Design of a New Non-Coherent Cross-QAM-Based M-ary DCSK Communication System

Abstract

In this paper, a new non-coherent cross-quadrature amplitude modulation (XQAM)-based M-ary differential chaos shift keying (XQAM-MDCSK) system is proposed. In such a system, an autocorrelator is adopted at the receiver to obtain the channel compensation value. This framework can be extended to various amplitude phase shift keying-based MDCSK systems, such as star QAM-based MDCSK (star QAM-MDCSK) and square QAM-based MDCSK (SQAM-MDCSK) systems. Moreover, the bit error rate (BER) expression of the proposed XQAM-MDCSK system is derived over a multipath Rayleigh fading channel. Results show that the proposed XQAM-MDCSK system can achieve better BER performance and a lower peak-to-average power ratio (PAPR) compared to the star QAM-MDCSK system. Furthermore, we also show that the performance of the proposed system can be close to that of a system with perfect channel state information (CSI).

1. Introduction

Differential chaos shift keying (DCSK) modulation is a non-coherent chaotic modulation [1] which is resistant to multipath fading and has low complexity. Due to these advantages, DCSK systems have been extensively investigated in different communication scenarios, e.g., underwater acoustic communications [2], simultaneous wireless information and power transfer systems [3,4], and power line communication [5,6]. In addition, similar to [7,8,9], some DCSK-based multiple-input multiple-output (MIMO) systems have been proposed to improve the transmission reliability under multipath fading channels [10,11,12].

The coherent chaos shift keying (CSK) system requires a complex synchronization circuit and channel estimator, which are difficult to implement [1]. However, in a DCSK system, the synchronization circuit and channel estimator are not required. Distinguished from the CSK system, the DCSK system represents each information bit by two chaotic signals, i.e., the chaotic-reference signal and the information-bearing signal [1,3]. Hence, compared with the CSK system, the DCSK system needs to transmit the reference signal so as to waste half a bit of energy, thus leading to low data rate and low energy efficiency. To boost the data rate and the performance, a quadrature chaos shift keying (QCSK) system has been studied in [13]. In such a system, it can obtain approximate BER performance with the DCSK system and double the data rate. Furthermore, a phase shift keying (PSK)-based M-ary DCSK (PSK-MDCSK) system and its multiresolution version have been designed in [14], while its adaptive scheme has been designed in [15]. For the PSK-MDCSK system, it inherits non-coherent feature of the DCSK system and can provide excellent error performance and high data rate. To further improve performance, a square quadrature amplitude modulation (QAM)-based MDCSK (SQAM-MDCSK) system and its hierarchical version have been conceived in [16] and [17], respectively. Although the SQAM-MDCSK system can obtain better BER performance compared with the PSK-MDCSK system, the former has a higher PAPR and requires channel state information (CSI) to demodulate the received signal. To reduce the PAPR, a star QAM-based MDCSK (star QAM-MDCSK) system has been designed in [18], which can offer close or better BER performance than the SQAM-MDCSK system. Although the star QAM-MDCSK system can achieve the rate and channel adaptation, the minimal Euclidean distance of the star QAM constellation is relatively small, thus resulting in worse BER performance. Moreover, the SQAM-MDCSK system is inadequate for changing channel conditions and rate requirements, thus introducing odd powers of 2 constellations. Cross QAM (XQAM) with an odd power of 2 constellations is a good choice which provides better performance over rectangular QAM [19]. From the above discussion, the existing framework needs to send the pilot signals at the transmitter and requires a channel estimator at the receiver, such as star QAM-MDCSK and SQAM-MDCSK systems.

With the above motivation, to bypass the channel estimation and achieve better performance, a non-coherent XQAM-based MDCSK (XQAM-MDCSK) system is designed in this paper. The proposed framework utilizes an autocorrelator to obtain the channel compensation value. This framework not only requires not any channel estimator and can well adapt to the varying channel, but also can be applied to the existing star QAM-MDCSK and SQAM-MDCSK systems. Moreover, the theoretical expression of BER for the proposed XQAM-MDCSK system over multipath Rayleigh fading channels is derived. The results indicate that, compared with the star QAM-MDCSK system, the proposed XQAM-MDCSK system has better BER performance and lower PAPR.

2. Non-Coherent XQAM-$\mathit{M}$DCSK System

Figure 1 illustrates the proposed non-coherent XQAM-MDCSK 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 $s_i=\left(a_i,b_i\right),\left(i=1,...,M\right)$, for example, XQAM-32DCSK and XQAM-8DCSK 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 $x_r\left(t\right)=\sqrt{E_p}{\sum }_{k=1}^{\beta }{c}_{x,k}p\left(t-k{T}_c\right)$ and the information-bearing signal is written as $x_i\left(t\right)=\sqrt{E_p}{\sum }_{k=1}^{\beta }{m}_{i,k}p\left(t-k{T}_c\right)$, where $p\left(t\right)$ and $E_p$ are a pulse and the energy of the pulse, respectively. Hence, the transmitted signal is given by

$$\begin{array}{c}x\left(t\right)=x_r\left(t\right)cos(2\pi f_{1}t+{\mathrm{\Phi }}_1)+x_i\left(t\right)cos(2\pi f_{2}t+{\mathrm{\Phi }}_2),\end{array}$$

(1)

where ${\mathrm{\Phi }}_1$ and ${\mathrm{\Phi }}_2$ 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

$$\begin{array}{c}r\left(t\right)={\sum }_{l=1}^L{\alpha }_l\delta \left(t-{\tau }_l\right)\otimes x\left(t\right)+n\left(t\right),\end{array}$$

(2)

where $\delta \left(·\right)$, ⊗ 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, $n\left(t\right)$ is an additive Gaussian white noise (AWGN) with a mean of zero and a variance of $N_0/2$. Then, the received signal $r\left(t\right)$ is first separated into a reference signal and an information-bearing signal using a cosine carrier modulation frequency. The separated signals are written by ${\tilde{x}}_r\left(t\right)=r\left(t\right)cos\left(2\pi f_{1}t+{\mathrm{\Phi }}_1\right)$ and ${\tilde{x}}_i\left(t\right)=r\left(t\right)cos\left(2\pi f_{2}t+{\mathrm{\Phi }}_2\right)$, 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 $z=\left(z_a,z_b\right)$, 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., $h={\sum }_{l=1}^L{\alpha }_l^2$, 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 $\widehat{h}={\displaystyle \sum _{k=1}^{\beta }}{\left({\displaystyle \sum _{l=1}^L}{a}_l{c}_{x,k-{\tau }_l}+n_k\right)}^2$. The equivalent decision vector is calculated as $\tilde{z}=\left({\tilde{z}}_a,{\tilde{z}}_b\right)=\frac{z}{\widehat{h}{E}_p}$ for the demodulation. Finally, the decision boundaries of the XQAM-MDCSK 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-MDCSK [18] and SQAM-MDCSK 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-MDCSK and star QAM-MDCSK systems. In addition, because the proposed framework can obtain the channel compensation value every symbol duration for demodulation, it can well adapt to the varying channel. However, in every frame including pilot symbols and data, the SQAM-MDCSK and star QAM-MDCSK systems first estimate the CSI by using pilot symbols, then the CSI is used for demodulation of the data. Hence, the existing systems cannot adapt well to the varying channel.

3. Performance Analysis

In this section, the BER expression of the proposed XQAM-MDCSK system over multipath Rayleigh fading channel is analyzed. The average symbol energy can be calculated as $E_s=E_p+E_p\frac{1}{M}{\sum }_{i=1}^M(a_i^2+b_i^2)=\left[\frac{(31M-32)d^2+48}{48}\right]E_p$, where d stands for the minimum distance of the XQAM-MDCSK constellation. According to [14,16], the mean and variance of the decision vector $\tilde{z}$ are $E\left({\tilde{z}}_a\right)=a_i$, $E\left({\tilde{z}}_b\right)=b_i$ and $Var\left[{\tilde{z}}_a\right]=Var\left[{\tilde{z}}_b\right]=\frac{h{E}_s{N}_0/2+\beta N_0^2/4}{h^2{E}_p^2}={\overline{\sigma }}^2$, respectively, where $E(·)$ and $Var(·)$ denote the expectation and variance operators, respectively. For convenience, L-path Rayleigh fading channels are used and their channel gains are equal. The probability density function (PDF) of the instantaneous symbol signal-to-noise rate (SNR) ${\gamma }_s=h{E}_s/N_0$ is obtained as $f\left({\gamma }_s\right)=\frac{{\gamma }_s^{L-1}}{\left(L-1\right)!{\overline{\gamma }}_c^L}exp\left(-\frac{{\gamma }_s}{{\overline{\gamma }}_c}\right)$, where ${\sum }_{l=1}^{L}E\left(a_l^2\right)=1$ and ${\overline{\gamma }}_c=\left(E_s/N_0\right)E\left({\alpha }_j^2\right)$, $j=l,\dots ,L$. Hence, the BER expression of the proposed XQAM-MDCSK system can be written as

$$\begin{array}{cc}{P}_b& =\underset{0}{\overset{+\infty }{\int }}{P}_b\left({\gamma }_s\right)f\left({\gamma }_s\right)d{\gamma }_s,\hfill \end{array}$$

(3)

where $P_b\left({\gamma }_s\right)$ is the conditional BER and ${\gamma }_s={log}_2\left(M\right)E_b/N_0$.

To make the expression more concise, we can define the following expression as

$$\begin{array}{cc}q\left(i\right)& =Q\left(\frac{id}{\overline{\sigma }}\right),\hfill \end{array}$$

(4)

where $Q\left(x\right)$ is a Q-function.

According to [20], we also define the following expression as

$$\begin{array}{cc}q\left(i,j;\rho \right)& =\frac{2}{\pi }\underset{0}{\overset{{\mathrm{\Phi }}_1}{\int }}exp\left(-{\left(\frac{id}{\sqrt{2}sin\theta \overline{\sigma }}\right)}^2\right)d\theta \hfill \\ & +\frac{2}{\pi }\underset{0}{\overset{{\mathrm{\Phi }}_2}{\int }}exp\left(-{\left(\frac{jd}{\sqrt{2}sin\theta \overline{\sigma }}\right)}^2\right)d\theta \left(i⩾0,j⩾0\right),\hfill \end{array}$$

(5)

where ${\mathrm{\Phi }}_1={tan}^{-1}\left(\frac{i\sqrt{1-{\rho }^2}}{j-\rho i}\right)+\frac{\pi }{2}\left(1-sgn\left(\left(j-\rho i\right)/j\right)\right)$, ${\mathrm{\Phi }}_2={tan}^{-1}\left(\frac{j\sqrt{1-{\rho }^2}}{i-\rho j}\right)+\frac{\pi }{2}\left(1-\right.$$\left.sgn\left(\left(i-\rho j\right)/i\right)\right)$ and $\rho =-\frac{1}{\sqrt{2}}$. In addition, one has $q\left(-i,j;\rho \right)=2q\left(j\right)-q\left(i,j;-\rho \right)$, $q\left(i,-j;\rho \right)=2q\left(i\right)-q\left(i,j;-\rho \right)$ and $q\left(-i,-j;\rho \right)=4-2q\left(i\right)-2q\left(j\right)+q\left(i,j;\rho \right)$.

3.1. XQAM-MDCSK ($M\ge 32$)

For the XQAM-MDCSK constellation ($M=32,128,...$), $P_{b\_i_2}\left({\gamma }_s\right)$, $P_{b\_q_2}\left({\gamma }_s\right)$, and $P_{b\_i_3}\left({\gamma }_s\right)$ are special, while the BER expressions of the other bits can be derived according to the method in [21]. For the bit $i_1$, the decision boundary is the y-axis, while the decision domain is the x-axis for the bit $q_1$, the conditional BER of $i_1$ is equal to the one of $q_1$, which is given by

$$\begin{array}{cc}{P}_{b\_i_1}\left({\gamma }_s\right)& =P_{b\_q_1}\left({\gamma }_s\right)=\frac{4}{M}\left[\frac{3}{4}\sqrt{\frac{M}{2}}\sum _{t=1}^{\frac{2\sqrt{2M}}{8}}q\left(2t-1\right)+\frac{1}{2}\sqrt{\frac{M}{2}}\sum _{t=1+\frac{2\sqrt{2M}}{8}}^{\frac{3\sqrt{2M}}{8}}q\left(2t-1\right)\right].\hfill \end{array}$$

(6)

For the bit $i_2$, its decision boundary is a square in the center, which can be divided into inside and outside the square. The conditional BER of $i_2$ can be calculated as

$$\begin{array}{cc}{P}_{b\_i_2}\left({\gamma }_s\right)& =\frac{4}{M}\left[2\sum _{t=1}^{2^{m-2}}\left(q\left(2t-1\right)-q\left(2t-1+2^{m+1}\right)\right)\left(2^{m-1}-\sum _{t=1}^{2^m}q\left(2t-1\right)\right)\right.\hfill \\ & \left.+\left[\frac{M}{8}-{\left(2^{m-1}-\sum _{t=1}^{2^m}q\left(2t-1\right)\right)}^2\right]\right].\hfill \end{array}$$

(7)

For the bit $q_2$, the judgment domain is divided into two parts. The first part is the boundary of two horizontal judgments, and the second part is the concave and convex judgment boundaries near the y-axis. The conditional BER of $q_2$ can be calculated as

$$\begin{array}{cc}{P}_{b\_q_2}\left({\gamma }_s\right)& =\frac{4}{M}\left[\left[2^{m-2}\times \sum _{t=1}^{2^{m-2}}\left(q\left(2t-1\right)-q\left(2^{m+1}+2t-1\right)\right)+3\times \left(2^{m-2}\right)\left(\sum _{t=1}^{2^{m-2}}q\left(2t-1\right)\right.\right.\right.\hfill \\ & +\sum _{t=1+2^{m-2}}^{2^{m-1}}q\left(2t-1\right)\left.\left.-\sum _{t=1+2^{m-1}}^{3\times 2^{m-2}}q\left(2t-1\right)\right)\right]+\left[\left(2^{m-2}-\sum _{t=1}^{2^{m-2}}\left(q\left(2t-1\right)\right.\right.\right.\hfill \\ & \left.\left.-q\left(2t-1+2^{m+1}\right)\right)\right)\times \left(\sum _{t=1}^{2^{m-2}}q\left(2t-1\right)\right.+\sum _{t=1+2^{m-2}}^{2^{m-1}}q\left(2t-1\right)\left.-\sum _{t=1+2^{m-1}}^{3\times 2^{m-2}}q\left(2t-1\right)\right)\hfill \\ & \left.\left.+\left(\sum _{t=1}^{2^m}{\left(-1\right)}^{t-1}q\left(2t-1\right)\right)\times \left(2^{m-2}-\sum _{t=1+2^{m-1}}^{2^m}q\left(2t-1\right)\right)\right]\right].\hfill \end{array}$$

(8)

For the bit $i_3$, the judgment domain is also divided into two parts. The first part is two open rectangles, and the second part is four oblique triangular regions, for example $X_1$, $X_2$, $X_3$, and $X_4$ in Figure 2a. The conditional BER of $i_3$ can be calculated as

$$\begin{array}{cc}{P}_{b\_i_3}\left({\gamma }_s\right)& =\frac{4}{M}\left[\left[\left(\sum _{t=1}^{2^{m-1}}q\left(2t-1\right)\right)\left(2^{-1}-\sum _{t=1}^{2^m}q\left(2t-1\right)\right)+\left(\sum _{t=1}^{2^{m-2}}\left(q\left(2t-1\right)-q\left(2t-1+2^{m+1}\right)\right)\right)\right.\right.\hfill \\ & \times \left(2^{m-2}+\sum _{t=1}^{2^{m-1}}q\left(2t-1+2^{m-1}\right)\right)+2^{m-1}\sum _{t=1}^{2^m}q\left(2t-1\right)+\left(2^{m-1}-\sum _{t=1}^{2^m}q\left(2t-1\right)\right)\hfill \\ & \left.\times \left(\sum _{t=1}^{2^m}{\left(-1\right)}^{⌊\frac{t-1}{2^{m-1}}⌋}q\left(2t-1\right)\right)\right]+\left[\frac{1}{4}\sum _{t=1}^{2^m}\sum _{v=1}^{2^{m-1}}q\left(2t-1,\frac{-2t+2v+2^{m-1}}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right)\right.\hfill \\ & +\frac{1}{4}\sum _{t=1}^{2^{m-2}}\sum _{v=1}^{2^m}\left(q\left(1-2t,\frac{2t+2v-2}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right)+q\left(2t-1+2^{m+1},\frac{-2t-2v+2}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right)\right)\hfill \\ & \left.\left.-\frac{1}{4}\sum _{t=1}^{2^m}\sum _{v=1}^{2^{m-1}}\left(q\left(2t-1,\frac{-2t+2v-2^{m-1}}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right)+q\left(2t-1,\frac{-2t+2v+3\times 2^{m-1}}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right)\right)\right]\right].\hfill \end{array}$$

(9)

Due to the symmetry, $P_{b\_q_3}\left({\gamma }_s\right)$ is equal to $P_{b\_i_4}\left({\gamma }_s\right)$, which is given by

$$\begin{array}{cc}{P}_{b\_i_4}\left({\gamma }_s\right)& =P_{b\_q_3}\left({\gamma }_s\right)=\sqrt{\frac{2}{M}}\sum _{t=1}^{\sqrt{\frac{M}{32}}}\left[1-g_t^4+\left(2g_t^4-1\right)\left[\sum _{v=1}^6{\left(-1\right)}^{v+1}\right.\right.\left.\left.\times q\left(\left(2v-1\right)\frac{\sqrt{2M}}{8}-2t+1\right)\right]\right]\hfill \\ & +\sqrt{\frac{9}{2M}}\sum _{t=1+\sqrt{\frac{M}{32}}}^{\sqrt{\frac{9M}{32}}}\left[1-g_t^4\right.+\left(2g_t^4-1\right)\left[\sum _{v=1}^6{\left(-1\right)}^{v+1}\right.\times \left.\left.q\left(\left(2v-1\right)\frac{\sqrt{2M}}{8}-2t+1\right)\right]\right],\hfill \end{array}$$

(10)

and $P_{b\_q_{k-1}}\left({\gamma }_s\right)$ is equal to $P_{b\_i_k}\left({\gamma }_s\right)$, which is given by

$$\begin{array}{cc}{P}_{b\_i_k}\left({\gamma }_s\right)& =P_{b\_q_{k-1}}\left({\gamma }_s\right)=\sqrt{\frac{2}{M}}\sum _{t=1}^{\sqrt{\frac{M}{32}}}\left[g_t^k+\left(1-2g_t^k\right)\left[\sum _{v=1}^{3\times 2^{\left(k-3\right)}}{\left(-1\right)}^{v+1}\right.\right.\left.\left.\left.\times q\left[\left(2v-1\right)\sqrt{2M}{2}^{1-k}\right.-2t+1\right]\right]\right]\\ & +\sqrt{\frac{9}{2M}}\sum _{t=1+\sqrt{\frac{M}{32}}}^{\sqrt{\frac{9M}{32}}}\left[g_t^k+\left(1-2g_t^k\right)\left[\sum _{v=1}^{3\times 2^{\left(k-3\right)}}{\left(-1\right)}^{v+1}\right.\right.\times \left.\left.\left.q\left[\left(2v-1\right)\sqrt{2M}{2}^{1-k}\right.-2t+1\right]\right]\right]\begin{array}{cc}& \left(k⩾5\right),\end{array}\\ \end{array}$$

(11)

where $g_t^k$ is the kth bit in the Gray code of the tth symbol.

3.2. XQAM-8DCSK

For the XQAM-8DCSK constellation in Figure 2b, the conditional BER of the bits $i_1$ ($q_1$) and $i_2$ is given by

$$\begin{array}{ll}{P}_{b\_i_1}\left({\gamma }_s\right)& =P_{b\_q_1}\left({\gamma }_s\right)\\ & =\frac{1}{16}\left[q\left(-1,1;-0.5\right)+q\left(1,-\sqrt{3};0.5\right)\right.\\ & +q\left(-1-\sqrt{3},1+\sqrt{3};-0.5\right)+q\left(0,2+\sqrt{3};-0.5\right)\\ & +q\left(0,1;0.5\right)+q\left(-1,1+\sqrt{3};-0.5\right)\\ & \left.+q\left(1,0;0.5\right)+q\left(1+\sqrt{3},0;-0.5\right)\right],\\ P_{b\_i_2}\left({\gamma }_s\right)& =\frac{1}{4}+\frac{1}{8}\left[2q\left(1,-1;0\right)+q\left(-1,1;-0.5\right)\right.\\ & +q\left(-1-\sqrt{3},1+\sqrt{3};-0.5\right)+q\left(1+\sqrt{3},0;-0.5\right)\hfill \\ & +q\left(1,\sqrt{3};-0.5\right)-q\left(1,0;-0.5\right)-q\left(0,-1;-0.5\right)\hfill \\ & \left.-q\left(-1,1+\sqrt{3};-0.5\right)-q\left(0,2+\sqrt{3};-0.5\right)\right].\hfill \end{array}$$

(12)

Hence, based on the above derivations, $P_b\left({\gamma }_s\right)$ can be expressed as

$$\begin{array}{c}{P}_b\left({\gamma }_s\right)=\frac{1}{{log}_{2}M}\left(\sum _{k=1}^{m+1}{P}_{b\_i_k}\left({\gamma }_s\right)+\sum _{k=1}^m{P}_{b\_q_k}\left({\gamma }_s\right)\right),\end{array}$$

(13)

where $M=8,32,128,\dots$ and $m=\frac{\left({log}_{2}M-1\right)}{2}$.

4. Results and Discussion

In this section, the performance of the proposed XQAM-MDCSK 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 $E\left[a_1^2\right]=E\left[a_2^2\right]=E\left[a_3^2\right]=1/3$ 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-MDCSK ($M=8,32$) 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-MDCSK system and the star QAM-MDCSK system over a multipath Rayleigh fading channel, where $M=8,32$. In this figure, it can be seen that the proposed XQAM-MDCSK system can achieve better performance than the star QAM-MDCSK system. With a BER of ${10}^{-5}$, the proposed XQAM-8DCSK system can provide an approximately 2 dB gain over the star QAM-8DCSK system, while the proposed XQAM-32DCSK system can offer an approximately 1 dB gain over the star QAM-32DCSK system. To further verify the effect of the multipath fading channel on the performance, the parameters of a multipath fading channel are set to $E\left[{\alpha }_1^2\right]=1/2,E\left[{\alpha }_2^2\right]=2/5,E\left[{\alpha }_3^2\right]=1/10$, and ${\tau }_1=0,{\tau }_2=2,{\tau }_3=5$. Figure 4b compares the BER performance of the proposed XQAM-MDCSK system and the star QAM-MDCSK system over the above channel. It can be seen that the same BER performance gap between XQAM-MDCSK and the star QAM-MDCSK systems can be obtained. Although the proposed XQAM-MDCSK system outperforms the star-QAM-MDCSK one, the former requires a slightly higher complexity when compared with the latter at the demodulation from the perspective of the decision boundaries.

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 $PAPR=\underset{1⩽k⩽\beta }{max}\left\{{\left(a_i{c}_{x,k}+b_i{c}_{y,k}\right)}^2+c_{x,k}^2\right\}/\left\{\left({\sum }_{k=1}^{\beta }{\left(a_i{c}_{x,k}+b_i{c}_{y,k}\right)}^2+c_{x,k}^2\right)/\beta \right\}$. We consider the complementary cumulative distribution function (CCDF) of PAPR, $CCDF=Pr(PAPR>PAP{R}_0)$, i.e., to simulate the probability that $PAPR$ exceeds a certain $PAP{R}_0$. 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-MDCSK ($M=8,32$), 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.

5. Conclusions

In this paper, a new non-coherent XQAM-MDCSK system has been proposed by using an autocorrelator. The proposed framework not only retains the non-coherent characteristic of the DCSK system, but also can be utilized in the star QAM-MDCSK and SQAM-MDCSK systems. Moreover, the BER expression of the proposed XQAM-MDCSK system has been analyzed over a multipath Rayleigh fading channel. Compared to the star QAM-MDCSK system, the theoretical and simulated results show that the proposed XQAM-MDCSK system has better a BER performance and lower PAPR. In particular, the proposed XQAM-8DCSK system can provide an approximately 2 dB gain over the star QAM-8DCSK system, while the proposed XQAM-32DCSK system can offer an approximately 1 dB gain over the star QAM-32DCSK system. Furthermore, results show that the performance of the proposed system can approach that of the system with the perfect CSI.