Secure Mimo Communication System with Frequency Hopping Aided OFDM-DCSK Modulation

Abstract

In this paper, a multiple-input multiple-output (MIMO) communication system with frequency hopping (FH) aided orthogonal frequency division multiplexing differential chaotic shift keying (OFDM-DCSK) modulation is proposed. Our objective is to improve the security of MIMO communication system which is encoded by space time block coding (STBC). In order to combat the eavesdropping or malicious attacks due to the broadcast characteristics of wireless communication system, we propose to use DCSK and FH modules to encrypt the information, and hide the user data in the chaotic sequences, where the initial value of chaotic sequences and the method of generating FH module are only shared among legitimate users. Moreover, we derive the bit error rate (BER) and the secrecy capacity of the scheme in additive white Gaussian noise (AWGN) channel and Rayleigh fading channel. Simulation results show that the proposed scheme can effectively improve the security of MIMO communication system, which can be seen from the BER of eavesdroppers and legitimate users, and the secrecy capacity of the proposed scheme and the benchmark schemes.

1. Introduction

In recent years, with the increase of communication equipments, the single-input single-output (SISO) communication system has been unable to meet the needs of more efficient and faster communication [1]. In order to solve this problem, the multiple-input multiple-output (MIMO) communication system using a space resource has become an effective way. Because the performance gain of MIMO communication system can increase linearly with the increase of the number of antennas, the MIMO communication has not only been widely used in the radio frequency (RF) field but also aroused the interest of researchers which are interested in how to apply this scheme to visible light communication (VLC) [2,3].

At present, the MIMO communication system which is based on space time code (STC) [4] technology has been widely used in 4G and 5G communication systems. At the same time, in order to improve the reliability of the communication system, researchers further transform space time block coding (STBC) [2] in STC into orthogonal STBC or combine it with spatial multiplexing (SM) [1,3]. Moreover, in recent years, the orthogonal time-frequency space (OTFS) modulation has been proposed to be applied in the MIMO communication system [5], which has advantages of overcoming Doppler interference and low computational complexity [6,7]. However, compared with the orthogonal frequency division multiplexing (OFDM) scheme, the OTFS might induce security performance degradations [8].

In the last ten years, the chaotic communication in the SISO communication system has attracted much attention due to its good characteristics of non-periodic, wide-band, noise-like, and sensitivity to initial values [9]. Moreover, chaotic communication also has low probability of intercept [10]. The chaotic communication can be divided into coherent chaotic communication and non-coherent chaotic communication according to whether the chaotic synchronization process is needed at the receiver. As defined in the classification of chaotic communication, non-coherent chaotic communication attracts more attention because it does not need a chaotic synchronization process [10]. In non-coherent chaotic communication, differential chaotic phase-shift keying (DCSK) has attained more attention.

In recent years, in order to further improve the security of the DCSK, some scholars propose to scramble chaotic sequences [11], other scholars propose to use the internal correlation of chaotic sequences to encode and decode information bits [12]. In addition, some scholars also propose to use time slot sequences to allocate chaotic reference sequences and information-carrying sequences [13]. However, these methods require complex delay line circuits in the process of implementation; at the same time, because of the need to transmit chaotic reference sequences, the communication efficiency will be reduced [14].

In order to remove the delay line circuit of DCSK and improve communication efficiency, some scholars proposed to use OFDM technology to overcome this defect [15,16]. However, the direct transmission of chaotic reference sequences will degrade the security performance of signal transmissions [17]. Then, Ref. [18] proposed to apply the frequency hopping (FH) to the OFDM system to assist DCSK (FH-OFDM-DCSK) to overcome the defects of DCSK [18].

However, few research studies have been done to utilize the FH aided OFDM-DCSK scheme to enhance the security of MIMO transmissions. A model combining MIMO with DCSK has been proposed in [19,20], while according to [21], MIMO communication system which combined STBC with DCSK (STBC-DCSK) are divided into two types: those that require channel state information (CSI) and those that do not. Most scholars who still study the STBC-DCSK model focus on how to use the characteristics of DCSK to improve the reliability of MIMO communication system and use MIMO communication technology to improve the communication efficiency of DCSK [19,22]. For example, some scholars propose to directly add frequency hopping to improve the reliability of STBC-DCSK [23]. Although these models can make use of the sensitivity of DCSK to initial values, noise-like characteristics and the simple scrambling method to improve the security of MIMO communication system to a certain extent. However, since these schemes need to transmit chaotic reference sequences directly during the communication process, the security of MIMO communication system is still threatened.

Inspired by [18,21,22], in this paper, we propose the scheme which uses the FH-OFDM-DCSK to improve the security of MIMO communication system which combines STBC and DCSK and needs CSI.

In this paper, we propose a MIMO communication system assisted by FH-OFDM-DCSK (MIMO-FH-OFDM-DCSK). In this design, we use DCSK and FH modules to improve the security of MIMO communication system which is encoded by STBC [24]. On the basis of using DCSK and FH modules, the proposed scheme hides the user data in the chaotic sequences. At the same time, the proposed scheme inherits the advantage of FH-OFDM-DCSK scheme in eliminating the delay line circuit. We then derive the theoretical bit error rate (BER) and secrecy capacity expressions for the proposed system. The simulation results over both additive white Gaussian noise (AWGN) channel and Rayleigh fading channel are also provided to validate the effectiveness of the proposed scheme and the superior performance to benchmark scheme.

Briefly, the major contributions include:

1.

We propose to use FH-OFDM-DCSK to improve the security of the MIMO communication system. The initial values of chaotic sequences and the method of generating FH module are shared only between legitimate users. Thus, eavesdroppers or malicious users can not recover the received unknown information. Thus, the scheme proposed in this paper can improve the security of the MIMO communication system.

2.

We analyze the theoretical BER and secrecy capacity for the proposed scheme. Then, simulation results are provided to verify the effectiveness of the scheme proposed in this paper and that the proposed MIMO communication system can achieve higher secrecy capacity.

The rest of this paper is organized as follows: In Section 2, the principle of the MIMO-FH-OFDM-DCSK system is introduced. Then, theoretical performances are analyzed in Section 3. Section 4 shows simulation results to provide the BER of legitimate users and eavesdroppers over the AWGN channel and Rayleigh fading channel as well as the secrecy capacity of the proposed scheme and the benchmark scheme, and Section 5 presents our conclusion.

2. MIMO-FH-OFDM-DCSK System

In this section, we will introduce how to apply the FH-OFDM-DCSK to the MIMO communication system.

Figure 1 illustrates the transmitter and receiver structure of the MIMO-FH-OFDM-DCSK system with two transmitting antennas and one receiving antenna. To enhance the security performance of the MIMO communication system, we propose to apply the FH-OFDM-DCSK scheme in the MIMO communication system.

2.1. Transmitter

At the transmitter, firstly, we use the binary phase shift keying (BPSK) to modulate the user data. Secondly, the BPSK symbols $[a_1,a_2,\dots ]$ are encoded by STBC, which can be expressed as [19]:

$$\begin{array}{cc}\hfill \mathbf{A}& =\left. [\begin{array}{cc}{a}_1& -a_2^{*}\\ a_2& a_1^{*}\end{array}\right],\hfill \end{array}$$

(1)

where ${(·)}^{*}$ represents the conjugate operation. For the matrix $\mathbf{A}$, in the row direction, the symbols are delivered from the same antenna, while in the column direction, the symbols are delivered in the same time slot.

After composing the serial data stream that each antenna needs to send, the transmitter will conduct the serial to parallel (S/P) conversion and perform the FH-OFDM-DCSK modulation. As shown in Figure 2, each data stream delivered via the antenna 1 of the MIMO communication system is first modulated by the chaotic sequences. Then, we use the FH module to make chaotic modulation symbols hop in the frequency domain.

Subsequently, the symbols generated by the FH module are modulated using the inverse fast Fourier transform (IFFT) through the OFDM module. After the parallel to serial (P/S) conversion and adding a cyclic prefix (CP), the resultant signals are transmitted to the wireless channels [18].

To be more explicit, let ${\mathbf{b}}_1$ represent the first row of symbols in matrix $\mathbf{A}$, which can be expressed as follows:

$$\begin{array}{c}\hfill {\mathbf{b}}_1=[a_1,-a_2^{*}].\end{array}$$

(2)

Similarly, we can obtain an expression for ${\mathbf{b}}_2$:

$$\begin{array}{c}\hfill {\mathbf{b}}_2=[a_2,a_1^{*}].\end{array}$$

(3)

The symbols modulated by DCSK can be expressed as:

$$\begin{array}{c}\hfill {\mathbf{g}}_{\mathbf{k}}={\mathbf{b}}_1\times c_k,\end{array}$$

(4)

where $c_k$ represents the chaotic sequence, and ${\mathbf{g}}_{\mathbf{k}}=[g_{0,k},g_{1,k},\cdots ,g_{n,k},\cdots ,g_{N-1,k}]$ represents the symbols modulated by DCSK at antenna 1 and $g_{N-1,k}=a_{N-1}\times c_k$, where N is the number of subcarriers, $a_{N-1}$ represents the ($N-1$)th symbol in ${\mathbf{b}}_1$, $g_{0,k}=c_k$ denotes the kth chaotic reference symbol, and $g_{N-1,k}$ denotes the kth chaotic symbol carrying symbol $a_{N-1}$. Note that the chaotic sequences are generated by the second order Chebyshev polynomial function (CPF) [16], which can be expressed as follows:

$$\begin{array}{c}\hfill c_{k+1}=1-2c_k^2,(0\le k\le \beta -1),\end{array}$$

(5)

where $\beta$ is the length of chaotic sequences. The modulation process of the second row of symbols in matrix $\mathbf{A}$ is similar to the first row of symbols in matrix $\mathbf{A}$.

With reference to [18], the FH module is defined as a matrix represented by $\mathbf{U}$. Without loss of generality, here we set the length of the chaotic sequences to be the same as that of the number of the subcarriers, i.e., $\beta =N$. Then, after the FH, we can obtain the symbols as:

$$\begin{array}{c}\hfill {\mathbf{F}}_1={\mathbf{G}}_1·\mathbf{U},\end{array}$$

(6)

where the symbols modulated by DCSK at antenna 1 are represented by ${\mathbf{G}}_1$ = [${\mathbf{g}}_{\mathbf{0}}^T$, ${\mathbf{g}}_{\mathbf{1}}^T$, ⋯, ${\mathbf{g}}_{\mathbf{k}}^T$, ⋯, ${\mathbf{g}}_{\beta -\mathbf{1}}^T$], ${(·)}^T$ represents the transposition operation. ${\mathbf{F}}_1$ represents the symbols obtained after ${\mathbf{G}}_1$ hopping in antenna 1, $\mathbf{U}$ represents the hopping matrix used at antenna 1, the generation process of $\mathbf{U}$ can be expressed as follows [18]:

$$\begin{array}{cc}\hfill \mathbf{U}& =\left. [\begin{array}{cc}\mathbf{0}& {\mathbf{E}}_m\\ {\mathbf{E}}_{N-m}& \mathbf{0}\end{array}\right],\hfill \end{array}$$

(7)

where ${\mathbf{E}}_m,{\mathbf{E}}_{N-m}$ denote the identity matrices of order m and order $N-m$, respectively; thus, $\mathbf{U}$ can be equivalent to an identity matrix ${\mathbf{E}}_N$ that has undergone several elementary transformations. For ${\mathbf{E}}_m$, the value of m depends on chaotic sequence of ${\mathbf{c}}_k$, since $0\le k\le \beta -1, \beta =N$, then have $c_{k,m}>c_{k,q}, 0\le q\le \beta -1$, $m=m+1$, and the initial value of m is 0. Note that the initial value of chaotic sequences and the method of generating FH module are only shared between legitimate transceivers [18].

Similarly, for antenna 2, the same FH matrix is applied. Let ${\mathbf{F}}_2$ represent the symbols obtained after ${\mathbf{G}}_2$ hopping in antenna 2, then we have:

$$\begin{array}{c}\hfill {\mathbf{F}}_2={\mathbf{G}}_2·\mathbf{U},\end{array}$$

(8)

where ${\mathbf{G}}_2$ is the symbols modulated by DCSK, after conducting the chaotic modulation with the same chaotic sequences as the system with antenna 1.

Subsequently, in the proposed scheme, the symbols that are modulated by FH in each antenna are modulated by OFDM using IFFT [15], and the expression is given as follows:

$$\begin{array}{c}\hfill {\mathbf{T}}_1=\frac{1}{\sqrt{N}}\sum _{n=0}^{N-1}{\mathbf{F}}_1{exp}^{j\frac{2\pi n}{N}},\end{array}$$

(9)

where ${\mathbf{T}}_1$ represents the OFDM modulated symbols in antenna 1. These symbols are then sent to the antenna and passed into the channel.

2.2. Receiver

Figure 3 shows the process of FH-OFDM-DCSK demodulation after the receiver receiving the symbols $\mathbf{R}$. At the legitimate receiver, the user first removes the CP of the received data and performs S/P conversion on the received symbols. Then, use the fast Fourier transform (FFT) to implement OFDM demodulation [15]. The symbols after the FFT operation can be represented by $\mathbf{r}$. $\mathbf{r}$ can be expressed as follows:

$$\begin{array}{cc}\hfill \mathbf{r}& =\frac{1}{\sqrt{N}}\sum _{i=0}^{N-1}\mathbf{R}{exp}^{-j\frac{2\pi i}{N}}=\mathbf{H}·\mathbf{F}+\mathbf{n},\hfill \end{array}$$

(10)

where $\mathbf{n}$ represents the AWGN, $\mathbf{H}$ is the channel response matrix, and $\mathbf{F}=\left. [\begin{array}{c}{\mathbf{F}}_1\\ {\mathbf{F}}_2\end{array}\right]$. For the considered $2\times 1$ MIMO communication system, we have:

$$\begin{array}{c}\hfill \mathbf{H}=\left. [\begin{array}{cc}{h}_1& h_2\end{array}\right],\end{array}$$

(11)

where $h_1$ and $h_2$ respectively denote the channel impulse response with two channels. After the OFDM demodulation, legitimate users will recover the correct chaotic sequences by the de-hopping matrix that is generated by the FH module which is hidden in chaotic sequences. Note that only legitimate users know how the FH rule for performing the de-hopping operation. Next, the de-hopping matrix is used to de-hop $\mathbf{r}$, which can be expressed as follows:

$$\begin{array}{c}\hfill \mathbf{r}·{\mathbf{U}}^{-1}=\mathbf{H}·\mathbf{F}·{\mathbf{U}}^{-1}+\mathbf{n}·{\mathbf{U}}^{-1},\end{array}$$

(12)

where ${(·)}^{-1}$ represents the inverse operation of a matrix.

Next, the chaotic demodulation is performed with the initial values of the chaotic sequences which are only shared by legitimate users which can be expressed as follows [17]:

$$\begin{array}{c}\hfill yn=\Re \{\sum _{k=0}^{\beta -1}conj\left(g_{0,k}\right)·g_{n,k}\},\end{array}$$

(13)

where $yn$ represents the nth equivalent symbol after chaotic demodulation, $g_{0,k}$ denotes the kth chaotic reference symbol, $g_{n,k}$ denotes the kth chaotic symbol carrying information used to transmit $yn$, and $\Re \left\{·\right\}$ represents the real part of the symbol. Let us take $y1$, $y2$ for example, the symbols after the chaotic demodulation [14] can be expressed as follows:

$$\begin{array}{cc}\hfill y_1& =\Re \{\sum _{k=0}^{\beta -1}(h_1{a}_1+h_2{a}_2)c_k^2+c_k\xi \hfill \\ \hfill & \phantom{=}+\frac{(h_1{a}_1+h_2{a}_2)c_k\xi }{(h_1^{*}+h_2^{*})}+\frac{\xi {\xi }^{*}}{(h_1^{*}+h_2^{*})}\},\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill y_2& =\Re \{\sum _{k=0}^{\beta -1}(h_1(-a_2^{*})+h_2\left(a_1^{*}\right))c_k^2+c_k\xi \hfill \\ \hfill & \phantom{=}+\frac{(h_1(-a_2^{*})+h_2\left(a_1^{*}\right))c_k\xi }{(h_1^{*}+h_2^{*})}+\frac{\xi {\xi }^{*}}{(h_1^{*}+h_2^{*})}\},\hfill \end{array}$$

(15)

where $\xi$ is the complex Gaussian white noise whose real part and imaginary part that followed the Gaussian distribution with a mean value of zero.

Meanwhile, the equivalent noise $N_k$ of the kth equivalent symbol can be expressed as:

$$\begin{array}{c}\hfill N_1=\Re \left\{\sum _{k=0}^{\beta -1}\frac{(h_1{a}_1+h_2{a}_2)c_k\xi }{(h_1^{*}+h_2^{*})}+c_k\xi +\frac{\xi {\xi }^{*}}{(h_1^{*}+h_2^{*})}\right\},\end{array}$$

(16)

$$\begin{array}{c}\hfill N_2=\Re \left\{\sum _{k=0}^{\beta -1}\frac{(h_1(-a_2^{*})+h_2\left(a_1^{*}\right))c_k\xi }{(h_1^{*}+h_2^{*})}+c_k\xi +\frac{\xi {\xi }^{*}}{(h_1^{*}+h_2^{*})}\right\}.\end{array}$$

(17)

Finally, after the FH-OFDM-DCSK demodulation, we perform STBC decoding on the demodulated symbols, so that legitimate users can obtain the estimated symbols. The decoding process is expressed as follows:

$$\begin{array}{c}\mathbf{y}=\left. [\begin{array}{c}{y}_1\\ y_2\end{array}\right],\end{array}$$

(18)

$$\begin{array}{c}\left. [\begin{array}{c}{\widehat{a}}_1\\ {\widehat{a}}_2\end{array}\right]={\left({\mathbf{H}}^H\mathbf{H}\right)}^{-1}{\mathbf{H}}^H·\mathbf{y},\end{array}$$

(19)

$$\left. [\begin{array}{c}{\widehat{a}}_1\\ {\widehat{a}}_2\end{array}\right]=\left. [\begin{array}{c}\Re \left\{\sum _{k=0}^{\beta -1}{c}_k^2{a}_1\right\}\\ \Re \left\{\sum _{k=0}^{\beta -1}{c}_k^2{a}_2\right\}\end{array}\right]+\left. [\begin{array}{cc}{h}_1^{*}& h_2\\ h_2^{*}& -h_1\end{array}\right]·\left. [\begin{array}{c}{N}_1\\ N_2^{*}\end{array}\right],$$

(20)

where $\mathbf{y}$ represents the symbols for which STBC decoding is required, ${\widehat{a}}_1$ and ${\widehat{a}}_2$ represent the symbols estimated by legitimate users, and ${(·)}^H$ represents the conjugate transpose operation of a matrix.

3. Performance Analysis

In this section, we assume that the chaotic sequences’ length $\beta$ is large enough so that the Gaussian approximation (GA) method can be applied [25] to derive the theoretical BER and secrecy capacity expressions over the AWGN channel and the Rayleigh fading channel.

3.1. Ber Performance

Firstly, according to the FH-OFDM-DCSK demodulation and STBC decoding process in Section 2.2, we are able to derive the energy per bit $E_b$ as:

$$\begin{array}{c}\hfill E_b=\Re \left\{\sum _{k=0}^{\beta -1}{c}_k^2\right\}.\end{array}$$

(21)

Then, substitute $E_b$ into Equation (20), and the estimates of received symbols can be expressed as:

$$\left. [\begin{array}{c}{\widehat{a}}_1\\ {\widehat{a}}_2\end{array}\right]=\left. [\begin{array}{c}{E}_b{a}_1\\ E_b{a}_2\end{array}\right]+\left. [\begin{array}{cc}{h}_1^{*}& h_2\\ h_2^{*}& -h_1\end{array}\right]·\left. [\begin{array}{c}{N}_1\\ N_2^{*}\end{array}\right].$$

(22)

Let $\mathrm{E}\{·\}$ represent the mean and $\mathrm{var}\{·\}$ represents the variances. Next, we use the method of numerical analysis to deduce the numerical characteristics of the estimated symbol, and the derivation process is as follows:

$$\begin{array}{cc}\hfill \mathrm{E}\left\{{\widehat{a}}_1\right\}& =E_b\mathrm{E}\left\{a_1\right\},\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill \mathrm{var}\left\{{\widehat{a}}_1\right\}& =\mathrm{E}\left\{a_1^2\right\}-\mathrm{E}{\left\{a_1\right\}}^2=\mathrm{var}\left\{h_1^{*}{N}_1\right\}+\mathrm{var}\left\{h_2{N}_2^{*}\right\},\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill \mathrm{var}\left\{h_1^{*}{N}_1\right\}& =\frac{\beta \mathrm{E}\left\{{\xi }^{*}\xi \right\}\mathrm{E}\{{\left(h_1^{*}\right)}^2\}}{\mathrm{E}\{{(h_1^{*}+h_2^{*})}^2\}}+E_b\times \mathrm{E}\left\{{\xi }^2\right\}\times \mathrm{E}\{{\left(h_1^{*}\right)}^2\}\hfill \\ \hfill & \phantom{=}\times \mathrm{E}\{(\frac{{(h_1{a}_1+h_2{a}_2)}^2}{{(h_1^{*}+h_2^{*})}^2}+1)\}\hfill \\ \hfill & =\frac{\beta N_0^2\mathrm{E}\{{\left(h_1^{*}\right)}^2\}}{2\mathrm{E}\{{(h_1^{*}+h_2^{*})}^2\}}+\frac{E_b{N}_0\mathrm{E}\{{\left(h_1^{*}\right)}^2\}}{2}\times \mathrm{E}\{(\frac{{(h_1{a}_1+h_2{a}_2)}^2}{{(h_1^{*}+h_2^{*})}^2}+1)\},\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill \mathrm{var}\left\{h_2{N}_2^{*}\right\}& =\frac{\beta \mathrm{E}\left\{{\xi }^{*}\xi \right\}\mathrm{E}\{{\left(h_2\right)}^2\}}{\mathrm{E}\{{(h_1+h_2)}^2\}}+E_b\times \mathrm{E}\left\{{\xi }^2\right\}\times \mathrm{E}\{{\left(h_2\right)}^2\}\hfill \\ \hfill & \phantom{=}\times \mathrm{E}\{(\frac{{(h_1^{*}(-a_2)+h_2^{*}{a}_1)}^2}{{(h_1+h_2)}^2}+1)\}\hfill \\ \hfill & =\frac{\beta N_0^2\mathrm{E}\{{\left(h_2\right)}^2\}}{2\mathrm{E}\{{(h_1+h_2)}^2\}}+\frac{E_b{N}_0\mathrm{E}\{{\left(h_2\right)}^2\}}{2}\times \mathrm{E}\{(\frac{{(h_1^{*}(-a_2)+h_2^{*}{a}_1)}^2}{{(h_1+h_2)}^2}+1)\},\hfill \end{array}$$

(26)

where $N_0$ represents the noise power.

Furthermore, $\Gamma$ represents the signal-to-noise ratio (SNR) can be determined as:

$$\begin{array}{c}\hfill \Gamma =\frac{{\left(\mathrm{E}\left\{{\widehat{a}}_1\right\}\right)}^2}{\mathrm{var}\left\{{\widehat{a}}_1\right\}}.\end{array}$$

(27)

Based on the derived numerical characteristics, we are able to derive the theoretical BER expressions over the selective Rayleigh fading channel, flat Rayleigh fading channel, and AWGN channel. For the selective Rayleigh fading channel, the theoretical BER can be derived as:

$$\begin{array}{cc}\hfill & BE{R}_{Selective}=\frac{1}{2}\mathrm{erfc}(\sqrt{\frac{\Gamma }{2}})=\frac{1}{2}\mathrm{erfc}({(\frac{\mathrm{E}{\left\{{\widehat{a}}_1\right\}}^2}{2\mathrm{var}\{{\widehat{a}}_1\}})}^{\frac{1}{2}})\hfill \\ \hfill & =\frac{1}{2}\mathrm{erfc}\left(\right((\frac{\frac{\beta N_0^2}{2\mathrm{E}\left\{{(h_1^{*}+h_2^{*})}^2\right\}}}{E_b^2}+\frac{(\frac{E_b{N}_0}{2})\times (\frac{\mathrm{E}\{{(h_1+h_2)}^2\}}{\mathrm{E}\{{(h_1^{*}+h_2^{*})}^2\}}+1)}{E_b^2}\hfill \\ \hfill & \phantom{=}+\frac{\frac{E_b{N}_0}{2}\times (\frac{\mathrm{E}\{{(-h_1^{*}+h_2^{*})}^2\}}{\mathrm{E}\{{(h_1^{*}+h_2^{*})}^2\}}+1)}{E_b^2}\hfill \\ \hfill & \phantom{=}+\frac{\frac{\beta N_0^2}{2\mathrm{E}\{{(h_1^{*}+h_2^{*})}^2\}}}{E_b^2})\times \frac{2}{\left(\right|h_1{|}^2+|h_2{{|}^2)}^2}{)}^{-\frac{1}{2}}).\hfill \end{array}$$

(28)

where $\mathrm{erfc}(·)$ represents complementary error function, and $|·|$ denotes complex number modulo. For the flat Rayleigh fading channel, let channel impulse response $h_k=h$, and the theoretical BER can be derived as:

$$\begin{array}{cc}\hfill & BE{R}_{flat}=\frac{1}{2}\mathrm{erfc}\left(\right((\frac{\beta }{8\mathrm{E}\{{\left(h^{*}\right)}^2\}{(\frac{E_b}{N_0})}^2}\hfill \\ \hfill & +\frac{\frac{\mathrm{E}\{h^2\}}{\mathrm{E}\{{\left(h^{*}\right)}^2\}}+2}{2\frac{E_b}{N_0}}+\frac{\beta }{8\mathrm{E}\{{\left(h^{*}\right)}^2\}{(\frac{E_b}{N_0})}^2})\times \frac{1}{{2\left|h\right|}^2}{)}^{-\frac{1}{2}}).\hfill \end{array}$$

(29)

For the AWGN channel, we set $h_k=h=1$. Then, we can derive the expression of theoretical BER as:

$$\begin{array}{cc}\hfill BE{R}_{AWGN}& =\frac{1}{2}\mathrm{erfc}(\sqrt{\frac{\Gamma }{2}})\hfill \\ \hfill & =\frac{1}{2}\mathrm{erfc}(\sqrt{\frac{8{(\frac{E_b}{N_0})}^2}{(\beta +6\frac{E_b}{N_0})}}).\hfill \end{array}$$

(30)

3.2. Secrecy Capacity

Next, we calculate the mutual information $I_n(Y_E;X)$ recovered by eavesdroppers. Without loss of generality, we assume that 0 and 1 are equally distributed, and that the transmissions over each subchannel are statistically independent. The mutual information can be derived as [26]:

$$\begin{array}{cc}\hfill I_n(Y_E;X)& =H_n\left(Y_E\right)-H_n\left(Y_E\right|X)\hfill \\ \hfill & =1+p_e{log}_2{p}_e+(1-p_e){log}_2(1-p_e),\hfill \end{array}$$

(31)

where $H_n(·)$ represents the entropy operation, $p_e$ represents the BER of eavesdroppers, X represents the transmitted information, and $Y_E$ represents the information retrieved by eavesdroppers.

Then, we derive the information leakage L expression as [26]:

$$\begin{array}{c}\hfill L=\frac{1}{N}\sum _{n=0}^{N-1}{I}_n(Y_E;X).\end{array}$$

(32)

Finally, the expression of secrecy capacity $C_{secrecy}$ can be derived as:

$$\begin{array}{c}\hfill C_{secrecy}=\frac{1}{N}\sum _{n=0}^{N-1}{I}_n(Y_L;X)-L,\end{array}$$

(33)

where $I_n(Y_L;X)$ still represents mutual information, but it represents mutual information of legitimate users, $Y_L$ represents the information retrieved by legitimate users, and $p_b$ represents the legitimate users’ BER.

4. Simulation Results

In this section, firstly, simulation results are provided to investigate the effectiveness of the proposed scheme. Next, we simulate the BER of legitimate users and eavesdroppers as well as the secrecy capacity of the proposed scheme and the benchmark scheme. Simulation results prove that the scheme proposed in this paper can effectively prevent eavesdroppers from obtaining useful information, thus improving the security of the MIMO communication system.

In the simulation, we assume that the propagation channels are the AWGN channel and Rayleigh fading channel. In addition, perfect CSI can be obtained at the receiver. There are two transmit antennas at the transmitter and one receive antenna at the receiver. Then, we assume that different subcarrier number N and chaotic sequences’ length $\beta$ are used in the simulation. In addition, we assume that (1) the eavesdroppers can steal the STBC code information; and (2) the eavesdroppers can steal STBC coding information and chaotic reference sequences. In both cases, the secrecy capacity of STBC and STBC-DCSK [19,20] will be compared with the proposed scheme.

Firstly, in Figure 4, we simulate the theoretical and simulated BER of the proposed system with different $\beta$ and different N. It can be observed that, over the AWGN channel, the theoretical BER matches the simulated ones, which verify the effectiveness of the proposed scheme. It is noticeable that, when N is the same in Figure 4a, the BER of the proposed scheme decreases with the increase of $\beta$. The reason is that the increase of chaotic sequences’ length leads the increase of interference between different chaotic sequences. Additionally, we assume that the equalization is applied, the multi carrier interferences are negligible, and $\beta$ is the same; it can be seen from Figure 4b that, in this assumption, the BER of the proposed scheme is basically the same as the increase of N.

In order to verify that the proposed scheme can effectively improve the security of MIMO communication system, we simulate the BER of legitimate users and eavesdroppers over different channels and $\beta =N$. It can be seen from the BER of eavesdroppers in Figure 5 and Figure 6, either the AWGN channel or the Rayleigh channel, the BER of eavesdroppers is very high when recovering the user data. This is because eavesdroppers do not know the initial values of the chaotic sequences, the method of generating FH module and the way FH module hidden in the chaotic sequences, so eavesdroppers cannot use the chaotic sequences to generate the correct FH module. Without the correct FH module, eavesdroppers could not correctly de-hop, so they could not obtain the correct user data. Therefore, the proposed scheme can effectively improve the security of MIMO communication system.

Then, to demonstrate the improved security performance of the MIMO communication system, we simulate the information secrecy capacity of the scheme proposed in this paper and the benchmark scheme (STBC). As shown in Figure 7,

Table 1. Secrecy capacity comparison between the proposed system and the benchmark scheme (STBC) in the AWGN channel.

	Secrecy Capacity
	MIMO-FH-OFDM-DCSK			Alimouti
Eb/N0 (dB)	$\mathbf{\beta }=\mathit{N}=\mathbf{32}$	$\mathbf{\beta }=\mathit{N}=\mathbf{64}$	$\mathbf{\beta }=\mathit{N}=\mathbf{128}$
2	0.332	0.217	0.117	0.000
3	0.501	0.311	0.190	0.000
4	0.620	0.448	0.271	0.000
5	0.756	0.553	0.399	0.000
6	0.870	0.718	0.537	0.000

and

Table 2. Secrecy capacity comparison between the proposed system and the benchmark scheme (STBC) in the Rayleigh channel.

	Secrecy Capacity
	MIMO-FH-OFDM-DCSK			Alimouti
Eb/N0 (dB)	$\mathbf{\beta }=\mathit{N}=\mathbf{32}$	$\mathbf{\beta }=\mathit{N}=\mathbf{64}$	$\mathbf{\beta }=\mathit{N}=\mathbf{128}$
8	0.453	0.353	0.188	0.000
9	0.561	0.496	0.252	0.000
10	0.769	0.673	0.380	0.000
11	0.873	0.800	0.543	0.000
12	0.946	0.919	0.690	0.000

, thanks to the FH and chaotic modulation, the secrecy capacity of the proposed scheme is significantly higher than the benchmark MIMO system. Moreover, we can see from Table 1 and Table 2 that, after the eavesdropper has stolen the information of MIMO communication system with STBC encoding only, whether it is AWGN channel or Rayleigh fading channel, the secrecy capacity of the benchmark scheme (STBC) is smaller than that of the proposed scheme at any Eb/N0 (dB). In addition, we can notice that the secrecy capacity also increases when the value of Eb/N0 (dB) increases.

Finally, we compare the secrecy capacity of the proposed scheme with that of the benchmark scheme (STBC-DCSK) under two kinds of channels when the eavesdroppers steal the initial value of chaotic sequences, and the results are shown in Figure 8. As can be seen from the figure, even if the eavesdroppers steal the initial value of chaotic sequences, by adding the FH module, the scheme proposed in this paper can achieve higher secrecy capacity.

In addition, we can see from

Table 3. Secrecy capacity comparison between the proposed system and the benchmark scheme (STBC-DCSK) [19,20] in AWGN channel.

	Secrecy Capacity
	MIMO-FH-OFDM-DCSK			STBC-DCSK
Eb/N0 (dB)	$\mathbf{\beta }=\mathit{N}=\mathbf{32}$	$\mathbf{\beta }=\mathit{N}=\mathbf{64}$	$\mathbf{\beta }=\mathit{N}=\mathbf{128}$	$\mathbf{\beta }=\mathit{N}=\mathbf{128}$
4	0.627	0.423	0.287	0.000
5	0.777	0.569	0.366	0.000
6	0.910	0.722	0.553	0.000
7	0.956	0.871	0.685	0.000
8	0.987	0.952	0.846	0.000

and

Table 4. Secrecy capacity comparison between the proposed system and the benchmark scheme (STBC-DCSK) [19,20] in the Rayleigh channel.

	Secrecy Capacity
	MIMO-FH-OFDM-DCSK			STBC-DCSK
Eb/N0 (dB)	$\mathbf{\beta }=\mathit{N}=\mathbf{32}$	$\mathbf{\beta }=\mathit{N}=\mathbf{64}$	$\mathbf{\beta }=\mathit{N}=\mathbf{128}$	$\mathbf{\beta }=\mathit{N}=\mathbf{128}$
11	0.697	0.565	0.347	0.000
12	0.844	0.703	0.454	0.000
13	0.910	0.815	0.592	0.000
14	0.973	0.937	0.776	0.000
15	0.998	0.950	0.882	0.000

that, if the eavesdroppers steal the key information of STBC-DCSK scheme and chaotic reference sequences, the user data delivered in the MIMO communication system will still be cracked. With the proposed design, the MIMO communication system can achieve better security performance than the one which only uses the DCSK technology.

5. Conclusions

In order to improve the security of MIMO communication system which is encoded by STBC, we propose to use FH-OFDM-DCSK to enhance the security performance of the MIMO communication system. In our design, the DCSK and FH modules are used to encrypt the information, while the user data are hidden in the chaotic sequences, where the initial value of chaotic sequences and the method of generating FH module are only shared among legitimate users. At the same time, the proposed scheme inherits the advantage of FH-OFDM-DCSK scheme in eliminating the delay line circuit. The theoretical BER and secrecy capacity of the proposed scheme are derived. Simulation results demonstrate the effectiveness of the proposed scheme. By comparing the BER of legitimate users and eavesdroppers and the secrecy capacity of the proposed scheme and the benchmark scheme, it can be seen that the proposed scheme can effectively improve the security of the MIMO communication system. The potential future work can be extended to reduce the complexity.