=2\), since a zero-constellation point is equivalent to the idle state of RF tag [26]. Although several constellation points can be overlapped in Figure 4b, symbols can be detectable as long as they are uniquely mapped in \(K_{1}\)-dimensional signal space in Equation (11), even achieving modulation diversity gain [54] for reliable AmBC. Thus, high-order modulation can be implemented to typical binary PSK (BPSK) backscatter modulator, enabling a high data rate for a small-form factor passive RF devices such as biosensors and wearables. Figure 4: Example of mapping function \(\psi_{M}\) for AmBC. (**a**) PSK-based backscatter modulator; (**b**) sparse-coded backscatter modulator. **Remark 1**. : _The multidimensional mapping function can be designed for any values of \(K_{1}\) and \(M\) with the following rules:_  
* _In each dimension, there are three constellation points (i.e.,_ \(-\sqrt{\alpha},0,\sqrt{\alpha}\)_)._
* _A parameter set_ \(\{K_{1},M\}\) _satisfies the inequality_ \(M\leq 3^{K_{1}}\)_._
* _Multidimensional symbols are mapped with Gray code labeling._  
An example of \(M\)-ary mapping function in vector space \(\mathbb{R}^{1\times K_{1}}\) is expressed in Figure 5. Interestingly, the previous example in Figure 4 is the projection of multidimensional symbol to the axes \(b_{1}^{\star},\cdots,b_{K_{1}}^{\star}\). We can observe that the parameter \(K_{1}\) constitutes the axes of the signal space and \(M\) represents the number of constellation points in the vector space. Since the number of symbols RF tag can represent in each dimension is only three, all of the constellation points exist in dashed rectangular lattices in Figure 5 and the number of symbols \(M\) is less than \(3^{K_{1}}\) for unique mapping of \(M\)-ary data. In addition, we adopt lattice constellations along with Gray code labeling to lower bit-error rate (BER), which may be suboptimal but efficient and practical heuristic for solving a mapping function design problem known to be NP-hard (Non-deterministic Polynomial-time hard) [33]. Based on these rules, we can generalize the mapping function \(\psi_{M}\) even for higher values of \(K_{1}\) and \(M\) based on the projection of the multidimensional constellation for the small-form factor AmBC applications. However, due to practical requirements of quality-of-service (QoS) and sparsity issue for low-complexity decoding, we consider small values of \(K_{1}\) and \(M\). In the sequel, \(K_{1}\)-length modulated symbols \(\mathbf{b}_{n}\) are spread into \(K\)-length sparse codewords using one-way factor graph (Tanner graph) \(\widetilde{\mathbf{G}}\)[44]. By doing so, dimension of signal is extended to \(2K\)-dimension [32; 33; 39; 42] with \(M^{N}\) points in a main constellation, in contrast to conventional AmBC modulators typically laid on two-dimensional Euclidean spaces [22] with \(M\) constellation points (e.g., 4-QAM [9; 38], 16-QAM [8; 52], 16-PSK [2], 32/64-QAM [53]). As a result, the number of tags allowed to transmit is increased to \(N=\binom{K}{K_{1}}\). The overloading factor [32], ratio of the number of RF tags and the time slots can be defined by  
\[\lambda=N/K\geq 1. \tag{14}\]  
In addition, the duty cycle of tag is extended to \(D=K_{1}/K\), which is \(K_{1}\)-fold increase compared to the time-division-based AmBC (TD-AmBC) with \(D=1/K\)[2; 27]. Although \(N_{1}=\binom{K-1}{K_{1}-1}\) tags are interfering in an arbitrary time slot, the MAI can be efficiently suppressed by exploiting sparse code structure. Although there are many possible choices of \(K_{1}\), we choose \(K_{1}=2\) for simplicity of system modeling. The setting can achieve maximum sparsity as well as reasonable error rate for backscatter detection [39]. In addition, it can reduce complexity of modulation and detection algorithms, which will be discussed later. (For example, if \(K_{1}\) becomes larger, the 'for' loops in Algorithm 1, expressed in  
Figure 5: Example of multidimensional mapping function.