## 3 System Model
### Receiver Structure
For the outputs of the maximum a-posterior probability (MAP) decoder, the first iteration assumes \(\overline{\mathrm{x}_{\mathrm{k}}}\) to be 0 and \(\mathrm{v}_{\mathrm{k}}\) to be 1. The signal \(\mathrm{Z}_{\mathrm{k}}\) of a specific user for the mean vector \(\mathrm{X}_{\mathrm{k}^{\prime}}\) can be obtained as follows [19; 20]:  
\[\mathrm{Z}_{\mathrm{k}}=\mathrm{Y}-\sum_{ \begin{subarray}{c}\mathcal{K}^{ \prime}\
eq\mathrm{k}\\ \mathcal{K}\in\mathcal{K}\end{subarray} }\!\!\!\!\!\!\sqrt{\mathrm{P}_{\mathrm{ k}^{\prime}}}\wedge_{\mathrm{k}^{\prime}}\overline{\mathrm{X}_{\mathrm{k}^{ \prime}}} \tag{8}\]  
The result of MMSE equalization is the frequency domain estimate of equalized symbols as follows:  
\[\widehat{\mathrm{X}_{\mathrm{k}}}=\mathrm{G}_{\mathrm{k}}^{\mathrm{H}}\mathrm{ Z}_{\mathrm{k}}-(\mathrm{G}_{\mathrm{k}}^{\mathrm{H}}\wedge_{\mathrm{k}}- \frac{1}{\mathrm{S}}\sum_{\mathrm{s}=0}^{\mathrm{S-1}}(\mathrm{G}_{\mathrm{k }}^{\mathrm{H}}\wedge_{\mathrm{k}}))\overline{\mathrm{X}_{\mathrm{k}}}\sqrt{ \mathrm{P}_{\mathrm{k}}} \tag{9}\]  
where \((. )^Simulation results shown in Figure 4 show the comparison of PPI and RI under AWGN and Vehicular A multipath channel environment with the mentioned multi-carrier multiple access schemes. The results reflect that the BER performance of PPI and RI for both multi-carrier schemes remains the same under AWGN channel condition. Addition of frequency diversity due to additional FFT operation in SC-FDMA-IDMA system results in better performance in comparison to OFDM-IDMA under Vehicular A multipath channel environment. Both sub-carrier mapping schemes including LFDMA and DFDMA result in the same BER performance for SC-FDMA-IDMA system with PPI and RI mechanism of interleaving. The above simulation results justify the appropriateness of PPI with suggested multi-carrier schemes under multipath channel conditions. Figure 5 illustrates the performance comparison between progressive pattern interleaving (PPI) and random interleaving (RI) across both multi-carrier schemes in the Vehicular A channel environment, considering different numbers of users, namely 16 and 32. These simulations utilize PPI and RI interleaving techniques to differentiate between multiple users, while other parameters remain consistent with those previously specified. However, for a large number of users, both interleaving schemes struggle to maintain acceptable bit error rate (BER) performance in the multipath environment of the Vehicular A channel, attributed to the non-orthogonal multiple access (NOMA) access method employed in the system. To address this issue, channel scheduling algorithms [19] in conjunction with multi-carrier schemes yield improved performance. It can be concluded from results that PPI interleaver performs equally good as RI interleaver with large number of users also under multipath channel conditions. Figure 4: BER performance of PPI and RI with OFDM-IDMA, LFDMA, DFDMA in AWGN and Vehicular A (Veh. A) channels. To gain further insight, simulations of PPI and RI have also been analyzed for multiple iteration counts with both multi-carrier schemes under a multipath scenario. Figure 6 shows the performance of both interleavers with the above-mentioned multiple access schemes for multiple iteration counts set at 2, 5, and 6 for MUD implementation at receiver. OFDM-IDMA performance is inferior to SC-FDMA-IDMA due to inherent frequency diversity in SC-FDMA-IDMA. Furthermore, it is observed that with iterative MUD implemented at the receiver of both schemes, an increase in iteration count improves the BER performance. This improvement saturates after a 5-iteration count, and further increase in iteration count simply increases the implementation complexity of the iterative MUD. Therefore, the optimum number of iteration counts has been considered as five in these simulations. Simulations conclude that PPI performs the same as RI under fewer iteration counts as well for the multipath channel environment.