Performance of a SIMO-CDSK System over Rayleigh Fading Channels

Abstract

This paper proposes a single-input multiple-output (SIMO) architecture for correlation delay shift keying (CDSK) modulation technique, and the bit error rate (BER) formula is derived under the assumption of the proposed system over Rayleigh fading channels. The new system employs multiple antennas at the receiver end to form a SIMO structure so as to obtain a diversity gain. Theoretical analysis and simulations show that, at a higher signal-to-noise ratio (SNR), the proposed SIMO-CDSK architecture has an outstanding bit error rate (BER) performance in contrast to the conventional single-input single-output (SISO) CDSK and GCDSK communication system; for the given SNR, the diversity gain of the proposed system will be improved with the number of receiver antennas increasing; for different SNRs, the best performance of the proposed system can be obtained by selecting the reasonable spreading factor; because the performance of SIMO-CDSK system is independent of the time delay, the proposed system has better security than GCDSK system.

1. Introduction

Chaotic maps/systems, with the favorable properties of being aperiodic, deterministic, noise-like, and wideband, have been widely employed in various modulation applications. During the last two decades or so, lots of digital communication schemes using chaos have been suggested and studied, which utilize either coherent or noncoherent detections for information extraction [1]. Because chaotic synchronization in coherent detection performs poorly, demodulation in chaos-based communication schemes with coherent receivers using this synchronization technique becomes a real challenge in noisy environments. Therefore, noncoherent schemes seem more feasible and appealing in practical environments. In 1996, Kolumbán et al. proposed the noncoherent differential chaos shift keying (DCSK) which is the most suitable scheme in wireless communications due to its good noise performance [2, 3]. However, the prices that it has to pay are low attainable data rate and weakened information security. In 2000, Sushchik et al. proposed the noncoherent correlation delay shift keying (CDSK) whose bandwidth utilization ratio and information security are much better than DCSK, but it is inferior in its bit error rate (BER) performances [4]. For this reason, the generalized CDSK (GCDSK) has been introduced so as to offer solution with improved bandwidth efficiency and BER but leading to a more complicated system design in hardware [5]. Furthermore, frequency-modulated CDSK (FM-CDSK) [6] and CDSK with Henon chaotic map [7] have been researched in recent years; these schemes further improve the security and practicality of CDSK.

The use of multiple antennas at all terminals or only the receiver end, which are called multiple input multiple output (MIMO) or single-input multiple-output (SIMO) techniques, demonstrates an effective enhancement in link reliability and bandwidth efficiency [8]. As a potential application, applying MIMO technique to chaos-based communication system has been considered. Due to the advantage of low BER, SIMO-DCSK system and MIMO-DCSK system have been carefully studied aiming to improve the performance of DCSK system over fading channels [9, 10]. In addition, a SIMO FM-DCSK UWB system has been proposed to further improve the performance in such environment [11].

However, little research has been conducted focusing on the performance improvement of CDSK modulation combined with multiantennas technique. To address this problem, an advanced CDSK system, single-input multiple-output CDSK (SIMO-CDSK) system, is proposed in present paper. Here, multiple antennas are allocated to the receiver end and provide several independent copies of signals from the transmitter. The BER formula is also derived under the assumption of the proposed system over Rayleigh fading channels. Theoretical analysis and simulations show that, at a higher signal-to-noise ratio (SNR), the proposed SIMO-CDSK architecture has an outstanding bit error rate performance in contrast to the conventional single-input single-output CDSK (SISO-CDSK) system and GCDSK system; for the given SNR, the diversity gain of the proposed system will be improved with the number of receiver antennas increasing; for different SNRs, the best performance of the proposed system can be obtained by selecting the reasonable spreading factor; because the performance of SIMO-CDSK system is independent of the time delay, the proposed system has better security than GCDSK system.

The remainder of this paper is organized as follows. In Section 2, the principle of CDSK modulation is briefly reviewed, while SIMO-CDSK system is described. In Section 3, performance analysis on SIMO-CDSK system over Rayleigh fading channels is carried out, with a close form BER expression derived. Simulation results are shown in Section 4 and conclusions are given in Section 5.

2. System Model

In this section, a single-input multiple-output CDSK system is introduced based on the principle of CDSK modulation. In the rest of this paper, it is assumed that all the independent channels satisfy the same Rayleigh fading channel condition and the same Gaussian-distributed random noise with the power spectral density of 𝑁₀. Also, equal-gain combiner (EGC) is adopted at the receivers in this system because it not only can improve the error performance but also can be easily implemented.

2.1. Modulation of CDSK

CDSK was proposed by Sushchik et al. to transmit information and to increase the efficiency of bandwidth [4].

As seen in Figure 1, in this scheme, the chaotic signal is employed as the carrier for transmission. When the chaotic signal is generated by a discrete time chaotic circuit, the transmitted signal is the sum of a chaotic sequence 𝑥_𝑘 and of the delayed chaotic sequence 𝑥_𝑘−𝐿 multiplied by the 𝑙th transmitted information signal 𝑑_𝑙=±1. During the 𝑙th bit period, as seen from Figure 1(a), the output of the transmitter 𝑠_𝑘 is 𝑠_𝑘=𝑥_𝑘+𝑑_𝑙𝑥_𝑘−𝐿, (1) where 𝐿 is the time delay. Assume that the transmitted signal will suffer from Rayleigh fading and the influence of Gaussian-distributed random noise. According to Figure 1(b), the received signal corresponding to the 𝑙th transmitted information signal 𝑑_𝑙 is given by 𝑟_𝑘=ℎ_𝑘𝑠_𝑘+𝜉_𝑘, (2) where ℎ_𝑘 is the independent and Rayleigh distribution random variable and 𝜉_𝑘 is a sequence of independent and identically distributed zero-mean Gaussian noise variables. During the period of the 𝑙th transmitted information signal, the correlator output of CDSK 𝑦_𝑙 is given by the sum 𝑦_𝑙=^2𝑙𝛽_{()𝑘=2𝑙−1𝛽+1}𝑟_𝑘𝑟_𝑘−𝐿, (3) where 2𝛽 is the spreading factor. The estimated information is decided as “+1” or “−1” depending on the following rules: 𝑑=+1,𝑦_𝑙≥0,−1,𝑦_𝑙<0. (4)

(a)

(b)

Figure 1 

CDSK modulation: (a) transmitter and (b) receiver.

2.2. SIMO-CDSK System Model

For digital chaotic communication, CDSK has advantages in bandwidth utilization ratio and information security. However, the inferiority in bit error rate limits its application and popularization. The use of multiple antennas at the receiver end, which is called single-input multiple-output (SIMO) technique, demonstrates a significant improvement in link reliability and bandwidth efficiency. Motivated by the virtue of CDSK and SIMO technology, SIMO-CDSK system is proposed to enhance the performance of the conventional SISO-CDSK system. Figure 2 shows the block diagram of the SIMO-CDSK system, where (a) is the transmitter and (b) is the receiver.

(a)

(b)

Figure 2 

SIMO-CDSK system: (a) transmitter and (b) receiver.

In this proposed system, we consider that multiple antennas are allocated to the receiver end. In order to generate the transmitted signal, the modulator is shown in Figure 2(a), which appears to be the identical version of CDSK modulation. The channels constructed by the transmitter-receiver antenna pairs are independent, meaning that the channel state remains constant during each transmission period.

In SIMO-CDSK system, we assume that all channels satisfy the same Rayleigh fading channel condition and the additive white Gaussian noise which is independent from channel. Consequently, according to Figure 2(b), the received signals at the 𝑖th receiver antenna which corresponding to the 𝑙th transmitted information signal 𝑑_𝑙 can be expressed as follow: 𝑟_𝑖,𝑘=ℎ_𝑖,𝑘𝑠_𝑘+𝜉_𝑖,𝑘, (5) where ℎ_𝑖,𝑘 and 𝜉_𝑖,𝑘 are parameters of the channel which are between the transmitter antenna and the 𝑖th receiver antenna, ℎ_𝑖,𝑘 is the independent and Rayleigh distributed random variable, and 𝜉_𝑖,𝑘 is a sequence of independent and identically distributed Gaussian noise. With expressions shown in (5), under the Rayleigh fading channel model, the 𝑖th correlator output of SIMO-CDSK system 𝑦_𝑖,𝑙 is 𝑦_𝑖,𝑙=^2𝑙𝛽_{()𝑘=2𝑙−1𝛽+1}ℎ_𝑖,𝑘𝑠_𝑘+𝜉_𝑖,𝑘ℎ_{𝑖,𝑘−𝐿}𝑠_𝑘−𝐿+𝜉_{𝑖,𝑘−𝐿}, (6) where 2𝛽 is the spreading factor. To obtain the performance gain, as seen from Figure 2(b), the equal-gain combiner (EGC) is adopted at the receiver in this proposed system. Therefore, by equally combining the correlator outputs, the sum of correlator outputs 𝑦_𝑙 is used to make the decision for 𝑑_𝑙. Under the Rayleigh fading channel model, the expression of 𝑦_𝑙 can be derived as 𝑦_𝑙=^𝑛_𝑖=1𝑦_𝑖,𝑙, (7) where 𝑛 is the number of receiver antennas. As shown in (4), the estimated information is decided as “+1” or “−1” depending on 𝑦_𝑙.

3. Performance Analysis

In this section, the BER expression of SIMO-CDSK system is derived over Rayleigh fading channels. To simplify the calculation of the bit error rate, in the following analysis, we assume that the time delay 𝐿 is greater than the spreading factor 2𝛽.

With the expression of 𝑦_𝑙, correlator output in (7) can be rewritten as 𝑦_𝑙=^𝑛_𝑖=1^2𝑙𝛽_{()𝑘=2𝑙−1𝛽+1}ℎ_𝑖,𝑘ℎ_{𝑖,𝑘−𝐿}𝑠_𝑘𝑠_𝑘−𝐿+^𝑛_𝑖=1^2𝑙𝛽_{()𝑘=2𝑙−1𝛽+1}ℎ_𝑖,𝑘𝑠_𝑖,𝑘𝜉_{𝑖,𝑘−𝐿}+ℎ_{𝑖,𝑘−𝐿}𝑠_{𝑖,𝑘−𝐿}𝜉_𝑖,𝑘+^𝑛_𝑖=1^2𝑙𝛽_{()𝑘=2𝑙−1𝛽+1}𝜉_𝑖,𝑘𝜉_{𝑖,𝑘−𝐿}=𝐴+𝐵+𝐶, (8) where 𝐴=^𝑛_𝑖=1^2𝑙𝛽_{()𝑘=2𝑙−1𝛽+1}ℎ_𝑖,𝑘ℎ_{𝑖,𝑘−𝐿}𝑠_𝑘𝑠_𝑘−𝐿,𝐵=^𝑛_𝑖=1^2𝑙𝛽_{()𝑘=2𝑙−1𝛽+1}ℎ_𝑖,𝑘𝑠_𝑖,𝑘𝜉_{𝑖,𝑘−𝐿}+ℎ_{𝑖,𝑘−𝐿}𝑠_{𝑖,𝑘−𝐿}𝜉_𝑖,𝑘,𝐶=^𝑛_𝑖=1^2𝑙𝛽_{()𝑘=2𝑙−1𝛽+1}𝜉_𝑖,𝑘𝜉_{𝑖,𝑘−𝐿}. (9)(10) As depicted in (8), 𝐴 means the signal component which is used to make the decision for 𝑑_𝑙, and 𝐵 means the interference component which is produced by the correlation between the transmitted signal and white Gaussian noise. 𝐶 means the interference component which is merely produced by the white Gaussian noise.

In this work, we assume that parallel channels are slow fading, the period of ℎ_𝑖,𝑘 stays constant which is much greater than the time delay. Thus, we can then represent the fading phenomenon using a random variable whose value is fixed; that is, ℎ_𝑖,𝑘=ℎ_{𝑖,𝑘−𝐿}=ℎ_𝑖, (11) where 𝑖 means the 𝑖th channel. For large 2𝛽 and a given chaotic map (e.g., chebyshev map [12]), we have ^2𝑙𝛽_{()𝑘=2𝑙−1𝛽+1}𝑥_𝑘𝑥_𝑘−𝐿=0. (12)

Assuming “+1” is transmitted (i.e., 𝑑_𝑙=+1), the following statistics are easily obtained: 𝐸𝑦_𝑙∣𝐻,𝑑_𝑙=+1=^𝑛_𝑖=12𝛽ℎ²_𝑖𝐸𝑥²_𝑘,Var𝑦_𝑙∣𝐻,𝑑_𝑙=+1=2𝛽^𝑛_𝑖=1ℎ⁴_𝑖3𝐸𝑥²_𝑘²+Var𝑥²_𝑘+2^𝑛_𝑖=1ℎ²_𝑖𝐸𝑥²_𝑘𝑁₀𝑁+𝑛⁰2², (13) where 𝐻 means the set of channel fading parameters and 𝐸[⋅] and Var[⋅] represent the expectation and variance operators, respectively. Since Var[𝑦_𝑙∣𝐻,𝑑_𝑙=+1] is mainly affected by the white Gaussian noise when 2𝛽 is relatively small, the value which is calculated by (3(𝐸[𝑥²_𝑘])²+Var[𝑥_𝑘]) is relatively small; (12) may be approximated as Var𝑦_𝑙∣𝐻,𝑑_𝑙=+1≈2𝛽^𝑛_𝑖=1ℎ²_𝑖²3𝐸𝑥²_𝑘²+Var𝑥²_𝑘+2^𝑛_𝑖=1ℎ²_𝑖𝐸𝑥²_𝑘𝑁₀𝑁+𝑛⁰2². (14)

The case of sending a symbol of “−1” may be computed in a likewise fashion; that is, 𝐸𝑦_𝑙∣𝐻,𝑑_𝑙=+1=−𝐸𝑦_𝑙∣𝐻,𝑑_𝑙,𝑦=−1Var_𝑙∣𝐻,𝑑_𝑙=+1=Var𝑦_𝑙∣𝐻,𝑑_𝑙=−1. (15)

Using (15) and assuming that follows a normal distribution under the given conditions, based on Gaussian approximation method [13], the conditional BER may be computed as BER𝑦_𝑙=1∣𝐻4𝐸𝑦erfc^𝑙∣𝐻,𝑑_𝑙=+1Var𝑦_𝑙∣𝐻,𝑑_𝑙+1=+14𝑦erfc−𝐸_𝑙∣𝐻,𝑑_𝑙=−1Var𝑦_𝑙∣𝐻,𝑑_𝑙,=−1 (16) where erfc(⋅) is the complementary error function.

The chebyshev map is used to generate chaotic sequence 𝑥_𝑘 and we have 𝐸[𝑥²_𝑘]=0.5 and Var[𝑥²_𝑘]=0.125. If 𝐸_𝑏 is the bit energy and is represented by 𝐸_𝑏=4𝛽𝐸𝑥²_𝑘, (17) for large 2𝛽, the conditional BER may be simplified as BER𝑦_𝑙∣𝛾_𝑏≈12erfc𝛾_𝑏7𝛾2𝛽^𝑏+8+4𝑛𝛽𝛾_𝑏⁻¹, (18) where 𝛾_𝑏=𝐸_𝑏𝑁₀^𝑛_𝑖=1ℎ²_𝑖. (19) Because ℎ_𝑖 is the Rayleigh distributed random variable, the probability density function (pdf) of 𝛾_𝑏 can be obtained as 𝑓𝛾_𝑏=𝑓𝛾₁⊗𝑓𝛾₂⊗⋯⊗𝑓𝛾_𝑛, (20) where 𝛾_𝑖=𝐸_𝑏𝑁₀ℎ²_𝑖,𝑓𝛾_𝑖=𝛾_𝑖⁻¹exp−𝛾_𝑖𝛾_𝑖,𝛾_𝑖=𝐸_𝑏𝑁₀𝐸ℎ²_𝑖, (21) where 𝐸[ℎ²_𝑖] is the average power gain for the 𝑖th channel. We assume that all independent channels satisfy the same Rayleigh fading channel condition; the pdf of 𝛾_𝑏 can be calculated as 𝑓𝛾_𝑏=𝛾_𝑏^𝑛−1𝛾_𝑖^𝑛(𝑛−1)!exp−𝛾_𝑏𝛾_𝑖, (22) where 𝑛 is the number of receiver antennas. Finally, the BER of SIMO-CDSK system can be obtained by averaging the conditional BER; that is, BER=₀^+∞𝑓𝛾_𝑏BER𝑦_𝑙∣𝛾_𝑏𝑑𝛾_𝑏. (23) This formula will be used in the next section for evaluating the bit error rate performance of the proposed system.

4. Simulation Results

In this section, some numerical calculation, approximation, and simulation results of BER are presented at various conditions. To compare them with those analytical results in Section 3, Chebyshev map with degree of 𝜇=3 is adopted here. The parameters used in SIMO-CDSK system are the same as those in the conventional SISO-CDSK system and GCDSK system where the distinct chaotic sequences are generated by 𝑥_𝑘+1=4𝑥³_𝑘−3𝑥_𝑘 with initial value of 𝑥₀=0.1 and the average power gain of each Rayleigh fading channel 𝐸[ℎ²_𝑖]=1. In order to facilitate the illustration, in the following figures, the conventional SISO-CDSK system is expressed by CDSK(1, 1), the SIMO system is expressed by CDSK(1, 𝑛), 𝑛 means the number of receiver antennas, and the GCDSK system is expressed by GCDSK.

Figure 3 plots relationships between BER performance and SNR for SISO-CDSK system and SIMO-CDSK system with the spreading factor 2𝛽=100 and the time delay 𝐿=200, respectively. Theory and simulation results in this figure show that the analytical estimates which perform in Section 3 agree well with the simulation results; with the increase in the number of receiver antennas, at the fixed 𝐸_𝑏/𝑁₀ value, the proposed system could obtain more diversity gain; this is because, in (18), erfc(⋅) is the decreasing function and the exception of correlator output grows faster than the square of double variance if the number of receiver antennas increases; when the 𝐸_𝑏/𝑁₀ value increases, further performance gain can also be expected by the proposed system.

Figure 3 

Relationships between BER performance and SNR for SISO-CDSK system and SIMO-CDSK system with the spreading factor 2𝛽=100 and the time delay 𝐿=200, respectively.

Figure 4 plots the relationships between BER performance and the spreading factor 2𝛽 for SISO-CDSK system and SIMO-CDSK system with 𝐿=2𝛽, and 𝐸_𝑏/𝑁₀=10 dB and 15 dB, respectively. Simulation results in this figure show that, for different 𝐸_𝑏/𝑁₀ values, the best performance of all systems can be obtained by selecting the critical point of spreading factors; for example, for 𝐸_𝑏/𝑁₀=10 dB, the critical point is nearly 20; for 𝐸_𝑏/𝑁₀=15 dB, the critical point is nearly 40; if the spreading factor is greater than the critical point, at the fixed 𝐸_𝑏/𝑁₀ value, BER performance for all systems becomes worse, which can be attributed to the increasing negative contribution from interference components in the correlator output which are shown in (10).

Figure 4 

Relationships between BER performance and the spreading factor 2𝛽 for SISO-CDSK system and SIMO-CDSK system with 𝐿=2𝛽 and 𝐸𝑏/𝑁0=10 dB and 15 dB, respectively.

Figure 5 plots the relationships between BER performance and the time delay 𝐿 for SISO-CDSK system and SIMO-CDSK system with the spreading factor 2𝛽=100 and 𝐸_𝑏/𝑁₀=10 dB and 15 dB, respectively. Simulation results in this figure show that, at the fixed 𝐸_𝑏/𝑁₀ value and 2𝛽, the diversity gain obtained by SIMO-CDSK system will not be affected by 𝐿. This is because the bit error rate (BER) formula, shown in (23), is independent of 𝐿. Therefore, we could choose a random number as the time delay for increasing the security of transmission.

Figure 5 

Relationships between BER performance and the time delay 𝐿 for SISO-CDSK system and SIMO-CDSK system with the spreading factor 2𝛽=100 and 𝐸𝑏/𝑁0=10 dB and 15 dB, respectively.

Figure 6 plots the relationships between BER performance and SNR for SISO-CDSK system, SIMO-CDSK system, and GCDSK system. We assume that there are 3 delay blocks in the transmitter of GCDSK. In Figure 6(a), the spreading factor 2𝛽 considered here are 100 and 200, respectively, and the time delay 𝐿 is set to 10. Simulation results in this figure show that, at the fixed 𝐸_𝑏/𝑁₀ value, for different spreading factors, the BER performance of the proposed system is always superior to that of SISO-CDSK system and GCDSK system. In Figure 6(b), the spreading factor 2𝛽 considered here is 100 and the time delay 𝐿 is set to 10 and 100, respectively. In this figure, we can observe that, at the fixed 𝐸_𝑏/𝑁₀ value, for different time delays, the proposed system has an outstanding bit error rate performance in contrast to the conventional SISO-CDSK and GCDSK communication system. Moreover, from the simulated curve, it is also very interesting to observe that, since the intrasignal interference increases with the time delay, the BER performance of GCDSK becomes worse when 𝐿 increases, while the performance of SISO-CDSK system and the proposed system is not affected by 𝐿. Therefore, the proposed system has better security than that of GCDSK system.

(a)

(b)

Figure 6 

Relationships between BER performance and SNR for SISO-CDSK system, SIMO-CDSK system, and GCDSK system.

5. Conclusion

In this paper, a single-input multiple-output (SIMO) architecture of correlation delay shift keying (CDSK) modulation technique is proposed as an enhanced version of CDSK, and its BER formula is also derived under the assumption of the proposed scheme over Rayleigh fading channels.

In the proposed scheme, multiple receiver antennas provide several independent copies of signals from the transmitter, and the equal-gain combiner (EGC) is adopted at the receiver to estimate the information symbol.

Theoretical analysis and simulations show that, at a higher signal-to-noise ratio (SNR), the proposed SIMO-CDSK architecture has an outstanding bit error rate performance in contrast to the conventional SISO-CDSK and GCDSK communication system; for the given SNR, the diversity gain of the proposed system will be improved with the number of receiver antennas increasing; for different SNRs, the best performance of the proposed system can be obtained by selecting the reasonable spreading factor; because the performance of SIMO-CDSK system is independent of the time delay, the proposed system has better security than that of GCDSK system.

Acknowledgments

This work was supported in part by the Natural Science Foundation of China under Grant 60874091, by the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant 20103223110003, and by the Six Projects Sponsoring Talent Summits of Jiangsu Province under Grant SJ209006.