# Design and Performance Analysis of MISO-ORM-DCSK System over Rayleigh Fading Channels

Authors: Gang Zhang; Chuan gang Wang; Guo quan Li; Jin zhao Lin
Journal: International Journal of Antennas and Propagation
 (2016)

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## Abstract

A novel chaotic communication system, named Orthogonality-based Reference Modulated-Differential Chaos Shift Keying (ORM-DCSK), is proposed to enhance the performance of RM-DCSK. By designing an orthogonal chaotic generator (OCG), the intrasignal interference components in RM-DCSK are eliminated. Also, the signal frame format is expanded so the average bit energy is reduced. As a result, the proposed system has less interference in decision variables. Furthermore, to investigate the bit error rate (BER) performance over Rayleigh fading channels, the MISO-ORM-DCSK is studied. The BER expressions of the new system are derived and analyzed over AWGN channel and multipath Rayleigh fading channel. All simulation results not only show that the proposed system can obtain significant improvement but also verify the analysis in theory.


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## Body

## 1. Introduction

Chaos based digital communication systems have been proposed and studied in recent years [1–7]. The differential chaos shift keying (DCSK) and its constant power version called frequency-modulated DCSK (FM-DCSK) are widely studied. However, the drawbacks of DCSK or FM-DCSK are low data rate and poor security [1, 2]. In order to address those problems, NR-DCSK, OFDM-DCSK, and short reference DCSK are proposed in [8–10]. Based on the robustness of DCSK against linear and nonlinear channel distortions, Kaddoum et al. studied DCSK for PLC applications. On the other hand, smart antenna technology can eliminate the multipath wave propagation [11]. Therefore, MIMO-DCSK and STBC-DCSK are proposed to improve the performance of DCSK system [12–14]. Furthermore, the reference modulated DCSK (RM-DCSK) is proposed in [15]. In RM-DCSK system, the chaotic signal sent in each time slot not only carries one bit of information but also serves as the reference signal of the information bit transmitted in its following slot. For this reason, the attainable data rate of RM-DCSK is doubled in comparison to DCSK. However, the intrasignal interference is produced in decision variables at the receiver. Therefore, compared to DCSK, BER performance of RM-DCSK is not improved.In this paper, the ORM-DCSK system is proposed as an improved version for RM-DCSK. A novel orthogonal chaotic generator (OCG) is designed to generate orthogonal chaos signal. Also, the signal frame format is extended. Furthermore, two transmit antennas are used in the transmitter of the ORM-DCSK and BER formula is derived.The remainder of this paper is organized as follows: in Section2, the principle of MISO-ORM-DCSK is presented. In Section 3, the BER expressions are analyzed. In Section 4, simulation results are shown to evaluate the performance of the new system. Finally, summaries are given in Section 5.


## 2. MISO-ORM-DCSK

In this section, the baseband implementation of MISO-ORM-DCSK is introduced and some details are explained.


### 2.1. Signal Format

Figure1 shows the frame format of ORM-DCSK system. Different with RM-DCSK, there are N adjacent time slots in each frame, in which average bit energy (labeled as Eb) decreases from 1.5βE(xi2) to (1+1/N)βE(xi2). The time slot is divided into β chips to carry the chaotic sequences and bl is the lth information bit in kth frame. Concretely, Xk is modulated by the information bit b1∈{+1,-1} in the first time slot of kth frame. In the second time slot, b2b1Xk is sent. In the Nth time slot, bN⋯b1Xk+X~k+1 is sent. Here, X~k+1 can be converted into the reference sequence of k+1th frame. In Figure 1, Xk=(1/MT)·[xk(1),xk(2),…,xk(β)] and MT is the number of transmit antennas.Figure 1
Signal frame format of ORM-DCSK.


### 2.2. Transmitter Structure

Figure2 shows the block diagram of orthogonal chaotic generator (OCG). It is necessary to make the chaotic carriers transmitted in neighboring two frames orthogonal to erase the intrasignal interference components included in decision variables clearly. We assume that the spreading factor is six (β=6). In the first-time slot period, as shown in Figure 2, the switch is connected to bottom. The chaotic sequence U=[a,b,c] is multiplied by w22=[+1,-1]. In other words, Xk=U·w22=[a,-a,b,-b,c,-c]. During second- to N-1th-slots period, the switch is suspended. In the Nth-time slot period, the switch is connected to bottom. The chaotic sequence V=[x,y,z] is multiplied by w21=[+1,+1]. X~k+1=V·w21=[x,x,y,y,z,z]. In the first-time slot period of k+1th frame, the switch is connected to bottom again. Xk+1=V·w22=[x,-x,y,-y,z,-z].Figure 2
Block diagram of OCG.The chaotic sequences generated by the OCG satisfy(1)Xk⊥X~k+1,X~k+1⊥Xk+1,Xk+1=X~k+1W,Xk+1⊥XkW.Here W=[+1,-1⋯+1,-1]︸β is a modification function.Figure3 shows the block diagram of ORM-DCSK transmitter. There are two transmitting antennas (MT=2) and one receiving antenna in MISO-ORM-DCSK system. The two transmitting antennas are independent. The signal can be sent by number 1 antenna and the replica of the signal can be sent by number 2 antenna. The transmit signal and replica can be sent at the same time in theory. However, it may be hardly implementable at present due to plenty of delay lines in transmitter. To replace the delay circuit, some excellent algorithms could be employed here, which are provided in [16]. As described in Figure 1, the transmitted signal in kth frame is denoted by (2)S1=b1Xk,Sl=blSl-1l=2,…,N-1,SN=bNSN-1+X~k+1.Figure 3
Block diagram of the proposed system transmitter.


### 2.3. Receiver Structure

The receiver structure of ORM-DCSK is depicted in Figure4, which has similar appearance to RM-DCSK detector. In our receiver, to demodulate b1, we make the switch up. The modification function W is used to adjust the relationship between signal in first bit period of kth frame and signal in its former time slot. When decoding bl(l=2,3,…,N) of kth frame, we make the switch down. Furthermore, the information extraction could work quite well if the receiver knows the starting time of each time slot. As a result, there is a timing synchronization circuit in our receiver, as done in RM-DCSK. To acquire perfect bit synchronization, many traditional timing techniques could be adopted, like data-aided timing synchronization algorithm designed for DCSK [17].Figure 4
Block diagram of the proposed system receiver.


### 2.4. Channel Model

The channel between each transmit antenna and the receive antenna is two-ray Rayleigh quasi-static block faded channel.ξ denotes a noise sample following a Gaussian distribution with zero mean and variance N0/2. Further, αm,1 and αm,2 denote the gains of the two paths between mth(m=1,2) transmit antenna and the receive antenna, which are independent, Rayleigh distributed random variables. It is assumed that the channel state remains constant during each bit period.According to different channel gains, we consider the following two cases:(3)Case1:Eα12=Eα22=12,Case2:Eα12=23,Eα22=13At the receiver, we denote the received signal vector during the bit duration byrl(l=1,…,N). Then, rl can be given by(4)rl=∑m=1MTαm,1sm,lτm,1+∑m=1MTαm,2sm,lτm,2+ξl.Here, sm,lτm,i(i=1,2) is delayed signal transmitted during lth bit duration and ξl is the additive white Gaussian noise vector.As described in Figure1, the chaotic wavelet sent in each time slot not only carries one bit of data but also serves as the reference signal of the data bit transmitted in its next time slot. Then, rl-1 can be denoted as the reference segment for rl. In this case, the outputs of the demodulators should be given by (5)c1=r1·r0·w,cl=rl·rl-1l=2,…,N.In (5), r0 is signal of last slot time of previews frame.Finally, based on the following rule, the estimated information bit is decided as “+1” or “-1”:(6)b^l=+1,ci>0,-1,ci<0.


## 2.1. Signal Format

Figure1 shows the frame format of ORM-DCSK system. Different with RM-DCSK, there are N adjacent time slots in each frame, in which average bit energy (labeled as Eb) decreases from 1.5βE(xi2) to (1+1/N)βE(xi2). The time slot is divided into β chips to carry the chaotic sequences and bl is the lth information bit in kth frame. Concretely, Xk is modulated by the information bit b1∈{+1,-1} in the first time slot of kth frame. In the second time slot, b2b1Xk is sent. In the Nth time slot, bN⋯b1Xk+X~k+1 is sent. Here, X~k+1 can be converted into the reference sequence of k+1th frame. In Figure 1, Xk=(1/MT)·[xk(1),xk(2),…,xk(β)] and MT is the number of transmit antennas.Figure 1
Signal frame format of ORM-DCSK.


## 2.2. Transmitter Structure

Figure2 shows the block diagram of orthogonal chaotic generator (OCG). It is necessary to make the chaotic carriers transmitted in neighboring two frames orthogonal to erase the intrasignal interference components included in decision variables clearly. We assume that the spreading factor is six (β=6). In the first-time slot period, as shown in Figure 2, the switch is connected to bottom. The chaotic sequence U=[a,b,c] is multiplied by w22=[+1,-1]. In other words, Xk=U·w22=[a,-a,b,-b,c,-c]. During second- to N-1th-slots period, the switch is suspended. In the Nth-time slot period, the switch is connected to bottom. The chaotic sequence V=[x,y,z] is multiplied by w21=[+1,+1]. X~k+1=V·w21=[x,x,y,y,z,z]. In the first-time slot period of k+1th frame, the switch is connected to bottom again. Xk+1=V·w22=[x,-x,y,-y,z,-z].Figure 2
Block diagram of OCG.The chaotic sequences generated by the OCG satisfy(1)Xk⊥X~k+1,X~k+1⊥Xk+1,Xk+1=X~k+1W,Xk+1⊥XkW.Here W=[+1,-1⋯+1,-1]︸β is a modification function.Figure3 shows the block diagram of ORM-DCSK transmitter. There are two transmitting antennas (MT=2) and one receiving antenna in MISO-ORM-DCSK system. The two transmitting antennas are independent. The signal can be sent by number 1 antenna and the replica of the signal can be sent by number 2 antenna. The transmit signal and replica can be sent at the same time in theory. However, it may be hardly implementable at present due to plenty of delay lines in transmitter. To replace the delay circuit, some excellent algorithms could be employed here, which are provided in [16]. As described in Figure 1, the transmitted signal in kth frame is denoted by (2)S1=b1Xk,Sl=blSl-1l=2,…,N-1,SN=bNSN-1+X~k+1.Figure 3
Block diagram of the proposed system transmitter.


## 2.3. Receiver Structure

The receiver structure of ORM-DCSK is depicted in Figure4, which has similar appearance to RM-DCSK detector. In our receiver, to demodulate b1, we make the switch up. The modification function W is used to adjust the relationship between signal in first bit period of kth frame and signal in its former time slot. When decoding bl(l=2,3,…,N) of kth frame, we make the switch down. Furthermore, the information extraction could work quite well if the receiver knows the starting time of each time slot. As a result, there is a timing synchronization circuit in our receiver, as done in RM-DCSK. To acquire perfect bit synchronization, many traditional timing techniques could be adopted, like data-aided timing synchronization algorithm designed for DCSK [17].Figure 4
Block diagram of the proposed system receiver.


## 2.4. Channel Model

The channel between each transmit antenna and the receive antenna is two-ray Rayleigh quasi-static block faded channel.ξ denotes a noise sample following a Gaussian distribution with zero mean and variance N0/2. Further, αm,1 and αm,2 denote the gains of the two paths between mth(m=1,2) transmit antenna and the receive antenna, which are independent, Rayleigh distributed random variables. It is assumed that the channel state remains constant during each bit period.According to different channel gains, we consider the following two cases:(3)Case1:Eα12=Eα22=12,Case2:Eα12=23,Eα22=13At the receiver, we denote the received signal vector during the bit duration byrl(l=1,…,N). Then, rl can be given by(4)rl=∑m=1MTαm,1sm,lτm,1+∑m=1MTαm,2sm,lτm,2+ξl.Here, sm,lτm,i(i=1,2) is delayed signal transmitted during lth bit duration and ξl is the additive white Gaussian noise vector.As described in Figure1, the chaotic wavelet sent in each time slot not only carries one bit of data but also serves as the reference signal of the data bit transmitted in its next time slot. Then, rl-1 can be denoted as the reference segment for rl. In this case, the outputs of the demodulators should be given by (5)c1=r1·r0·w,cl=rl·rl-1l=2,…,N.In (5), r0 is signal of last slot time of previews frame.Finally, based on the following rule, the estimated information bit is decided as “+1” or “-1”:(6)b^l=+1,ci>0,-1,ci<0.


## 3. Performance Analysis

In this section, BER expression is derived over Rayleigh fading channels. Based on Figure1 and (5), it can be easily found that the statistical properties of cl(l=1,2,…,N) are different. Then, we must calculate the BER performance of each bit, which differs from RM-DCSK.The logistic mapxi+1=1-2xi2,  -1<xi<1,  xi≠0,±0.5, is used with E(xi)=0 and Exi2=0.5,Var(xi2)=0.125, where E(·)and Var(·) denote the expectation operator and variance operator, respectively. We can get the average bit energy Eb=(1+1/N)βE(xi2).To further simplify the analysis, we can assume thatτm,1=0,τm,2=τ, and multipath time delay is much smaller than the spreading factor; that is, 0<τ≪β. Thus, (4) can be rewritten as(7)rl=∑m=1MTαm,1sm,l+∑m=1MTαm,2sm,lτ+ξlTherefore, without loss of generality, we consider the error rate of the bit b2. Substituting (7) into (5) for l=2, we can get (8)c2=r2·r1=1MTb2b1Xk∑m=1MTαm,1+b2b1Xkτ∑m=1MTαm,2+ξ2·1MTb1Xk∑m=1MTαm,1+b1Xkτ∑m=1MTαm,2+ξ1=A+B+ξ1·ξ2︸C,where(9)A=1MTb2b1Xk∑m=1MTαm,1+b2b1Xkτ∑m=1MTαm,2·b1Xk∑m=1MTαm,1+b1Xkτ∑m=1MTαm,2,B=1MTb2b1Xk∑m=1MTαm,1+b2b1Xkτ∑m=1MTαm,2ξ1+1MTb1Xk∑m=1MTαm,1+b1Xkτ∑m=1MTαm,2ξ2.Assumingb2=1, the instantaneous mean of the decision variable is E(c2)=E(A)+E(B)+E(C). Here,(10)EA=∑m=1MTαm,12+∑m=1MTαm,22NEbN+1MT,EB=EC=0.The variance of decision variable can be easily expressed asVar(c2)=Var(A)+Var(B)+Var(C). Here,(11)VarA=β∑m=1MTαm,14+∑m=1MTαm,248MT2+2∑m=1MTαm,12∑m=1MTαm,22N2Eb2βN+12MT2VarB=∑m=1MTαm,12+∑m=1MTαm,22NEbN0N+1MTVarC=βN024.Based on the above results, the mean and variance of c2 are(12)Ec2∣b2=+1=∑m=1MTαm,12+∑m=1MTαm,22NEbN+1MT.Varc2∣b2=+1=β∑m=1MTαm,14+∑m=1MTαm,248MT2+2∑m=1MTαm,12∑m=1MTαm,22N2Eb2βN+12MT2+∑m=1MTαm,12+∑m=1MTαm,22NEbN0N+1MT+βN04Similarly, ifb2=-1 is sent, the mean and variance of c2 are equal to(13)Ec2∣b2=-1=-Ec2∣b2=+1Varc2∣b2=-1=Varc2∣b2=+1.Based on the above results, the BER for decodingbl is computed as(14)BERc2γb=12erfcEc2∣b2=+12Varc2∣b2=+1=12erfcΛ1β+2Γγb+βΓ22γb2-1/2.Similarly, for l∈3,4,…,N-1 the BER is(15)BERclγb=BERc2γb.For l=1, we can get(16)c1=r1·r0·w=1MTb1Xk∑m=1MTαm,1+b1Xkτ∑m=1MTαm,2+ξ1·1MTb0Xk-1+X~kw∑m=1MTαm,1+b0Xk-1τ+X~kτwτ∑m=1MTαm,2+ξ0w=D+E+ξ1·ξ0w︸F,where(17)D=1MTb1Xk∑m=1MTαm,1+b1Xkτ∑m=1MTαm,2·b0Xk-1w+Xk∑m=1MTαm,1+b0Xk-1τwτ+Xkτ∑m=1MTαm,2,E=1MTb1Xk∑m=1MTαm,1+b1Xkτ∑m=1MTαm,2ξ0w+1MTb0Xk-1w+Xk∑m=1MTαm,1+b0Xk-1τwτ+Xkτ∑m=1MTαm,2ξ1.Similarly, we can get(18)ED=∑m=1MTαm,12+∑m=1MTαm,22NEbN+1MT,(19)EE=EF=0,(20)VarD=β∑m=1MTαm,14+∑m=1MTαm,248MT2+4∑m=1MTαm,12∑m=1MTαm,22N2Eb2βN+12MT2,(21)VarE=3∑m=1MTαm,12+∑m=1MTαm,22NEbN02N+1MT,(22)VarF=βN024.So, the mean and variance of c1 are(23)Ec1∣b1=+1=∑m=1MTαm,12+∑m=1MTαm,22NEbN+1MT,(24)Varc1∣b1=+1=β∑m=1MTαm,14+∑m=1MTαm,248MT2+4∑m=1MTαm,12∑m=1MTαm,22N2Eb2βN+12MT2+3NEbN0∑m=1MTαm,12+∑m=1MTαm,222N+1MT+βN04.Based on (23)-(24), the BER for decoding b1 is computed as(25)BERc1γb=12erfcEc1∣b1=+12Varc1∣b1=+1=12erfcΛ2β+3Γγb+βΓ22γb2-1/2.Here,(26)Λ1=1+2∑m=1MTαm,1∑m=1MTαm,22∑m=1MTαm,12+∑m=1MTαm,222,Λ2=1+6∑m=1MTαm,1∑m=1MTαm,22∑m=1MTαm,12+∑m=1MTαm,222,Γ=1+1Nγb=∑m=1MTαm,12EbMTN0︸γ1+∑m=1MTαm,22EbMTN0︸γ2.We denoteγ1¯=E(γ1) and γ2¯=E(γ2) to represent the average SNR per path. Moreover, for a multipath Rayleigh fading channel, the PDF of γb is given by (27)
f
γ
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.Based on (14)–(16), the BER is (28)BERγb=N-22NerfcΛ1β+2Γγb+βΓ22γb2-1/2+1NerfcΛ2β+3Γγb+βΓ22γb2-1/2.Finally, using (20) and (21), the BER of MISO-ORM-DCSK is derived as(29)BER=∫0∞BERγbfγbdγb.Moreover, we can get the theoretical lower bound of the BER whenN tends to infinity.(30)BERlowerBoundγb=limN→∞BERγb=12erfcΛ1β+2γb+β2γb2-1/2,BERlowerBound=∫0∞BERlowerBoundγbfγbdγb.On the other hand, (28) can be extended to the AWGN channel case by choosing αm,1=1 and αm,2=0. Hence, the BER would simplify to(31)BER=N-22Nerfc1β+ΓN0Eb+βΓ2N028Eb2-1/2+1Nerfc1β+3ΓN02Eb+βΓ2N028Eb2-1/2.


## 4. Simulation

In order to validate the BER performance of the MISO-ORM-DCSK and compare it with MISO-RM-DCSK, the BER is evaluated under AWGN and multipath fading channels. To facilitate the expression, the Monte Carlo simulations and theory results are labeled as “Sim” and “Theory” respectively.


### 4.1.AWGN Channel

Since the theoretical BERs for the 2nd toN-1th bits and the BER for Nth bit are different, Figure 5 shows BERs for N-1th bit and Nth bit in the AWGN channel of MISO-ORM-DCSK for β=256 and N=4. From Figure 5, it is shown that as Eb/N0 increases, the BERs become better. In addition, the error rate of c2 is superior to that of c1 in the proposed system. This can be explained as follows: the decision variable of c1 has an interference term, which does not exist in that of c2. It is visible that, with less intrasignal interference and noise interference components, MISO-ORM-DCSK is always better than MISO-RM-DCSK.Figure 5
Each BER in the AWGN channel for MISO-ORM-DCSK.Figure6 shows the relationship between BER performance and spreading factor β. It can also be noticed that ORM-DCSK is always better than RM-DCSK. In addition, it is obvious that the BER can maximize an optimal value by choosing the certain value of spreading factor; for example, the certain value is about 30 and 100 when Eb/N0=10 dB and Eb/N0=15 dB, respectively. BER starts to degrade if the spreading factor is beyond it, which is caused by fluctuations in Eb at small β and the noise-noise cross correlation at large β.Figure 6
BER performance versus spreading factorβ.Figure7 shows the relationships between analytical BER and Monte Carlo of MISO-ORM-DCSK in the AWGN channel. From Figure 7, it is easily found that clear matching between analytical expression and Monte Carlo simulations result can be obtained. This confirms that the Gaussian approximation works well when the spreading factor is relatively lager.Figure 7
Relationships between analytical BER and Monte Carlo results for MISO-ORM-DCSK and MISO-RM-DCSK withN=2.Figure8 plots the relationships between BER performance and N with Eb/N0=18 dB and β=256. The BER is smaller with larger N. We can get the theoretical lower bound of the BER when N tends to infinity. However, the complexity of system is increasing. We think N≤50 is the best choice and it can bring a trade-off between the BER and complexity.Figure 8
Relationships between BER andN for MISO-ORM-DCSK.


### 4.2. Rayleigh Fading Channel

Figure9 plots the relationships between BER performance and Eb/N0 for MISO-ORM-DCSK and MISO-RM-DCSK system with N=4. Through the examination of Figure 9, the proposed system achieves a performance gain of about 2 dB over the MISO-RM-DCSK at BER=1×10-4. In other words, the new system shows an enhanced robustness in Rayleigh fading channel.Figure 9
BER performance in Rayleigh fading channel with different gains.Figure10 shows the relationships between analytical BER and Monte Carlo results in Rayleigh fading channel when spreading factors β=128 and 512 and N=2. It is easily found that analytical results correspond with simulation results perfectly. This can be explained by identical reason in Figure 7 and the Gaussian approximation works well when the spreading factor is relatively lager. In addition, the BERs for all cases become worse as β increases, which can be attributed to the increasing negative contribution from interference components generated from the chaotic sequence and Gaussian noise.Figure 10
Relationships between analytical BER and Monte Carlo results in Rayleigh fading channel.


## 4.1.AWGN Channel

Since the theoretical BERs for the 2nd toN-1th bits and the BER for Nth bit are different, Figure 5 shows BERs for N-1th bit and Nth bit in the AWGN channel of MISO-ORM-DCSK for β=256 and N=4. From Figure 5, it is shown that as Eb/N0 increases, the BERs become better. In addition, the error rate of c2 is superior to that of c1 in the proposed system. This can be explained as follows: the decision variable of c1 has an interference term, which does not exist in that of c2. It is visible that, with less intrasignal interference and noise interference components, MISO-ORM-DCSK is always better than MISO-RM-DCSK.Figure 5
Each BER in the AWGN channel for MISO-ORM-DCSK.Figure6 shows the relationship between BER performance and spreading factor β. It can also be noticed that ORM-DCSK is always better than RM-DCSK. In addition, it is obvious that the BER can maximize an optimal value by choosing the certain value of spreading factor; for example, the certain value is about 30 and 100 when Eb/N0=10 dB and Eb/N0=15 dB, respectively. BER starts to degrade if the spreading factor is beyond it, which is caused by fluctuations in Eb at small β and the noise-noise cross correlation at large β.Figure 6
BER performance versus spreading factorβ.Figure7 shows the relationships between analytical BER and Monte Carlo of MISO-ORM-DCSK in the AWGN channel. From Figure 7, it is easily found that clear matching between analytical expression and Monte Carlo simulations result can be obtained. This confirms that the Gaussian approximation works well when the spreading factor is relatively lager.Figure 7
Relationships between analytical BER and Monte Carlo results for MISO-ORM-DCSK and MISO-RM-DCSK withN=2.Figure8 plots the relationships between BER performance and N with Eb/N0=18 dB and β=256. The BER is smaller with larger N. We can get the theoretical lower bound of the BER when N tends to infinity. However, the complexity of system is increasing. We think N≤50 is the best choice and it can bring a trade-off between the BER and complexity.Figure 8
Relationships between BER andN for MISO-ORM-DCSK.


## 4.2. Rayleigh Fading Channel

Figure9 plots the relationships between BER performance and Eb/N0 for MISO-ORM-DCSK and MISO-RM-DCSK system with N=4. Through the examination of Figure 9, the proposed system achieves a performance gain of about 2 dB over the MISO-RM-DCSK at BER=1×10-4. In other words, the new system shows an enhanced robustness in Rayleigh fading channel.Figure 9
BER performance in Rayleigh fading channel with different gains.Figure10 shows the relationships between analytical BER and Monte Carlo results in Rayleigh fading channel when spreading factors β=128 and 512 and N=2. It is easily found that analytical results correspond with simulation results perfectly. This can be explained by identical reason in Figure 7 and the Gaussian approximation works well when the spreading factor is relatively lager. In addition, the BERs for all cases become worse as β increases, which can be attributed to the increasing negative contribution from interference components generated from the chaotic sequence and Gaussian noise.Figure 10
Relationships between analytical BER and Monte Carlo results in Rayleigh fading channel.


## 5. Conclusion

In this paper, ORM-DCSK is proposed to improve the BER performance of RM-DCSK. The intrasignal interference components existing in RM-DCSK are eliminated clearly by designing an orthogonal chaotic generator. Without any cost in data rate, the proposed system not only shows excellent agreement between theoretical expressions and Monte Carlo simulations but also shows significance of BER improvement.However, the proposed system is slightly more complex than RM-DCSK due to additional delay and switch. This complexity cost is worthy and it provides a huge BER improvement. We believe that the proposed system has significant potential in chaotic communication environment.



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