# Design of a New Non-Coherent Cross-QAM-Based \(M\)-ary DCSK Communication System
## 4 Results and Discussion  
In this section, the performance of the proposed XQAM-\(M\)DCSK system over multipath Rayleigh fading channels is demonstrated to verify its superiority. A three-path Rayleigh fading channel is adopted, channel gains of which are \(E\big{[}a_{1}^{2}\big{]}=E\big{[}a_{2}^{2}\big{]}=E\big{[}a_{3}^{2}\big{]}=1/3\) with delays of \(\tau_{1}=0\), \(\tau_{2}=2\) and \(\tau_{3}=5\). Unless otherwise stated, the spreading factor is set to 320.  
Figure 3 depicts the theoretical and simulated BER curves of the proposed XQAM-\(M\)DCSK (\(M=8,32\)) system over a multipath Rayleigh fading channel. It can be noted that the theoretical results are relatively well fitted with the simulation.  
Figure 4a compares BER performance of the proposed XQAM-\(M\)DCSK system and the star QAM-\(M\)DCSK system over a multipath Rayleigh fading channel, where \(M=8,32\). In this figure, it can be seen that the proposed XQAM-\(M\)DCSK system can achieve better performance than the star QAM-\(M\)DCSK system. With a BER of \(10^{-5}\), the proposed XQAM-\(8\)DCSK system can provide an approximately 2 dB gain over the star QAM-\(8\)DCSK system, while the proposed XQAM-\(32\)DCSK system can offer an approximately 1 dB gain over the star QAM-\(32\)DCSK system. To further verify the effect of the multipath fading channel on the performance, the parameters of a multipath fading channel are set to \(E\big{[}a_{1}^{2}\big{]}=1/2\), \(E\big{[}a_{2}^{2}\big{]}=2/5\), \(E\big{[}a_{3}^{2}\big{]}=1/10\), and \(\tau_{1}=0,\ \tau_{2}=2,\ \tau_{3}=5\). Figure 4b compares the BER performance of the proposed XQAM-\(M\)DCSK system and the star QAM-\(M\)DCSK system over the above channel. It can be seen that the same BER performance gap between XQAM-\(M\)DCSK and the star QAM-\(M\)DCSK systems can be obtained. Although the proposed XQAM-\(M\)DCSK system outperforms the star-QAM-\(M\)DCSK one, the former requires a slightly higher complexity when compared with the latter at the demodulation from the perspective of the decision boundaries.  
Figure 4: BER performance comparisons of between the XQAM-MDCSK and star QAM-MDCSK systems with (**a**) quiproportional path gains and (**b**) unquiproportional path gains.  
Figure 3: Theoretical and simulated BER performance of the proposed XQAM-MDCSK system over a multipath Rayleigh fading channel.  
Figure 5 plots the PAPR performance of the proposed XQAM-MDCSK system and the star QAM-MDCSK system, where \(M=8,32\), and the spread factor is set to 100. The PAPR of the transmitted signal can be calculated as \(PAPR=\underset{1\leqslant k\leqslant\beta}{\max}\left\{\left(a_{i}c_{x,k}+b_{i}c _{y,k}\right)^{2}+c_{x,k}^{2}\right\}/\left\{\left(\sum_{k=1}^{\beta}\left(a_{ i}c_{x,k}+b_{i}c_{y,k}\right)^{2}+c_{x,k}^{2}\right)/\beta\right\}\). We consider the complementary cumulative distribution function (CCDF) of PAPR, \(CCDF=\Pr(PAPR>PAPR_{0})\), i.e., to simulate the probability that \(PAPR\) exceeds a certain \(PAPR_{0}\). Because the \(CCDF\) curve is consistent with the PAPR change, a lower \(CCDF\) curve indicates a lower PAPR. Compared with the star QAM-MDCSK system, the proposed XQAM-MDCSK system has a lower PAPR performance.  
To further verify the validation of the proposed framework, perfect CSI is adopted for channel compensation at the receiver. Figure 6 shows the BER performance of the systems with an autocorrelator (AC) and the systems with a perfect CSI, where XQAM-\(M\)DCSK (\(M=8,32\)), star QAM-MDCSK (\(M=8,32\)), and SQAM-MDCSK (\(M=16\)) systems are considered. It is evident that the performance of the system with AC can be close to that of the corresponding ones with perfect CSI.