# Novel Sparse-Coded Ambient Backscatter Communication for Massive IoT Connectivity
## 3 Sparse-Coded Load Modulation  
As described in the previous section, we design an AmBC modulator based on load impedances satisfying the following characteristics:  
* **Small-form Factor**: The size of the RF tags is very limited for IoT applications such as implanted biosensors and wearables, so they should have a small number of load impedances to reduce their size. The proposed SC-BMA can enable \(M\)-ary modulation [35; 36] for these small-form factor IoT devices, which is in sharp contrast to traditional \(M\)-ary backscatter modulators with massive number of load impedances (e.g., \(M\)-PSK [2], \(M\)-QAM [8; 52; 53]). * **Interference Exploitation**: Detection performance of AmBC is severely degraded by the interferences (i.e., MAI and ISI). To rather exploit these interferences for massive connectivity, we design the codeword structure capturing sparsity [39] of ambient EH and characteristics of DBC [18; 23] where two-way fading and severe ISI occur. Toward these goals, we describe the modulation and encoding models [32; 33; 39; 42], which include design of mapping function [35; 36] and sparse factor graph [39]. If tag \(n\) is activated, the tag generates \(\log_{2}M\)-length bit stream \(\mathbf{d}_{n}^{\dagger}\) as follows:  
\[\mathbf{d}_{n}^{\dagger}\in\{\mathbf{d}^{*}(1),\cdots,\mathbf{d}^{*}(M)\}, \quad\mathbf{d}^{*}(m)=\left[d_{1}^{*}(m)\cdots d_{\log_{2}M}^{*}(m)\right] \in\mathbb{B}^{1\times\log_{2}M},\quad\text{if }a_{n}=1. \tag{10}\]  
Figure 3: Example of duty-cycling operation per codeword of length \(\widetilde{K}=KL\). To modulate the binary vector \(\mathbf{d}_{n}^{\dagger}\) into \(M\)-ary backscatter symbol vector \(\mathbf{b}_{n}\), we define the mapping function \(\psi_{M}\) for \(K_{1}=2\) and \(M\in\{2,4,8\}\) as (Here, we consider relatively small values of \(K_{1}\) and \(M\) to show feasibility of high-order AmBC modulation. It is noteworthy that even higher-order modulation (e.g., \(\{K_{1}=3,M=16\}\), \(\{K_{1}=4,M=64\}\)) can be efficiently implemented at RF tag by appropriate design of the parameter set \(\{K_{1},M\}\) and the mapping function. ):  
\[\psi_{2} = \begin{cases}\sqrt{\alpha}[1,-1],&\text{for }\mathbf{d}_{n}^{ \dagger}=[0],\\ \sqrt{\alpha}[-1,1],&\text{for }\mathbf{d}_{n}^{\dagger}=[1],\\ \sqrt{\alpha}[1,0],&\text{for }\mathbf{d}_{n}^{\dagger}=[00],\\ \sqrt{\alpha}[0,1],&\text{for }\mathbf{d}_{n}^{\dagger}=[01],\\ \sqrt{\alpha}[0,-1],&\text{for }\mathbf{d}_{n}^{\dagger}=[10],\\ \sqrt{\alpha}[-1,0],&\text{for }\mathbf{d}_{n}^{\dagger}=[11],\end{cases} \quad \psi_{8}= \begin{cases}\sqrt{\alpha}[1,1],&\text{for }\mathbf{d}_{n}^{\dagger}=[000],\\ \sqrt{\alpha}[1,0],&\text{for }\mathbf{d}_{n}^{\dagger}=[001],\\ \sqrt{\alpha}[0,-1],&\text{for }\mathbf{d}_{n}^{\dagger}=[010],\\ \sqrt{\alpha}[1,-1],&\text{for }\mathbf{d}_{n}^{\dagger}=[011],\\ \sqrt{\alpha}[0,1],&\text{for }\mathbf{d}_{n}^{\dagger}=[100],\\ \sqrt{\alpha}[-1,1],&\text{for }\mathbf{d}_{n}^{\dagger}=[101],\\ \sqrt{\alpha}[-1,0],&\text{for }\mathbf{d}_{n}^{\dagger}=[111].\end{cases}  \tag{11}\]  
Then, the \(M\)-ary symbol vector \(\mathbf{b}_{n}\) can be expressed by  
\[\mathbf{b}_{n} =\psi_{M}(\mathbf{d}_{n}^{\dagger})\in\{\mathbf{b}^{*}(1),\cdots, \mathbf{b}^{*}(M)\},\quad\mathbf{b}^{*}(m)=[b_{1}^{*}(m),\cdots,b_{K_{1}}^{*} (m)]\in\mathbb{R}^{1\times K_{1}}, \tag{12}\] \[\sigma_{b}^{2} =\mathbb{E}\left[\frac{\|\mathbf{b}_{n}\|^{2}}{K_{1}}\right]= \begin{cases}\alpha,&\text{for }M=2,\\ 0.5\alpha,&\text{for }M=4,\\ 0.75\alpha,&\text{for }M=8.\end{cases}  \tag{13}\]  
As described in Figure 4, these mapping functions satisfy the small-form factor requirement in RF tags. For instance, in conventional phase-shift keying (PSK)-based backscatter modulator, \(M\) constellation points are present in signal constellation with phase difference \(2\pi/M\) in Figure 4a [2]. Thus, the modulator requires \(N_{Z}=M\) distinct load impedances, inevitably increasing the size of RF tags. On the other hand, the SC-AmBC modulator only has three constellation points even for higher-order modulation.