# Design of a New Non-Coherent Cross-QAM-Based \(M\)-ary DCSK Communication System
## 2 Non-Coherent XQAM-_M_DCSK System  
Figure 1 illustrates the proposed non-coherent XQAM-_M_DCSK system. At the transmitter, the chaotic reference signal \(c_{x}\) is generated via the chaos generator through the logistic mapping of length \(\beta\), where the logistic map is \(x_{k+1}=1-2x_{k}^{2}\). Then, the Hilbert transform is applied to \(c_{x}\) and thus the orthogonal chaotic signal \(c_{y}\) is obtained, where \(\sum_{k=1}^{\beta}c_{x,k}^{2}=1\) and \(\sum_{k=1}^{\beta}c_{y,k}^{2}=1\). Moreover, it converts the information bits \(b\) into XQAM symbols \(s_{i}=(a_{i},b_{i}),(i=1,...,M)\), for example, XQAM-32 DCSK and XQAM-8 DCSK in Figure 2. Thus, the constellation map can be obtained by linear combination of the orthogonal signals \(c_{x}\) and \(c_{y}\), which is given by \(m_{i}=a_{i}c_{x}+b_{i}c_{y}\). Through the pulse shaping, the reference signal is represented as \(x_{r}(t)=\sqrt{E_{p}}\sum_{k=1}^{\beta}c_{x,k}p(t-kT_{c})\) and the information-bearing signal is written as \(x_{i}(t)=\sqrt{E_{p}}\sum_{k=1}^{\beta}m_{i,k}p(t-kT_{c})\), where \(p(t)\) and \(E_{p}\) are a pulse and the energy of the pulse, respectively. Hence, the transmitted signal is given by  
\[x(t)=x_{r}(t)\cos(2\pi f_{1}t+\Phi_{1})+x_{i}(t)\cos(2\pi f_{2}t+\Phi_{2}), \tag{1}\]  
where \(\Phi_{1}\) and \(\Phi_{2}\) are the phase angles of the cosine carrier modulation, and the carrier frequencies \(f_{1}\) and \(f_{2}\) are orthogonal to each other. At the receiver, the received signal is denoted by  
\[r(t) = \sum_{l=1}^{L}\alpha_{l}\delta(t-\tau_{l})\otimes x(t)+n(t), \tag{2}\]  
where \(\delta(\cdot)\), \(\otimes\) and \(L\) are the Dirac function, the convolution operator, and the number of multipaths, respectively, \(\tau_{l}\) and \(\alpha_{l}\) are the path delay and Rayleigh channel coefficients of the \(l\)-th path, \(n(t)\) is an additive Gaussian white noise (AWGN) with a mean of zero and a variance of \(N_{0}/2\). Then, the received signal \(r(t)\) is first separated into a reference signal and an information-bearing signal using a cosine carrier modulation frequency. The separated signals are written by \(\tilde{x}_{r}(t)=r(t)\cos(2\pi f_{1}t+\Phi_{1})\) and \(\tilde{x}_{i}(t)=r(t)\cos(2\pi f_{2}t+\Phi_{2})\), respectively. The chaotic reference signal \(\tilde{c}_{x}\) and the information-bearing signal \(\tilde{m}_{i}\) are obtained by using two corresponding matched filters respectively. In addition, the quadrature reference signal \(\tilde{c}_{y}\) is gained by Hilbert transform. The decision vector \(z = (z_{a},z_{b})\), where \(z_{a}=\sum_{k=1}^{\beta}\tilde{m}_{i,k}\tilde{c}_{x,k}\) and \(z_{b}=\sum_{k=1}^{\beta}\tilde{m}_{i,k}\tilde{c}_{y,k}\) is attained. Notably, the CSI, i.e., \(h=\sum_{l=1}^{L}a_{l}^{2}\), can be acquired by using the least square estimator in [16; 18]. In the proposed system, an autocorrelator is adopted at the receiver. The channel compensation value can be calculated as \(\hbar=\sum\limits_{k=1}^{\beta}\left(\sum\limits_{l=1}^{L}\,a_{l}\,\,c_{x,k-\, \eta}+n_{k}\right)^{2}\). The equivalent decision vector is calculated as \(\widetilde{z}=(\widetilde{z}_{a},\widetilde{z}_{b})=\frac{\widetilde{z}_{a}}{ HE_{p}}\) for the demodulation. Finally, the decision boundaries of the XQAM-_M_DCSK constellation, e.g., in Figure 2, are used to obtain an estimate of the information bits \(\tilde{b}\). According to the above principle, the proposed framework adopts an autocorrelator to obtain the channel compensation value by using the reference signal, thus it does not require any channel estimator. However, in the existing framework, such as star QAM-_M_DCSK [18] and SQAM-_M_DCSK systems [16], it needs to send pilot information at the transmitter and demands a channel estimator to estimate the CSI at the receiver. In this sense, the proposed system has lower complexity in comparison to the existing SQAM-_M_DCSK and star QAM-_M_DCSK systems.