# CeRA-eSP: Code-Expanded Random Access to Enhance Success Probability of Massive MTC
## 3 System Model
### Frame Structure and the Legacy Uplink Radio Resources  
Figure 3 shows the frame structure and uplink radio resources of CeRA. The specified subframe according to the PRACH configuration is utilized as an RA slot for the preamble transmission. The RA slots of codeword length \(L_{c}\) form one virtual frame for the successive transmission of the preamble sequence. Devices randomly transmit one of the \(M\) preamble sequences in each RA slot, similar to the legacy RA scheme. Because a virtual frame consists of \(L_{c}\) RA slots, the devices transmit a total of \(L_{c}\) preamble sequences during one virtual frame duration. After transmission of the preamble codeword, the BS deduces the preamble codewords based on the preamble sequence observation in each RA slot. A preamble sequence observation refers to a process in which the BS confirms the results of the preamble sequences that are successfully received in each RAO. The BS deduces the preamble codewords by combining the preamble sequences observed in the virtual frame.  
The RA cycle defined in [30] was used in the system model to consider uplink radio resources for RA. The RA cycle has a length of \(T_{ra}\) ms and consists of a PRACH resource of \(T_{pr}\) ms and a PUSCH resource of \(T_{pu}=T_{ra}-T_{pr}\) ms. In the 5G IoT random access scenario, the RA cycle duration and PRACH duration are commonly set to \(T_{ra}=5\) ms and \(T_{pr}=1\) ms [5; 21]. The PRACH resource consists of 839 subcarriers at 1.25 kHz. Because the PRACH resource includes \(M\) orthogonal contention-based preamble sequences, the devices use the PRACH resource for the preamble transmission. The PUSCH resource consists of 72 subcarriers at 15 kHz. The PUSCH resource is us  
Figure 3: Frame structure of codeword-based random access (codeword length \(L_{c}=2\) and PRACH configuration = 6)  
Figure 2: State transition diagram of an IoT device.  
a connection request and data after transmitting the preamble. The maximum number of available transmissions in PUSCH, as described in [30], can be calculated as follows:  
\[N_{ST}=\left\lfloor\begin{array}{c}L_{RAC}\\ \hline\frac{\Theta_{max}}{\left\lceil\frac{\Theta_{max}}{\ell log_{2}(I)}\right\rceil }\end{array}\right\rfloor\cdot T_{pu}, \tag{2}\]  
where \(\mathcal{L}_{RAC}=6048\) denotes the number of selectable layers supported by sparse code multiple access in one RAO, \(\theta_{max}=160\) bits denotes the data transmission size for each device, \(r=1\) denotes the code rate, and \(I=4\) denotes the number of constellation points. The above-mentioned values are calculated according to the formula in [30] and follow the network standard of [27]. For the special case where the codeword length \(L_{c}=2\), as shown in Figure 3, the total number of subcarriers is \(T_{ra}=10\) and the number of PRACH subcarriers is \(T_{pr}=2\); thus, the number of PUSCH subcarriers is \(T_{pu}=8\). When \(N_{ST}\) is the maximum number of available transmissions in the PUSCH, it can support as many devices as \(N_{ST}=144\) using Equation (1). If the number of deduced preamble codewords exceeds the allocable PUSCH resources, the BS selects random preamble codewords up to \(N_{ST}\) to send RAR messages and sends PUSCH waiting messages to extra preamble codewords.