A Non-Orthogonal Modulation Based High Spectral Efficiency and High Security Multi-Carrier Differential Chaos Shift Keying for Maritime Communications

The Space-Air-Ground-Sea Integrated Networks (SAGSIN) will place higher requirements on both spectral efficiency and security for future maritime communications. To simultaneously address these two challenges, in this paper, a non-orthogonal modulation based multi-carrier differential chaos shift keying system (namely MCNO-DCSK) is proposed and demonstrated. The most remarkable feature of MCNO-DCSK is that the strict orthogonality between subcarriers is no longer required. In this way, frequency intervals between subcarriers can be much smaller than ever, which leads to a high spectral efficiency. Meanwhile, the frequency interval can be set more flexibly to make the symbol more difficult to be deciphered. Therefore, a high level of security can also be guaranteed. In this design, multiple modulated non-orthogonal subcarriers are sequentially delayed and directly superimposed in time domain to construct the MCNO-DCSK symbol. At the receiver, information bits are demodulated through solving a system of linear equations. The spectral efficiency, computational complexity, and security are analyzed, and the bit-error-rate expressions are derived. Moreover, as subcarriers are non-orthogonal and time delayed, the MCNO-DCSK will suffer severe interference over the multi-path channel. Therefore, the effect of the multi-path interference on the system performance is discussed, and a method to mitigate this interference is designed. Finally, simulation results are provided to verify the theoretical analysis and demonstrate the superiority of MCNO-DCSK.

SECTION 1 Introduction

Space-Air-Ground-Sea Integrated Networks (SAGSIN) will achieve a three-dimensional and comprehensive network connection, which is a key direction for the development of future communications technology ^[1]-​[6]. As an integral part of SAGSIN, the maritime communication network undertakes the information interaction of all maritime operations and also serves as a bridge for mutual communication between maritime nodes and ground, air, and space nodes, which is of great significance for the advancement of SAGSIN, as shown in Fig. 1. However, compared with satellite, space and ground based communications, maritime communications face a number of unique challenges in their development. Firstly, the spectrum resource for maritime communications is tighter than those for ground, which requires maritime communication technologies to have a higher spectrum efficiency than ground-based communications when facing large-scale access scenarios of marine nodes. Secondly, the maritime communication network carries a large amount of sensitive data, such as commercial information and national defense information, which makes the maritime communication technology more demanding on security and confidentiality. Finally, due to the special characteristics of the maritime environment, maritime nodes cannot receive a continuous and stable energy supply, thus requiring maritime communication technologies to be simple to implement and high energy efficient.

Fig. 1.

Diagram of the sagsin. yellow lines represent the communication link. various maritime infrastructures, such as unmanned surface vehicles (usv), buoys, fishing boats, liners, and freighters, are connected to the network through sagsin.

Recently, Multi-Carrier Differential Chaos Shift Keying systems (MC-DCSK) have been widely studied in a variety of scenarios ^[7]-​[11]. Firstly, the multi-carrier technology has a high spectral efficiency and communication rate. Secondly, DCSK is a non-coherent communication technique, which does not require complex chaotic synchronization circuits, and therefore has a lower complexity and energy consumption. Meanwhile, DCSK has many advantages, such as low probability of detection, anti-eavesdropping, and anti-multipath fading. Therefore, MC-DCSK fits well with the requirements of the maritime communication technologies. Kaddoum et al.^[12] firstly proposed the concept of MC-DCSK in 2013. In MC-DCSK, the reference and multiple information-bearing signals are assigned to different subcarriers. Compared with the traditional DCSK system, MC-DCSK simultaneously transmits the reference and information-bearing signals in time domain, and therefore has a high spectral efficiency. Then, the optimal power allocation and noise reduction method, and a general iterative receiver are proposed to improve the performance of MC-DCSK ^[13]-​[17]. In addition, the MC-DCSK is extended to multi-user scenarios through analog network coding scheme^[18]. To further increase the spectral efficiency of MC-DCSK, Li et al.^[19] and Kaddoum^[20] proposed an Orthogonal Frequency Division Multiplexing based DCSK (OFDM-DCSK) system. Due to the orthogonality in frequency domain, the guard band between each sub-channel in traditional MC-DCSK is eliminated, and the spectrum efficiency is increased. To improve the performance of OFDM-DCSK, Liu et al.^[21] utilized the carrier interferometry spreading codes to decrease the Peak-Average-Power-Ratio (PAPR) of OFDM-DCSK.

To increase the data transmission rate of MC-DCSK, a multi-resolution M-Ary DCSK system, in which a single chaotic carrier can carry multiple information bits via the constellation symbol, was proposed ^[22]-​[24]. Cai and Song^[25] derived the closed-form BER expressions of M-Ary DCSK systems over multipath Rayleigh fading channels, and a replica piecewise M-Ary DCSK scheme was designed to resist the impulse noise for power line communicatious^[27]. Huang et al.^[27] expanded the M-Ary DCSK modulation to MC-DCSK to increase the data transmission rate and energy efficiency, and reduce the PAPR of the whole system.

As the Index Modulation (IM) can map more information bits into various key parameters of the multi-carrier system, it is considered as a promising candidate in current or even future mobile communication networks to further increase the data transmission rate. Thus, in the past decade, a large volume of research on fusing IM and MC-DCSK have been published. Cai et al.^[28] proposed an M-ary DCSK with IM, in which the IM technique is used to select the transmission time slots of information bearing signals. Chen et al.^[29] demonstrated a Carrier-Index MC-DCSK (CI-MC-DCSK) system to convey more bits through the index of the activated subcarrier. A Generalized CI-MC-DCSK (GCI-MC-DCSK) with M-Ary modulation was proposed to increase the spectral efficiency and improve the BER performance of the chaos-based communication system^[30]. In addition, Cai et al.^[31] demonstrated a multi-carrier M-Ary DCSK system with code index modulation (namely CIM-MC- M-DCSK), in which the Walsh code is applied to code the reference signal and to fully utilize the system energy resources and carry additional information bits. Then, a CIM-MC-M-DCSK with Multiple-Input-Single-Output Simultaneous Wireless Information and Power Transfer scheme (MISO-SWIPT) was designed for e-health Internet of Things (IoTs)^[10]. Moreover, Cai et al.^[32] increased the data rate of IM-based M-ary multi-carrier DCSK system through orthogonal chaotic vector shift keying and a joint time-frequency index modulation assisted multiple-model^[33]. Recently, the Media-Based Modulation (MBM), which is one of the newest members of IM, is proposed to increase the data transmission rate of MC-DCSK through the index of different channel states achieved by reconfigurable antennas^[34].

To enhance the level of security of the DCSK, Lau et al.^[35] proposed a permutation transformation to destroy the similarity between the reference and data samples of the DCSK system. Then, Herceg et al.^[36] proposed a multi-level Permutation Index DCSK (PI-DCSK) architecture to eradicate the similarity between reference and data bearing signals through using different permutations. Qiu et al.^[37] exploited the frequency hopping to enhance the security of the OFDM-DCSK system. Then, Liu et al.^[38] proposed a pre-coding module, wherein the chaotic chips are interleaved and a chaos mask is added to enhance the security performance. Zhang et al.^[39] implemented an iterative chaotically shift-aided shuffling and overlapping operations to enhance the security. Recently, they have proposed a deep-learning based OFDM-DCSK transceiver to further enhance the security, where the reference chaotic sequence is no longer needed to be delivered^[40],

Although characteristics of MC-DCSK fit well with current needs of maritime communications, the future maritime communication network in SAGSIN will put forward higher demands than ever on the spectrum efficiency, energy efficiency, security, etc. In our previous work, we proposed a delay-and-superposition modulation based DCSK system, which can gather all information into a single subchannel to achieve a higher spectral efficiency^[41]. However, since it belongs to the single-carrier communication, the performance will deteriorate under frequency selective channels. Meanwhile, the system parameters are homogeneous and the security is relatively poor. Therefore, in this paper, to further improve the performance and security while maintaining a high spectral efficiency, we propose a Multi-Carrier Non-Orthogonal DCSK (MCNO-DCSK) system to make the DCSK more suitable for future maritime communication networks in SAGSIN.

In the proposed system, multiple modulated non-orthogonal subcarriers are sequentially delayed and directly superimposed in the time domain to form the symbol. At the receiver, the transmitted information bits are demodulated via solving a system of linear equations. Through this novel modulation technology, the frequency intervals of the proposed system can be even smaller than those of the OFDM-DCSK, indicating a higher spectral efficiency. What's more, this novel modulation technology can be easily extended to index modulations and other modern multi-carrier technologies. Briefly, the main contributions of this paper are as follows.

1. A high spectral efficiency and high security MC-DCSK system based on a non-orthogonal modulation is proposed. Thanks to this novel modulation technology, frequency intervals can be much smaller than ever, which significantly increases the spectral efficiency. Meanwhile, the non-orthogonal modulation will introduce a more complex parameter space, making deciphering more difficult and increasing system confidentiality. Therefore, the proposed MCNO-DCSK will simultaneously address multiple needs for future maritime communications.

2. The spectral efficiency, complexity, and the security of the proposed system are analyzed. Meanwhile, the BER expressions are derived. Furthermore, the interference suffered over the multi-path fading channel is discussed, and a method to mitigate this interference is designed. Then, simulation results are provided to verify the theoretical analysis.

3. The influence of system parameters on system performance are studied, and methods of improving BER performance are demonstrated. In addition, BER comparisons are implemented between MCNO-DCSK and other typical MC-DCSK benchmarks, such as MC-DCSK and OFDM-DCSK, to verify the superiority of the proposed system.

The reminder of this paper is organized as follows. In Section 2, the MCNO-DCSK system model and the principles of the transmitter and receiver is demonstrated. In Section 3, the spectral efficiency, computation complexity and security are analyzed, and the BER expressions over the Additive White Gaussian Noise (AWGN) channel is derived. For the multi-path channel, a channel estimation and equalization method is designed. Then, various simulation results and discussions are given in Section 4. Finally, some conclusions are drawn and the future work is given in Section 5.

SECTION 2 System Model

2.1 Transmitter

The transmitter of the proposed MCNO-DCSK is shown in Fig. 2a. $M$ serial data bits are transformed to $M$ paths of parallel bits $S=\left[s_1, \ldots, s_m, \ldots, s_M\right]^{\mathrm{T}}$, where $s_{m}=\{+1,\ -1\}$. Then, a signal matrix $A_{k}^{(M+1)\times\beta}$ is obtained, whose elements are given by\begin{align*} & a_{0 k}=x_k, k=1,2, \ldots, \beta, \\ & a_{m k}=s_m \cdot x_k, m=1,2, \ldots, M, k=1,2, \ldots, \beta \tag{1} \end{align*}

The first row of $A_{k}$ is the chaotic reference signal, and other rows are information-bearing signals. In this paper, the chaotic sequence is generated through the logistic map $x_{k+1}=1-2x_{k}^{2}(k=1,2,\ \ldots,\ \beta)$, where $x_{k}$ is the k-th sample of the chaotic sequence, and $\beta$ is the spreading factor.

Then, elements $a_{mk}(m=0,1,\ \ldots,\ M)$ are mapped on subcarriers with different frequencies. The subcarrier frequency $f_{0}$ is used to transmit the chaotic reference signal, and other subcarriers with frequencies $f_{1}-f_{M}$ are used to convey information-bearing signals. As shown in Fig. 2a, different from typical multi-carrier modulations, in MCNO-DCSK, except for the first carrier, all other subcarriers are delayed respect to its former subcarrier. Therefore, the subcarrier can be expressed as\begin{equation*} g_n(t)=\begin{cases} \sin \left(2 \pi f_n \cdot(t-n \cdot \tau)\right), t \in T; \\ 0, t \notin T \end{cases} \tag{2} \end{equation*} where $n=0,1, \ldots, M, T$ is the duration of $g_{n}(t)$, and $\tau$ is the time delay between neighbor subcarriers. This time delay can be set arbitrarily. In this paper, subcarriers are uniformly distributed, therefore $\tau$ can be calculated by\begin{equation*} \tau=\frac{T}{M+1} \tag{3} \end{equation*} where $M+1$ is the total number of subcarriers in one symbol. Then, all modulated subcarriers are directly superimposed in time domain,\begin{equation*} e_{k}(t)=\sum_{n=0}^{M}a_{nk}\cdot g_{n}(t) \tag{4} \end{equation*} where $e_{k}(t)$ is the k-th symbol of MCNO-DCSK. Figure 3 gives a clear demonstration of the principle of modulation. In this design, all subcarriers have the same duration, and the start of all subcarriers lies within the duration of the first subcarrier. It is worth noting that, according to Ref. [31], thanks to the credible digital system design, the time delay is easier to be implemented than that in the analog modulation scenario.

Fig. 2

Architecture of mcno-dcsk (s/p and p/s represent the serial-to-parallel and parallel-to-serial conversion, respectively).

After $\beta$ modulation processes, the frame of MCNO-DCSK is constructed, and a Cyclic Prefix (CP) and a pilot is inserted in front of the frame. In this design, a sequence of zero is applied as the CP, and a summation of all the unmodulated subcarriers is used as the pilot for channel estimation and equalization,\begin{equation*} P(t)=\sum_{n=0}^{M}g_{n}(t) \tag{5} \end{equation*}

Finally, the frame containing the CP and pilot are sent to the maritime wireless channel. The maritime wireless channel model used in this paper is the same as that in Ref. [41]. In this paper, the sea state level is 4.

2.2 Receiver

The receiver of the MCNO-DCSK is shown in Fig. 2b. Firstly, we ignore the influence of the wireless channel to demonstrate the principle of demodulation, and then we discuss the influence of the channel in the next section. At the receiver, the first subcarrier, $gs_{0}(t)=\sin(2\pi f_{0}t)$, is used to implement a coherent computing $J_{0k}$ with $e_{k}(t)$, \begin{align*} J_{0 k}= & \int_0^T g_0(t) \times e_k(t) \mathrm{d} t= \\ & \int_0^T \sin \left(2 \pi f_0 t\right) \times e_k(t) \mathrm{d} t= \\ & \int_0^T \sin \left(2 \pi f_0 t\right) \times\left\{a_{0 k} \sin \left(2 \pi f_0 t\right)+\right. \\ & a_{1 k} \sin \left[2 \pi f_1(t-\tau)\right]+\cdots+ \\ & \left.a_{M k} \sin \left[2 \pi f_1(t-M \tau)\right]\right\} \mathrm{d} t= \\ & a_{0 k} r_{00}+a_{1 k} r_{01}+\cdots+a_{M k} r_{0 M} \tag{6} \end{align*} where\begin{gather*} r_{0 m}=\int_0^T \sin \left(2 \pi f_0 t\right) \times \sin \left[2 \pi f_m(t-m \tau)\right] \mathrm{d} t, \\ m=0,1, \ldots, M \tag{7} \end{gather*}

Fig. 3

Principle of the frequency domain non-orthogonal modulation.

It can be seen that Eq. (6) is a linear equation about the modulated signal $a_{mk}(n=0,1,\ \ldots,\ M)$. Then, the second subcarrier, $g_{1}(t)=\sin(2\pi f_{1}(t-\tau))$, is used to implement a coherent computing with $e_{k}(t)$, \begin{align*} & J_{1 k}=\int_0^T g_1(t) \times e_k(t) \mathrm{d} t= \\ & \int_0^T \sin \left[2 \pi f_1(t-\tau)\right] \times e_k(t) \mathrm{d} t= \\ & \int_0^T \sin \left[2 \pi f_1(t-\tau)\right] \times\left\{a_{0 k} \sin \left(2 \pi f_0 t\right)\right. \\ & a_{1 k} \sin \left[2 \pi f_1(t-\tau)\right]+ \\ & \quad a_{2 k} \sin \left[2 \pi f_2(t-2 \tau)\right]+\cdots+ \\ & \left.\quad a_{M k} \sin \left[2 \pi f_M(t-M \tau)\right]\right\} \mathrm{d} t= \\ & a_{0 k} r_{10}+a_{1 k} r_{11}+\cdots+a_{M k} r_{1 M} \tag{8} \end{align*} where\begin{gather*} r_{1 m}=\int_0^T \sin \left[2 \pi f_1(t-\tau)\right] \times \sin \left[2 \pi f_m(t-m \tau)\right] \mathrm{d} t, \\ m=0,1, \ldots, M \tag{9} \end{gather*}

Similar to Eq. (6), Eq. (8) is also a linear equation about all the modulated signals. After $M+1$ times coherent computations using all subcarriers $g_{n}(t)$, a system of linear equations is obtained,\begin{equation*} \pmb {RA}_{k}=\pmb J_{k} \tag{10} \end{equation*} where\begin{equation*} \pmb{R}=\begin{bmatrix} r_{00} & r_{01} & \cdots & r_{0 M} \\ r_{10} & r_{11} & \cdots & r_{1 M} \\ \vdots & \vdots & \ddots & \vdots \\ r_{M 0} & r_{M 1} & \cdots & r_{M M} \end{bmatrix} \tag{11} \end{equation*}

\begin{gather*} \pmb{A}_k=\begin{bmatrix} a_{0 k} & a_{1 k} & \ldots & a_{M k} \end{bmatrix}^{\mathrm{T}}= \\ \\ \begin{bmatrix} x_k & s_1 x_k & \ldots & s_M x_k \end{bmatrix}^{\mathrm{T}} \tag{12} \\ \pmb{J}_k= \\ \begin{bmatrix} J_{0 k} & J_{1 k} & \ldots & J_{M k} \end{bmatrix}^{\mathrm{T}} \tag{13} \end{gather*}

According to Eqs. (7) and (9), the matrix $\pmb R$ can be easily constructed by using the subcarrier matrix $G$ at the receiver, i.e.,\begin{equation*} \pmb R=\pmb {GG}^{\mathrm{T}} \tag{14} \end{equation*} where\begin{equation*} \pmb{G}=\begin{bmatrix} g_0(t) & 0 & \cdots & 0 \\ 0 & g_1(t) & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & g_M(t) \end{bmatrix} \tag{15} \end{equation*}

It can be seen that the dimension of $G$ is $(M+1)\times 2T$. In addition, according to Eq. (15), $G$ is a row-full-rank matrix, i.e., $r(\pmb G)=M+1$. As a result, R is an $M+1$ order Toeplitz matrix and the rank of $\pmb R$ is $M+1$. Thus, the inverse matrix of $\pmb R$ exists. Accordingly, all the modulated information bits are obtained by solving Eq. (10), \begin{equation*} \pmb A_{k}=\pmb R^{-1}\pmb J_{k} \tag{16} \end{equation*}

After $\beta$ clock cycles, the chaotic reference sequence $X$ and information-bearing signal matrix $\pmb {Y}$ can be obtained,\begin{gather*} \pmb{X}=\begin{bmatrix} x_1 & x_2 & \cdots & x_\beta \end{bmatrix} \tag{17}\\ \pmb{Y}=\begin{bmatrix} s_1 x_1 & s_1 x_2 & \cdots & s_1 x_\beta \\ s_2 x_1 & s_2 x_2 & \cdots & s_2 x_\beta \\ \vdots & \vdots & \ddots & \vdots \\ s_M x_1 & s_M x_2 & \cdots & s_M x_\beta \end{bmatrix} \tag{18} \end{gather*}

Finally, the transmitted data can be acquired through matrix multiplication,\begin{equation*} \hat{s}=\text{sign}(XY^{\mathrm{T}}) \tag{19} \end{equation*} where sign $()$ is a sign function.

It is worth noting that, MCNO-DCSK is naturally a MC-DCSK system, the method of solving a system of linear equations is also applicable to other MC-DCSK systems, such as OFDM-DCSK and traditional MC-DCSK systems.

SECTION 3 System Performance Analysis

3.1 Spectrum Efficiency

The most significant feature of MCNO-DCSK is the non-orthogonality in the frequency domain, which can lead to a higher spectral efficiency than ever before. Since the information is demodulated by solving a system of linear equations instead of the Fast Fourier Transform (FFT), the subcarriers are no longer constrained by the strict orthogonality, and the frequency interval can be much narrower. Therefore, when the total number of subcarriers is fixed, the total bandwidth of MCNO-DCSK can be much narrower than ever, as shown in Fig. 4. In the traditional MC-DCSK system, spectrums of sub-channels must be separated, and a guard band is always needed to avoid the interference between neighbor channels, as shown in Fig. 4a. In OFDM-DCSK, the frequencies are orthogonal, thus the spectrums of the sub-channels can be overlapped, indicating a higher spectral efficiency (see Fig. 4b). As shown in Fig. 4c, for MCNO-DCSK, the frequency interval will be further narrowed, which means that more spectrum resources will be saved for conveying more information. Therefore, MCNO-DCSK is particularly suitable for communication systems with extremely tight spectrum resources, such as maritime communications.

According to Fig. 4a, the bandwidth of MC-DCSK is\begin{equation*} B_{\text{MC}-\text{DCSK}}=(M+1) \cdot\frac{2}{T} \tag{20} \end{equation*}

Here we ignore the guard band.

Suppose one sub-channel is used to transmit the chaotic reference signal, and other $M$ sub-channels are used to convey $M$ information bits. Therefore, the bit rate of MC-DCSK is\begin{equation*} \text { rate }_{\text {MC}-\text {DCSK}}=\frac{M}{\beta \cdot T} \tag{21} \end{equation*}

Fig. 4

Spectrums of mc-dcsk, ofdm-dcsk, and mcno-dcsk. the total number of subcarriers is fixed.

Then, the spectral efficiency of MC-DCSK is\begin{align*} \text{SE}_{\text{MC}-\text{DCSK}}= & \frac{\text { rate }_{\text{MC}-\text{DCSK}}}{B_{\text{MC}-\text{DCSK}}}= \\ & \frac{M /(\beta \cdot T)}{(M+1) \cdot \frac{2}{T}}=\frac{M}{2 \beta(M+1)} \tag{22} \end{align*}

In addition, assuming that an OFDM-DCSK symbol carries the same number of bits as MC-DCSK, the bandwidth and spectral efficiency of OFDM-DCSK are as follows:\begin{align*} & B_{\text{OFDM}-\text{DCSK}}=(M+2) \cdot \frac{1}{T} \tag{23}\\ & \text{SE}_{\text{OFDM}-\text{DCSK}}= \frac{\text { rate }_{\text{OFDM}}}{B_{\text{OFDM}} \text{DCSK}}= \\ & \frac{M /(\beta \cdot T)}{(M+2) \cdot \frac{1}{T}}=\frac{M}{\beta(M+2)} \tag{24} \end{align*}

For MCNO-DCSK, the duration of one symbol is $2T$. Therefore, the bit rate of MCNO-DCSK is\begin{equation*} \text { rate }_{\text{MCNO}-\text{DCSK}}=\frac{M}{\beta \cdot 2 T} \tag{25} \end{equation*}

Then, we assume that the frequency interval of MCNO-DCSK is less than 1. Therefore, the bandwidth and spectral efficiency of MCNO-DCSK is\begin{align*} B_{\text{MCNO}}-\text{DCSK} & =(2+M \cdot \delta) \cdot \frac{1}{T} \tag{26}\\ \text{SE}_{\text{MCNO}-\text{DCSK}}= & \frac{\text { rate }_{\text{MCNO}-\text{DCSK}}}{B_{\text{MCNO}-\text{DCSK}}}= \\ & \frac{M /(2 \beta \cdot T)}{(2+M \cdot \delta) \cdot \frac{1}{T}}=\frac{M}{2 \beta(2+M \cdot \delta)} \tag{27} \end{align*}

When $\delta$ equals to 1 or 2, MCNO-DCSK is equivalent to OFDM-DCSK and MC-DCSK, respeclively. Table 1 lists the bandwidth and spectral efficiency of above systems.

To clearly demonstrate the spectral efficiency of all the systems, Fig. 5 shows the influence of the total number of subcarriers on the Spectral Efficiency (SE) for different MC-DCSK systems. It is clearly shown that the SE of the proposed MCNO-DCSK is higher than those of other two MC-DCSK systems. Moreover, a small $\delta$ offers a higher SE. Noting that when $\delta$ does not equal to zero, the SE curves tend to be flat with the increase of M, indicating an upper limit of SE. When δ equals to zero, both the chaotic reference signal and information-bearing signal occupy the same frequency sub-channel. Thus, in this case, SE is a linear function about M. However, small frequency intervals will influence the form of the coefficient matrix R, and then will affect the BER performance of the whole system. In addition, due to the introduced time delay in the symbol, the time duration is inevitably increased, and the data transmission rate will therefore be slightly decreased. However, the spectral efficiency is significantly improved, because all subcarriers have the same frequency. Moreover, the increase in spectral efficiency is much greater than the decrease in the communication rate. For the current tight spectrum resources, a high spectral efficiency means that the communication network can allocate bandwidth to more users and communication services. Therefore, it is worthwhile to pay the price of a slight decrease in data transmission rate.

Fig. 5

Spectral efficiencies of mc-dcsk, ofdm-dcsk, and mcno-dcsk.

Table 1 Bandwidths and spectral efficiencies of various mc-dcsk systems.

3.2 Complexity

In this paper, complexity is defined as the total number of multiplications and additions required to generate symbols and demodulated information. MCNO-DCSK is essentially a traditional MC-DCSK^[12], and hence the complexities of both systems are similar. For MC-DCSK^[12], $\beta\times M$ multiplications and $\beta\times(M-1)$ additions are required to form a symbol. To demodulate $(M-1)$ information bits, $\beta\times(M-1)$ multiplications and $(\beta-1)\times(M-1)$ additions are required. Thus, the total complexity of MC-DCSK is $2\beta M-\beta$ multiplication and $(2\beta-1)\times(M-1)$ addition. The complexity of MCNO-DCSK is similar to that of MC-DCSK, but solving the system of linear equations introduces additional complexity. It is well known that the complexity of solving a system of equations of dimension $M$ is $O(M^{3})$. Therefore, the complexity of MCNO-DCSK is larger than that of MC-DCSK. In addition, OFDM-DCSK requires $\beta\times(M-1)$ Inverse FFT (IFFT) and FFT operations to demodulate $(M-1)$ information bits, and its complexity is also smaller than that of MCNO-DCSK. Although MCNO-DCSK inevitably increases the complexity, the process of solving the system equations is easy to be implemented by a common computer. More importantly, the spectral efficiency is greatly improved, and more information bits can be transmitted within a limited bandwidth.

3.3 Security

The proposed MCNO-DCSK can further improve the security. Since neighboring subcarriers are no longer orthogonal, the frequency spacing between subcarriers is no longer fixed, and can be a flexible parameter. This makes the FFT no longer useful. Also, for a given fixed bandwidth, a different frequency spacing $(\Delta f)$ will result in a different total number of subcarriers $(M+1)$, as shown in Fig. 6. These two parameters are crucial for the receiver to establish the demodulation equation set. Even if the eavesdropper has the demodulation method (solving the system of linear equations), it will be very difficult to decipher both $\Delta f$ and $M+1$. It is worth noting that in the MCNO-DCSK system, the frequency intervals between sub-channels can be different from each other, which further increases the difficulty of deciphering. At the same time, these two parameters can be changed periodically between legitimate users to prevent eavesdropping. Therefore, the proposed MCNO-DCSK can further improve the security.

Fig. 6

Spectrum of mcno-dcsk. the total bandwidth is fixed.

3.4 Derivation of Ber Expressions

In this section, BER expressions of MCNO-DCSK over the AWGN channel are derived. Considering the noise in the channel, the received signal is\begin{equation*} \pmb r_{k}=\pmb s_{k}+\pmb n_{k} \tag{28} \end{equation*} where $\pmb r_{k}, \pmb s_{k}$, and $\pmb n_{k}$ represents the received signal, transmitted signal, and the noise vectors, respectively. Therefore, the coherent computation becomes\begin{equation*} \pmb{G}^{\mathrm{T}} \pmb{r}_k=\pmb{G}^{\mathrm{T}}\left(\pmb{s}_k+\pmb{n}_k\right)=\pmb{G}^{\mathrm{T}} \pmb{s}_k+\pmb{G}^{\mathrm{T}} \pmb{n}_k=\pmb{J}_k+\Delta \pmb{J}_k \tag{29} \end{equation*}

We can see that the noise causes a deviation $\Delta \pmb J_{k}$. As a result, the solution of Eq. (10) will also have a deviation $\Delta \pmb A_{k}$, \begin{equation*} \pmb A_{k}+\Delta \pmb A_{k}=\pmb R^{-1}(\pmb J_{k}+\Delta \pmb J_{k}) \tag{30} \end{equation*}

Then, after $\beta$ coherent computations, the chaotic reference and information-bearing signals are as follows:\begin{gather*} \pmb{X}^{\prime}=\begin{bmatrix} x_1+\Delta x_1 & \cdots & x_\beta+\Delta x_\beta \end{bmatrix} \tag{31}\\ \pmb{Y}^{\prime}=\begin{bmatrix} s_1 x_1+\Delta s_1 x_1 & \cdots & s_1 x_\beta+\Delta s_1 x_\beta \\ \vdots & \ddots & \vdots \\ s_M x_1+\Delta s_M x_1 & \cdots & s_M x_\beta+\Delta s_M x_\beta \end{bmatrix} \tag{32} \end{gather*} where $s_{m}x_{k}$ and $x_{k}(m=1,2,\ \ldots,\ M,\ k=1,2,\ \ldots,\ \beta)$ are the elements of vector $\pmb A$, and $\Delta s_{m}x_{k}$ and $\Delta x_{k}$ are the elements of vector $\Delta \pmb A$. At the output of the correlator, the decision variable of the m-th sub-channel is\begin{equation*} D_m=\sum_{k=1}^\beta\left(s_m x_k+\Delta s_m x_k\right) \times\left(x_k+\Delta x_k\right) \tag{33} \end{equation*}

According to Eq. (6), when $\beta$ is large we have\begin{equation*} \sum_{k=1}^{\beta}x_{k}x_{j}\approx 0 \tag{34} \end{equation*}

Thus, according to Eq. (33) we have\begin{align*} D_m= & \sum_{k=1}^\beta s_m x_k x_k+\sum_{k=1}^\beta\left(s_m x_k \cdot \Delta x_k+x_k \cdot \Delta s_m x_k\right)+ \\ & \sum_{k=1}^\beta\left(\Delta s_m x_k \cdot \Delta x_k\right) \tag{35} \end{align*}

Similar to Eq. (6), in MCNO-DCSK, the ^chaoti^c reference signal is shared by other $M$ paths of information-bearing signals, so ^we have\begin{equation*} E_{b}=\frac{M+1}{M}\sum_{k=1}^{\beta}x_{k}^{2} \tag{36} \end{equation*} where $E_{b}$ represents the energy of a given bit. Meanwhile, since $\pmb {RA}=\pmb J$, we have\begin{equation*} \Delta A= \pmb R^{-1}\Delta \pmb J=\pmb R^{-1}\pmb G^{\mathrm{T}}n \tag{37} \end{equation*}

Then, $\Delta s_{m}x_{k}$ and $\Delta x_{k}$ can be expressed as\begin{align*} \Delta x_k & =\sum_{t=1}^T \pmb{U}(1, t) n_t \tag{38}\\ \Delta s_m x_k & =\sum_{t=1}^T \pmb{U}(m, t) n_t \tag{39} \end{align*} where $\pmb U=\pmb R^{-1}\pmb C^{\mathrm{T}}, \pmb C=[g_{0}(t),\ g_{1}(t),\ \ldots\,\ g_{M}(t)]$ represents the matrix containing all subcarriers. $n_{t}$ represents the tth sampling points during $T$. From Eqs. (25) and (26), $\Delta s_{m}x_{k}$ and $\Delta x_{k}$ have similar statistical distributions with Gaussian noise. Thus, we have\begin{gather*} E\left(\Delta s_m x_k\right)=E\left(x_k\right)=0 \tag{40}\\ \text{var}\left(x_k\right)=\left(\sum_{t=1}^T \pmb{U}(1, t)\right)^2 \cdot \frac{N_0}{2} \tag{41}\\ \text{var}\left(\Delta s_m x_k\right)=\left(\sum_{t=1}^T \pmb{U}(m, t)\right)^2 \cdot \frac{N_0}{2} \tag{42} \end{gather*} where $E(\cdot)$ and var $(\cdot)$ represent the expectation and variance operator, respectively, ^and $N_{0}$ is the Power Spectrum Density (PSD) of the noise. Substituting Eqs. (27–29) into Eq. (22), we can get\begin{align*} & E\left(D_m \vert s_m=+1\right)=\frac{M+1}{M} E_b \tag{43}\\ {var}\left(D_m\right)= & \frac{M+1}{M} E_b \frac{N_0}{2} \times \\ & {\left[\left(\sum_{t=1}^T \pmb{U}(1, t)\right)^2+\left(\sum_{t=1}^T \boldsymbol{U}(m, t)\right)^2\right] } \tag{44} \end{align*}

Thus, the BER of the m-th channel and the average BER of the whole system are expressed in Eq. (45) and Eq. (46), respectively, shown at the bottom of this page, where erfc $()$ represents the complementary error funclion.

3.5 Analysis and Mitigation of the Interference in the Multi-Path Fading Channel

In this paper, we do not derive the BER expression under multipath fading channels, because the multipath ^effect ^can seriously interfere with the demodulation process. Therefore, the BER of the system will always remain at a relatively high level even if the Signal-to-Noise Ratio (SNR) is high. It is analyzed as follows.

In a multipath fading channel, the receiver receives the original MCNO-DCSK signal and its multiple delayed copies. Since there are already time delays between multiple non-orthogonal subcarriers in the MCNO-DCSK symbol, the additional delay caused by the multipath effect will bring additional interference to the equation set solution process, which will lead to a huge deviation in the solution of the equation set. As a result, the BER performance ^will be severely deteriorated.

When $e_{k}(t)$ transmits through a multi-path fading channel, the received signal $r_{k}(t)$ is\begin{align*} & \frac{1}{2} \text{erfc}\left\{\left[\frac{M+1}{M}\left[\left(\sum_{t=1}^T 2 U(1, t)\right)^2+\left(\sum_{t=1}^T 2 U(m, t)\right)^2\right] \frac{N_0}{E_b}+\left(\frac{M+1}{M}\left[\left(\sum_{t=1}^T 2 U(1, t)\right)^2+\left(\sum_{t=1}^T 2 U(m, t)\right)^2\right]\right)^2 \frac{\beta}{2}\left(\frac{N_0}{E_b}\right)^2\right]^{-\frac{1}{2}}\right\} \tag{45}\\ & \text{BER}_{\text{MCNO}-\text{DCSK}-\text{AWGN}}=\frac{1}{M} \sum_{m=1}^M \text{BER}_{m-\text{th}-\text{AWGN}}= \\ & \frac{1}{M} \sum_{m=1}^M \frac{1}{2} \text{erfc}\left\{\left[\frac{M+1}{M}\left[\left(\sum_{t=1}^T 2 U(1, t)\right)^2+\left(\sum_{t=1}^T 2 U(m, t)\right)^2\right] \frac{N_0}{E_b}+\left(\frac{M+1}{M}\left[\left(\sum_{t=1}^T U(1, t)\right)^2+\left(\sum_{t=1}^T 2 U(m, t)\right)^2\right]\right)^2 \frac{\beta}{2}\left(\frac{N_0}{E_b}\right)^2\right]^{-\frac{1}{2}}\right\} \tag{46} \end{align*}

\begin{align*} r_k^{\prime}(t)= & \sum_{l=1}^L \alpha_l e_k\left(t-\tau_l\right)= \\ & a_{0 k} \sum_{l=1}^L \alpha_l \sin \left[2 \pi f_0\left(t-\tau_l\right)\right]+ \\ & a_{1 k} \sum_{l=1}^L \alpha_l \sin \left[2 \pi f_1\left(t-\tau-\tau_l\right)\right]+\cdots+ \\ & a_{M k} \sum_{l=1}^L \alpha_l \sin \left[2 \pi f_M\left(t-M \tau-\tau_l\right)\right] \tag{47} \end{align*} where $\alpha_{l}$ and $\tau_{l}$ represent the power fading and the time delay of the l-th path, and $L$ is the total number of paths. To discuss the multi-path effect, the noise is ignored here. In this case, the coherent computation becomes\begin{align*} & J_{i k}^{\prime}=\int_0^T \sin \left[2 \pi f_i(t-i \tau)\right] \times r_k^{\prime}(t)= \\ & \int_0^T \sin \left[2 \pi f_i(t-i \tau)\right] \times \\ & \\ & \left\{a_{0 k} \sum_{l=1}^L \alpha_l \sin \left[2 \pi f_0\left(t-\tau_l\right)\right]+\right. \\ & a_{1 k} \sum_{l=1}^L \alpha_l \sin \left[2 \pi f_1\left(t-\tau-\tau_l\right)\right]+\cdots+ \\ & \left.a_{M k} \sum_{l=1}^L \alpha_l \sin \left[2 \pi f_M\left(t-M \tau-\tau_l\right)\right]\right\} \mathrm{d} t= \\ & a_{0 k} r_{i 0}^{\prime}+a_{1 k} r_{i 1}^{\prime}+\cdots+a_{M k} r_{i M}^{\prime}, \\ & i=0,1, \ldots, M \tag{48} \end{align*} where\begin{align*} r_{i j}^{\prime}= & \int_0^T \sin \left[2 \pi f_i(t-i \tau)\right] \times \\ & \sum_{l=1}^L \alpha_l \sin \left[2 \pi f_1\left(t-j \tau-\tau_l\right)\right] \mathrm{d} t, \\ & i=0,1, \ldots, M, \quad j=0,1, \ldots, M \tag{49} \end{align*}

Comparing Eq. (40) with Eq. (7) or Eq. (9), the interference, such as the additional time delay and power fading induced by the channel, is imbedded in the elements of $\pmb J$, which is the right side of the equation $\pmb {RA}=\pmb J$. Nevertheless, for practice scenarios, the receiver always has no information about the channel state, and then constructs the coefficient matrix $\pmb R$ through subcarriers without any time delay and power fading information of the channel. As a result, it will induce a big error in the solution of the equation set, i.e., $\Delta A$ will be significantly enlarged. Therefore, the BER performance cannot be guaranteed in multi-path fading channels, even if the power of noise is low.

In other words, the BER will always remain a relatively high level even for a high SNR.

Therefore, to ensure the performance, the channel estimation and equalization technique is necessary. It is worth noting that, in MCNO-DCSK, a simpler channel estimation and equalization can be implemented rather than acquiring every channel coefficient in other coherent communication systems. Firstly, the pilot is utilized to implement the channel estimation. The FFT is implemented to the pilot before and after transmission, and the frequency response of the channel can be expressed as follows:\begin{equation*} H(f)=\frac{\text{FFT}(P^{\prime})}{\text{FFT}(P)} \tag{50} \end{equation*} where $H(f)$ represents the frequency response of the channel, FFT $(\cdot)$ is the FFT operator, and $P$ and $P^{\prime}$ are the pilot before and after transmission, respectively. Then, an FFT operation is done to the received signal $r_{k}^{\prime}(t)$, and the frequency response of $r_{k}^{\prime}(t)$ is obtained by a division as follows:\begin{equation*} E_{k}^{\prime}(f)=\frac{\text{FFT}(r_{k}^{\prime}(t))}{H(f)} \tag{51} \end{equation*}

Then, the time domain waveform of the received signal is achieved by an IFFT operation,\begin{equation*} \hat{e}_{k}^{\prime}(t)=\text{IFFT}(E_{k}^{\prime}(f)) \tag{52} \end{equation*} where IFFT $(\cdot)$ is the IFFT operator. Finally, $\hat{e}_{k}^{\prime}(t)$ is sent to the receiver to demodulate the transmitted information. Moreover, BER comparison with other typical MC-DCSK systems is implemented to verify the superiority of MCNO-DCSK.

SECTION 4 Simulation Result and Discussion

In this section, the BER performance of MCNO-DCSK under various system parameters are simulated.

4.1 Effect of Various Parameters on the Ber Performance

Firstly, as stated in Ref. [28], the spreading factor is the most basic parameter of the chaos-based communication system, so we study its effect on the BER performance. As shown in Fig. 7a, the effect of $\beta$ on the BER performance is very small. When $\beta$ increases, the BER increases a little. Meanwhile, the simulation results have the same trend with the theoretical curves, which means the validity of the theoretical analysis. Figure 7b shows the effect of $\beta$ on the BER performance at different $E_{b}/N_{0}$. We can see that the BER increases with $\beta$ at all $E_{b}/N_{0}$. The reason is similar to Ref. [28]. A larger $\beta$ will include more noise component, and cause the decision error.

Fig. 7

Ber performance of mcno-dcsk over awgn channel. (a) ber performance with $\pmb \tau=0.5, \pmb M=64$, and $\beta= 10, 20$, and 40. (b) BER performance versus the spreading factor $\pmb \beta$ with $\pmb \tau=0.5, \pmb M=64$, and $\pmb E_{\pmb b}/\pmb N_{\pmb 0}=4,8$, and 12 db.

As MCNO-DCSK permits a narrower frequency interval, the relationship between BER and the frequency interval $\delta$ is studied, and results are shown in Fig. 8. Firstly, simulation results match with the theoretical curves, which verifies our theoretical analysis. Secondly, although a narrower frequency interval offers a higher spectral efficiency, it can induce a deterioration of the system performance. When $\delta$ equals to 0.1, which means that the frequency interval reduces to only 10% of that of OFDM-DCSK, the BER maintains around $10{-1}$. It means that the communication almost fails. In our previous work^[41], we find that the BER is proportional to the norm of $\pmb R^{-1}$ in Eqs. (10) and (20). Here, the results shown in Fig. 8 can also be verified clearly from the aspect of $\Vert \pmb R^{-1}\Vert$. As shown in Table 2, at $\delta=0.4$ and $\delta=0.1\ \Vert \pmb R^{-1}\Vert$ is about eight times and fifty times of that at $\delta=0.5$, respectively. This means that when $\delta < 0.5$, the BER performance will significantly deteriorate. Moreover, Fig. 8 also reflects the effect of $\delta$ on the BER performance, which is similar with Fig. 7.

Fig. 8

Ber performance of mcno-dcsk under awgn channel with $\beta=10, \pmb M=64$, and $\delta=0.1,0.2,0.3,0.4$, and 0.5.

Table 2 Norm of $\pmb R {-1}$ at different $\pmb \delta$.

Meanwhile, to thoroughly illustrate the performance of MCNO-DCSK, Fig. 9 shows the BER performance of MCNO-DCSK over maritime multi-path channel with and without channel estimation and equalization, respectively. During the simulation, sea state level is set to 4, and the parameters of the maritime channel are the same as thoes in Ref. [41]. It can be clearly seen that the multi-path effect has a severe influence on the system performance. Without channel estimation and equalization, the BER maintains a relatively high level. After channel estimation and equalization, the BER performance is significantly improved.

For $\delta < 0.5$ scenarios, the key to improve the performance of MCNO-DCSK is decreasing $\Vert \pmb R^{-1}\Vert$. One approach is reducing the total number of sub-channels $M$, as changing $M$ will change the dimension of $\pmb R$, and then change $\Vert \pmb R^{-1}\Vert$. For instance, Fig. 10 shows the BER performance of MCNO-DCSK with various $\pmb M$ when $\delta$ equals to 0.1 and 0.5. It can be clearly seen that the BER performance is improved when $M$ decreased. The norm of $\pmb R^{-1}$ also verifies this phenomenon, as show in Table 3. However, reducing $M$ means decreasing the data transmission rate. Thus, for a specific BER performance, the total number of $M$ and frequency interval $\delta$ must be appropriately chosen.

Fig. 9

Ber performance of mcno-dcsk under maritime multipath channel with and without channel estimation and equalization. System parameters are $\pmb \beta=10, \pmb M=64$, and $\delta= 0.1, 0.2, 0.3, 0.4$, and 0.5.

Fig. 10

Ber performance of mcno-dcsk under maritime multipath channel. System parameters are $\delta$ 0.1 with $\pmb M=4,6,8,12$, and 64, and $\pmb \delta=0.5$ with $\pmb M=16,32$, and 64.

Another approach is changing the form of subcarriers. For example, subcarriers can be cosine waveforms,\begin{equation*} g_{n}(t)=\begin{cases} \cos(2\pi f_{n}(t-n\cdot\tau)),\ t\in T;\\ 0,\ t\not\in T \end{cases} \tag{53} \end{equation*}

As shown in Fig. 11, comparing with MCNO-DCSK with sinusoidal subcarriers, the BER performance of MCNO-DCSK with cosine subcarriers is improved, except for $\delta$ equals to 0.5. This phenomenon can also be explained from the aspect of $\Vert \pmb R^{-1}\Vert$. From Table 4 we can see that the norm of $\pmb R^{-1}$ is reduced when the subcarriers are cosine waveforms. Therefore, MCNO-DCSK with cosine subcarriers is more suitable for bandwidth-limited circumstance.

Fig. 11

Ber performance of mcno-dcsk with sinine and cosine subcarriers. System parameters are $\delta=0.1,0.2$, 0.3, 0.4, 0.5, and $\pmb M=\pmb {64}$.

Table 3 $\Vert \pmb R^{-1} \Vert$ of mcno-dcsk with various $\delta$ and $\pmb M$.

Table 4 $\Vert \pmb R^{-1}\Vert$ of mcno-dcsk with sinusoidal and cosine subcarriers.

Inspired by the results shown in Fig. 11, we study the MCNO-DCSK with cosine subcarriers and $\delta=0$, which means that the reference signal and all information-bearing chaotic signals are encapsulated into a single frequency sub-channel. From Fig. 12, we can see that when $M$ decreases to 40, the BER at $E_{b}/N_{0}=20$ dB is less than 3.8 ×10–³, which is the threshold of the Forward Error Correction (FEC) coding. Thus, through the MCNO modulation, one frequency sub-channel can contain up to 40 paths of chaotic information-bearing channels, indicating a significantly improved spectral efficiency.

Since time delay has been introduced in the MCNO-DCSK symbol, it is important to investigate the impact of delays in the channel as well as timing errors on the performance. Firstly, we use $\tau_{l}$ to represent the maximum time delay induced by the channel. In above studies, $\tau_{l}$ is assumed to be less than the time delay $\tau$ in the symbol. However, when $\tau_{l}$ is larger than $\tau$, i.e., $\tau_{l} > 7$, the system performance is deteriorated, as shown in Fig. 13. The reason is that when $\tau_{l} > \tau$, the sampling points of former subcarriers will introduce extra interference to the coefficient matrix R. As a result, the BER increases. One way to eliminate this interference is improving the duration of the whole symbol to ensure that $\pmb \tau_{l}$ is less than $\pmb \tau$, Then, the effect of timing error on the system performance is shown in Fig. 14. We can see that when the timing error is less than 3 sampling points, the BER performance maintains acceptable, while when the timing error is large than 3 sampling points, the BER performance is significantly deteriorated. Noting that during the simulation, the time delay $\tau$ between neighbor subcarriers is 10 sampling points. Therefore, to ensure the BER performance, the timing error must be less than 1/3 of the time delay between neighbor subcarriers.

Fig. 12

Ber performance of meno-dcsk with $\pmb \delta=0$ and m= 16, 24, 32, 40, 48, and 64. The subcarrier is cosine wave.

Fig. 13

Ber performance of mcno-dcsk under the situations $\pmb \tau_{l} <\pmb \tau$ and $\tau_{l} > \tau$.

4.2 Comparison With Other Typical Mc-Dcsk Benchmarks

In this section, we implement performance comparisons between MCNO-DCSK and other MC-DCSK benchmarks based on typical multi-carrier modulations, such as OFDM-DCSK^[19] and the traditional MC-DCSK^[12]. For fairness, the system parameters, such as the spreading factor, the total number of subcarriers, are all set to be the same, and all the systems apply the same channel models, channel estimation Eq. (50), channel equalization Eq. (51) and demodulation method Eq. (16).

Fig. 14

Ber performance of mcno-dcsk under different timing errors.

From Fig. 15, when $M$ of OFDM-DCSK and MCNO-DCSK are both 64, the BER performance of MCNO-DCSK is a little worse than that of OFDM-DCSK. However, the bandwidth of MCNO-DCSK is only half of that of OFDM-DCSK, indicating that the spectral efficiency of MCNO-DCSK is two times of that of OFDM-DCSK. Then, when $\delta$ equals to 0.5 and $M$ equals to 48 and 32, both the spectral efficiency and the BER performance of MCNO-DCSK are better than those of OFDM-DCSK. When δ equals to 0.1, from Eqs. (24) and (27), the spectral efficiency of MCNO-DCSK with M=5 is 1.6 times of that of OFDM-DCSK, and BER performances of MCNO-DCSK is much better. Thus, we can draw some conclusions as follows. For the same M, performances of MCNO-DCSK and OFDM-DCSK are similar, and the spectral efficiency of MCNO-DCSK is higher. When $M$ of MCNO-DCSK is less than that of OFDM-DCSK, both the BER performance and spectral efficiency of MCNO-DCSK are higher than those of OFDM-DCSK.

Secondly, when $M=64$ comparing with the traditional MC-DCSK^[6], we can see that the BER performance of MCNO-DCSK with $\delta=0.5$ is better, and the spectral efficiency is much higher than that of MC-DCSK. When $\delta=0.1s$, the BER performance of MCNO-DCSK with $M=6$ is similar to that of MC-DCSK with $M=64.s$ Although the data transmission rate is lower, the bandwidth of MCNO-DCSK is less than 1/10 of the bandwidth of MC-DCSK, which means that most of the spectrum resources are saved.

Fig. 15

Ber performance comparison of ofdm-dcsk and mcno-dcsk under various system parameters.

In addition, the PAPR of various MC-DCSKs is compared, as the PAPR can influence the structure of the receiver. Figure 16 shows the PAPR comparison among above systems. We can see that the PAPR of MCNO-DCSK is always lower than that of OFDM. The reason is as follows. In some regions of the MCNO-DCSK symbol, the superposition of positive and negative amplitudes of the carrier leads to energy cancellation, so that the peak power of MCNO-DCSK is lower than that of OFDM-DCSK, and the PAPR is therefore reduced.

SECTION 5 Conclusion

In this paper, a novel high spectral efficiency MCNO-DCSK for future maritime communications in SAGSIN is proposed and numerically demonstrated. Thanks to this novel non-orthogonal modulation technology, the frequency intervals between sub-channels can be much smaller than ever, which leads to a high spectral efficiency. Moreover, the non-orthogonality will further improve the security of the system. The influence of various system parameters, such as the spreading factor, the total number of subcarriers, and the frequency interval on the system performance, have been studied. We find that although a smaller frequency interval or a larger number of subcarriers will improve the spectral efficiency, it will deteriorate the system performance. It is because the system performance is proportional to the norm of R^-1. subcarriers with cosine form will solve this contradiction. With suitable system parameters, MCNO-DCSK can accommodate 40 paths of information-bearing signals in a single sub-channel, which indicates a high spectral efficiency. In addition, extending the symbol duration is able to counteract the effect of the delay in the multipath channel, and a sampling error within 1/3 of the delay between subcarriers will ensure that the system performance is not deteriorated. Finally, the non-orthogonality between subcarriers will simultaneously introduce a more flexible parameter space and improve the system security.

Fig. 16

Papr comparison between mcno-dcsk and ofdm-dcsk.

When comparing with other MC-DCSK benchmarks, the BER performance of MCNO-DCSK is about 3 dB better than that of the traditional MC-DCSK, and similar to that of OFDM-DCSK under the same total number of subcarriers, while the bandwidth of MCNO-DCSK is half of that of OFDM-DCSK. Meanwhile, there are multiple combination of parameters that enables both the spectral efficiency and the BER performance to be better than other MC-DCSK competitors.

Finally, this novel digital modulation method can be easily extended to various novel modulation technologies, such as index modulations, which will be our future work. In addition, future work will also focus on discovering more methods to decrease the norm of R-¹ to further improve the performance of the proposed system.

ACKNOWLEDGMENT

This work was supported by the National Key Research and Development Program of China (No. 2019YFE0111600), the National Natural Science Foundation of China (Nos. 62001077, 61801074, 61971083, and 51939001), the China Postdoctoral Science Foundation Funded Project (No. 2019M661075), the Dalian High Level Talent Innovation Support Plan (No. 2021RQ063), the Dalian Science and Technology Innovation Fund (No. 2019J11CY015), and the Liaoning Revitalization Talents Program (No. XLYC2002078).