# Anti-Multipath Performance Improvement of an M-ary Position Phase Shift Keying Modulation System
## 4 Simulation and Numerical Results
### Bit Error Rate of Normalized Symbol Joint Decision  
Under the condition that the sub-path can be separated from the main path, the current decision of the symbol can be understood as a joint decision between the sampled integrated value of the normalized sub-path and the sampled integrated value of the main path. This can be understood as the introduction of \"encoding\" in the transmitter. The coding mode is to encode the information of the current symbol into the following seventh symbol without affecting the information representation of the following seventh symbol, thereby ensuring that the current symbol demodulation performance is improved without lowering the symbol rate. The modulation parameters and channel conditions are the same as the Time delay estimation section. In the case where the main path gain is 0.8 and the sub path gain is 0.5, the demodulation performance of the amplitude integral decision, the symbol joint decision and the normalized symbol joint decision is compared. The comparison result is shown in Figure 8.  
Figure 8: Bit error rate comparison of three decision schemes. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Period of m-Sequence** & **Adjust Modulation Parameter Scheme** & **Spread Spectrum Scheme** \\ \hline
7 & 0.295 MHZ & 7.001 MHZ \\
15 & 0.295 MHZ & 11.831 MHZ \\
31 & 0.295 MHZ & 20.171 MHZ \\ \hline \hline \end{tabular}
\end{table} 
Table 3: \(-40\) dB Bandwidth of The Two Separated Multipath Schemes. As shown in Figure 8, the normalized symbol joint decision reduces the bit error rate of the demodulator compared to the amplitude integral decision and the symbol joint decision. In the case of low signal-to-noise ratio conditions, the normalized symbol joint decision is improved compared with the symbol joint decision in terms of bit error rate. As the signal-to-noise ratio increases, the noise signal gradually decreases. Therefore, the weakening effect of the normalized symbol joint decision on the noise is gradually reduced, and the performance improvement of the normalized symbol joint decision is gradually reduced relative to the joint decision of the symbol. When the signal-to-noise ratio is \(-4\) dB, the channel gain of the main path is fixed at 0.8. The effect of the change in the sub-path gain on the demodulation performance of the symbol joint decision and the normalized symbol joint decision is shown in Figure 9.  
As shown in Figure 9, after fixing the main path gain, the demodulator uses the normalized symbol joint decision with a lower bit error rate than the use of the symbol joint decision at different sub-path gains. When the sub-path gain is 0.1, the normalized symbol joint decision has the highest improvement in demodulation performance relative to the symbol joint decision, because the normalized symbol joint decision not only utilizes the symbol information of the sub-path to participate in the decision, but also minimizes the introduced noise. As the gain of the sub-path increases, the normalized value of the sub-path gain to the main path gain becomes closer to 1, that is, the normalized symbol joint decision introduces more and more noise when using the sub-path symbol information. This leads to a gradual decrease in the advantages of the normalized symbol joint decision and the symbol joint decision. Finally, when the main path gain and the sub path gain are the same, the demodulation performance of the two decision modes is the same. When the signal-to-noise ratio is \(-4\) dB, the channel gain of the sub-path is fixed at 0.1. The effect of the change in the main path gain on the demodulation performance of the symbol joint decision and the normalized symbol joint decision is shown in Figure 10.  
As shown in Figure 10, the conclusion is the same as when the main path gain is fixed. Compared with the sub-path gain, the bit error rate is the smallest when the main path gain is maximum.