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

The major drawback of the differential chaos shift keying (DCSK) system is that equal time and energy are spent on the reference and data signal. This paper presents the design and performance analysis of a short reference multifold rate DCSK (SRMR-DCSK) system to overcome the major drawback. The SRMR-DCSK system is proposed to enhance the data rate of the short reference differential chaos shift keying (SR-DCSK) system. By recycling each reference signal in SR-DCSK, the data slot carries N bits of data and by P times. As a result, compared with SR-DCSK, the proposed system has a higher data transmission rate and evaluates the energy efficiency with respect to the conventional DCSK system. To further improve the bit-error-rate (BER) performance over Rayleigh fading channels, the multiple-input single-output SRMR-DCSK (MISO-SRMR-DCSK) is also studied. The BER expression of the proposed system is derived based on Gaussian approximation (GA), and simulations in Rayleigh fading channels are performed. Simulation results show a perfect match with the analytical expression.


Introduction
Chaos based digital communication systems have been proposed and studied in recent years [1][2][3][4][5][6][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][9][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][13][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 Section 2, 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.

MISO-ORM-DCSK
In this section, the baseband implementation of MISO-ORM-DCSK is introduced and some details are explained.2.1.Signal Format.Figure 1 shows the frame format of ORM-DCSK system.Different with RM-DCSK, there are  adjacent time slots in each frame, in which average bit energy (labeled as   ) decreases from 1.5( 2  ) to (1 + 1/)( 2  ).The time slot is divided into  chips to carry the chaotic sequences and   is the th information bit in th frame.Concretely, X  is modulated by the information bit  1 ∈ {+1, −1} in the first time slot of th frame.In the second time slot,  2  1 X  is sent.In the th time slot,   ⋅ ⋅ ⋅  1 X  + X+1 is sent.Here, X+1 can be converted into the reference sequence of ( + 1)th frame.In Figure 1, X  = (1/√  ) ⋅ [  (1),   (2), . . .,   ()] and   is the number of transmit antennas.

Transmitter Structure.
Figure 2 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 ( = 6).In the first-time slot period, as shown in Figure 2, the switch is connected to bottom.The chaotic sequence U = [, , ] is multiplied by During second-to ( − 1)th-slots period, the switch is suspended.In the th-time slot period, the switch is connected to bottom.The chaotic sequence V = [, , ] is multiplied by In the first-time slot period of (+1)th frame, the switch is connected to bottom again.
The chaotic sequences generated by the OCG satisfy Here Figure 3 shows the block diagram of ORM-DCSK transmitter.There are two transmitting antennas (  = 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 th frame is denoted by (2) 2.3.Receiver Structure.The receiver structure of ORM-DCSK is depicted in Figure 4, which has similar appearance to RM-DCSK detector.In our receiver, to demodulate  1 , we make the switch up.The modification function W is used to adjust the relationship between signal in first bit period of th frame and signal in its former time slot.When decoding   ( = 2, 3, . . ., ) of th 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].

Channel Model.
The channel between each transmit antenna and the receive antenna is two-ray Rayleigh quasistatic block faded channel. denotes a noise sample following a Gaussian distribution with zero mean and variance  0 /2.Further,  ,1 and  ,2 denote the gains of the two paths between th ( = 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: At the receiver, we denote the received signal vector during the bit duration by r  ( = 1, . . ., ).Then, r  can be given by Here, s , ( = 1, 2) is delayed signal transmitted during th bit duration and   is the additive white Gaussian noise vector.
As described in Figure 1, 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, r −1 can be denoted as the reference segment for r  .In this case, the outputs of the demodulators should be given by In ( 5), r 0 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":
To further simplify the analysis, we can assume that  ,1 = 0,  ,2 = , and multipath time delay is much smaller than the spreading factor; that is, 0 <  ≪ .Thus, (4) can be rewritten as Therefore, without loss of generality, we consider the error rate of the bit  2 .Substituting ( 7) into ( 5) for  = 2, we can get where Assuming  2 = 1, the instantaneous mean of the decision variable is ( 2 ) = (A) + (B) + (C).Here, The variance of decision variable can be easily expressed as Var ( 2 ) = Var (A) + Var (B) + Var (C).Here, International Journal of Antennas and Propagation Based on the above results, the mean and variance of  2 are Similarly, if  2 = −1 is sent, the mean and variance of  2 are equal to Based on the above results, the BER for decoding   is computed as Similarly, for  ∈ {3, 4, . . .,  − 1} the BER is For  = 1, we can get where Similarly, we can get Var So, the mean and variance of  1 are Var Based on ( 23)-( 24), the BER for decoding  1 is computed as Here, We denote  1 = ( 1 ) and  2 = ( 2 ) to represent the average SNR per path.Moreover, for a multipath Rayleigh fading channel, the PDF of   is given by Based on ( 14)-( 16), the BER is Finally, using (20) and ( 21), the BER of MISO-ORM-DCSK is derived as Moreover, we can get the theoretical lower bound of the BER when  tends to infinity.
On the other hand, (28) can be extended to the AWGN channel case by choosing  ,1 = 1 and  ,2 = 0. Hence, the BER would simplify to

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.

AWGN Channel.
Since the theoretical BERs for the 2nd to (−1)th bits and the BER for th bit are different, Figure 5 shows BERs for ( − 1)th bit and th bit in the AWGN channel of MISO-ORM-DCSK for  = 256 and  = 4. From Figure 5, it is shown that as   / 0 increases, the BERs become better.In addition, the error rate of  2 is superior to that of  1 in the proposed system.This can be explained as follows: the decision variable of  1 has an interference term, which does not exist in that of  2 .It is visible that, with less intrasignal interference and noise interference components, MISO-ORM-DCSK is always better than MISO-RM-DCSK.Figure 6 shows the relationship between BER performance and spreading factor .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   / 0 = 10 dB and   / 0 = 15 dB, respectively.BER starts to degrade if the spreading factor is beyond it, which is caused by fluctuations in   at small  and the noise-noise cross correlation at large .
Figure 7 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 8 plots the relationships between BER performance and  with   / 0 = 18 dB and  = 256.The BER is smaller with larger .We can get the theoretical lower bound of the BER when  tends to infinity.However, the complexity of system is increasing.We think  ≤ 50 is the best choice and it can bring a trade-off between the BER and complexity.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  increases, which can be attributed to the increasing negative contribution from interference components generated from the chaotic sequence and Gaussian noise.

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.

Figure 3 :Figure 4 :
Figure 3: Block diagram of the proposed system transmitter.

Figure 9
plots the relationships between BER performance and   / 0 for MISO-ORM-DCSK and MISO-RM-DCSK system with  = 4. Through the examination of Figure9, 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 10 :
Figure 10: Relationships between analytical BER and Monte Carlo results in Rayleigh fading channel.