# Intelligent Beam-Hopping-Based Grant-Free Random Access in Secure IoT-Oriented Satellite Networks
## 3 Beam-Hopping-Based Grant-Free Random Access
### Resource-Hopping Mechanism  
The proposed hopping schemes facilitate the utilization of shared resources by multiple devices in a non-orthogonal manner. In this section, we virtualize the resource grid into  
Figure 2: The procedure of the proposed GFRA protocol. a matrix and put forward resource-hopping schemes as matrix transformations. The proposed resource-hopping schemes are as follows: (1) orthogonal frequency-hopping pattern, (2) random hopping pattern, and (3) group-divided hopping pattern. The number of devices in a single GFRA group \(G\) is assumed in all three schemes. Therefore, the proposed resource \(R\) hopping schemes are used to reduce resource collisions in IoT devices and increase user capacity in IoT scenarios. (1) Orthogonal Frequency-Hopping (OFH) Pattern: Orthogonal frequency-hopping (OFH) is a scheme for allocating spectrum resources using orthogonal frequency-hopping sequences, ensuring that each user's hopping sequences do not overlap in the frequency domain. OFH can significantly reduce interference between users and improve the utilization of spectrum resources. Orthogonal sequences ensure that the probability of frequency collisions between different users is low. However, in IoT, generating massive orthogonal sequences requires more computational resources. In this paper, however, we use an alternating iterative algorithm to reduce the frequency of OFH. It will be mentioned in the following algorithm. The load factor, denoted by the symbol \(L\), and the satellite provide a unique means of data transmission. Thus, no resource conflict is present in this scenario. Conversely, when \(G>R\), there may be orthogonal frequency-hopping sequences of size R with\(\lceil G/R\rceil=L\). For the \(Q^{l}\) OFH matrix, the element at \((i,j)\) is expressed as  
\[\left[Q^{l}\right]_{i,j}=\left(l\times i+j\right)\text{ mod }R \tag{3}\]  
where \(\left[\cdot\right]_{i,j}\) signifies the \((i,j)\) element of the matrix, and mod denotes the modulo operator. An OFH is characterized as a collection of devices that transmit packets over RBs in accordance with a uniform Latin square matrix. In the BH-GFRA system, (3) denotes the index of devices associated with the \(l\)-th OFH group, allocated to the \(i\)-th frequency and \(j\)-th time resource. In actuality, a single resource block may have numerous subcarriers and OFDM symbols, as observed in 3 GPP. Figure 3 illustrates an instance of the collection of OFH square matrices. Each device inside the BH-GFRA group possesses a distinct hopping pattern, hence effectively precluding the occurrence of resource conflicts within the group. Nonetheless, when there are \(L\) OFH groups, a device may unsuccessfully transmit the data to the satellite because of the interfering devices from distinct OFH groups. The likelihood that a single resource is interfered with by \(K\) devices, given an activation probability \(v\), is expressed as  
\[\alpha_{\text{group}}\left(K,v\right)=\binom{L-1}{K}v^{K}(1-v)^{\left(L-1-K \right)} \tag{4}\]  
From (4), the probability that \(\left(N_{0},\ldots,N_{L-1}\right)\) resources are collided by other \(L_{device-1}\) devices, respectively, is expressed as  
\[\Pr(N_{0},\ldots,N_{L-1}\mid v)=\binom{R}{N_{0}\ldots N_{L-1}}\prod_{K=0}^{L-1 }\left\{\alpha_{\text{group}}\left(K,v\right)\right\}^{N_{K}} \tag{5}\]  
where \(N_{0}+\ldots+N_{L-1}=R\). The anticipated quantity of resources impacted by \(K\) interfering devices for a specified activation probability \(v\) is alternatively expressed as  
\[\mathbb{E}(N_{K}\mid v)=R\cdot P_{\text{group}}\left(K,v\right) \tag{6}\]  
(2) Random Hopping Pattern: The patterns of arbitrary resource-hopping may be ascertained by an identifier.