## 5 Simulation Results  
In order to verify the error performance of the proposed coded modulation scheme over the non-linear satellite channel, the performance simulation and comparative analysis between the proposed NB-LDPC coded 7-QAM scheme and the NB-LDPC coded 8-PSK scheme are carried out in this section. The two NB-LDPC codes used to combine with the 7-QAM and the 8-PSK in the simulation are with the code rate of \(1/2\), the code length of 2304 symbols over GF(7) or GF(8), the row weight \(d_{c}\) of 4, and the column weight \(d_{v}\) of 2. In addition, the two codes share the same graph structure in the parity check matrix, that is, the position of non-zero elements in the matrix is the same and only the non-zero element sets are different. Therefore, the comparison is relatively fair because the performance of NB-LDPC codes is highly related to the graph structures. In addition, the iteration times for decoding are all set to 20. First, the SER of 8-PSK, 8-QAM and 7-QAM are simulated over the AWGN channel without channel coding. The simulation results and the average SEP of 7-QAM computed using Equation (21) are presented in Figure 6. It can be observed in Figure 6 that there is a coincidence between analytical and simulation results of 7-QAM at practical SNRs, which proves the correctness of Equation (21). In addition, at \(\text{SER}=10^{-6}\), compared with the demodulation thresholds of 8-PSK and 8-QAM, that of 7-QAM has nearly 3 dB and 1.4 dB gain, respectively. That is, the simulation results are the same as the theoretical analysis, which proves that the 7-QAM constellation has better anti-noise performance than 8-QAM and 8-PSK constellations. Second, the SERs of the proposed coded modulation scheme and the NB-LDPC coded 8-PSK modulation scheme are simulated over the AWGN channel. As the simulation results shown in Figure 7, compared with the SER of NB-LDPC coded 8-PSK modulation scheme, the proposed scheme also has nearly 0.5 dB gain at \(\text{SER}=10^{-5}\). Figure 6: SER performance of different modulations over the AWGN channel without channel coding and the average SEP of 7-QAM. Figure 7: SER of different modulations combined with NB-LDPC codes over the AWGN channel. Third, the error performance of the two schemes are simulated over the non-linear satellite channel, and the characteristic parameters in Equations (3) and (4) are set as \(\alpha_{\rm a}=2.1587\), \(\beta_{\rm a}=1.1517\), \(\alpha_{\rm p}=4.0033\), and \(\beta_{\rm p}=9.1040\) given in [32]. As the simulation results shown in Figure 8, under the non-linear satellite channel model, compared with the NB-LDPC coded 8-PSK modulation scheme, the proposed scheme is with 0.8 dB performance gain at frame error rate (FER) of \(10^{-2}\), and it also has a 0.6 dB performance gain at SER of \(10^{-5}\). Thereby, the 7-QAM constellation has a better error performance over the non-linear channel because of its larger phase difference, which makes it less sensitive to the AM/PM distortion. Fourth, the SERs of 7-QAM combined with two different LDPC codes are simulated over both the AWGN channel and the non-linear satellite channel for the sensitivity analysis of LDPC codes to the non-linear effects. We construct another NB-LDPC code over GF(7) with the same code length and code rate, and different graph structure of the parity check matrix. Specifically, in the parity check matrix of the second 7-ary LDPC code, \(d_{c}=5\), and \(d_{v}=2\) or 3. Therefore, the second code has better error correction performance than the LDPC code with \(d_{v}=2\). As shown in Figure 9, the performance differences between two schemes over the AWGN channel and the non-linear channel are generally the same. It verifies that the error performance of the scheme can be improved by using codes with better error correction performance and would be rarely interfered by the non-linear effect.