# Preamble Design and Collision Resolution in a Massive Access IoT System
## 3 Preamble Design
### Sub-Preamble Sequence Design
The successful resource allocation rate for RA-step 3 under the proposed scheme holds as

\[P_{res}=\min\{1,\frac{n_{PUSCH}}{Pr\{\xi_{s,m}\geq 1\}Pr\{\xi_{d,m}\geq 1\}Pr\{ \xi_{\mu}\geq 1\}\cdot\min\{n_{k},n_{p,s}n_{p,d}n_{\mu}\}}\} \tag{19}\]

Here, as \(\xi_{s,m}\), \(\xi_{d,m}\), and \(\xi_{\mu}\) both follow Binomial distributions, there holds

\[\begin{split} Pr\{\xi_{s,m}\geq 1\}=1-Pr\{\xi_{s,m}=0\}= \binom{n_{k}}{0}(1-\frac{1}{n_{p,s}})^{n_{k}}=(1-\frac{1}{n_{p,s}})^{n_{k}}\\ Pr\{\xi_{s,d}\geq 1\}=1-Pr\{\xi_{s,d}=0\}=\binom{n_{k}}{0}(1- \frac{1}{n_{p,d}})^{n_{k



Thus, the RA success probability under under the proposed scheme holds as

\[P_{suc} =p_{1}+p_{3} \tag{21}\] \[=(1-\frac{1}{n_{p_{s}}n_{p_{d}}n_{\mu}})^{n_{k}-1}\cdot P_{res}\] \[=(1-\frac{1}{n_{p_{s}}n_{p_{d}}n_{\mu}})^{n_{k}-1}\min\{1,\frac{n_ {PUSCH}}{(1-\frac{1}{n_{p_{s}}})^{\mu_{k}}(1-\frac{1}{n_{p_{d}}})^{\mu_{k}}(1- \frac{1}{n_{p_{s}}})^{\mu_{k}}\cdot\min\{n_{k},n_{p_{s}}n_{p_{d}}n_{\mu}\}}\}\]

## 6 Simulation Results

In this section, the performance of the proposed RA scheme is simulated and analyzed in terms of preamble detection probability, preamble collision probability, RA success probability, resource efficiency and TA capturing. The simulation parameters and their configurations are listed in Table 1. The simulation experiments are carried out on a MATLAB platform. In the simulation, preamble sequences are first generated for the conventional RA scheme and the proposed RA scheme respectively. Then different preambles with distinct TA values are transmitted to the gNB, suffering from a small-scale fading channel and additive white Gaussian noise. In convention RA scheme, the receive preamble sequence is detected by calculating the correlation value with all the local available preamble sequences. In the proposed RA scheme, the received sequence is divided into three sub-sequences, and each sub-sequence is detected by calculating the correlation values with all the corresponding available sub-preambles. The TA estimation is determined as the peak location of correlation. Once a TA is captured, the PUSCH resource block will be allocated to the user by the gNB in RA-step 3. An RA failure occurs when the TA cannot be captured or the PUSCH resource blocks are used up. To avoid the impact of randomness on the simulation results, we simulate each case for 10,000 times and output the mean results finally. ### Preamble Detection Probability

To evaluate the impact of sub-preambles on the preamble detection probability, the proposed RA scheme is compared with conventional RA scheme when total \(P\) devices select exclusive sub-preambles to access to a gNB on the same PRACH. The preamble detection performance of the proposed RA scheme may be degraded, since the length of designed preambles is shorter than the conventional ones. To test the negative effect of sub-preambles, we randomly select sub-preamble sequence with different indexes from the available sub-preamble set. Figure 4 depicts the average preamble detection probability for different case of the number of devices \(P\). The following observations are obtained: (1) When SNR is higher than \(-14\) dB, the preamble detection performance of proposed scheme is a little poorer than conventional RA scheme; (2) The detection performance variation caused by varying amount of RA-attempting devices can be neglected; (3) The difference in the average preamble detection probabilities of proposed RA scheme and conventional RA scheme is negligible. The above observations imply that the sub-preamble structure does not impair \begin{table}
\begin{tabular}{c c} \hline Number of available shared sub-preambles & 60 \\ \hline Number of available dedicated sub-preambles & 60 \\ \hline Length of ZC sequence & 839 \\ \hline Length of shared sub-preamble sequence & 431 \\ \hline Length of dedicated sub-preamble sequence & 199 \\ \hline Length of GI & 5 \\ \hline Number of communication devices on a PRACH & 1–1000 \\ \hline Signal strength (SNR) & \(-20\)\(\sim\)15 dB \\ \hline \end{tabular}
\end{table}