I. INTRODUCTION II. SYSTEM MODEL. This section reviews the related system models, based on which the proposed system is developed.

Transcription

1 196 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 58, NO. 1, JANUARY 2011 Performance of DCSK Cooperative Communication Systems Over Multipath Fading Channels Weikai Xu, Student Member, IEEE, Lin Wang, Senior Member, IEEE, and Guanrong Chen, Fellow, IEEE Abstract A differential chaos shift keying cooperative communication (DCSK-CC) system with two users is proposed in this paper, which has an orthogonal subchannel in broadcast phase and cooperative phase through orthogonal Walsh code sequences as its multiaccess scheme. The single relay cooperative network with decode-and-forward relay is investigated in the proposed system according to two cooperation protocols, namely, conventional cooperation and space-time cooperation. Unlike conventional CDMA cooperative communication (CDMA-CC) systems, quite surprisingly power control devices that consume more energy to mitigate near-far effects can be avoided in the proposed system, which is of great importance to energy-constrained networks such as wireless sensor networks. Simulation results demonstrate that, through a conventional cooperation mechanism, the proposed system has a prominent advantage of good bit-error-probability (BEP) performance over the CDMA-CC systems that have a single path correlation receiver, at the same data rate with a high SNR range over multipath Rayleigh fading channels. Meanwhile, it is found that conventional cooperation is a better cooperation strategy relative to space-time cooperation in the proposed system. In addition, a lower bound of BEP performance is derived and verified by simulations over independent three-ray Rayleigh fading channels. Index Terms Bit error probability (BEP), chaotic communication, differential chaos shift keying (DCSK) cooperative communication system, multipath Rayleigh fading channel, near-far effect. I. INTRODUCTION B Y using a chaotic carrier to spread a digital signal over a wide bandwidth spectrum, the resulting system inherits the benefits of spread-spectrum communications such as mitigation of multipath fading. Based on this observation, a number of chaos-based communication schemes have been proposed and analyzed in recent years [1] [16]. Among all the digital communication schemes proposed thus far, DCSK and frequency-modulated (FM )-DCSK show superior capability in terms of anti-interference over multipath fading channels. Compared with the direct-sequence spread-spectrum CDMA technique, the DCSK Manuscript received January 18, 2010; revised April 23, 2010; accepted June 16, Date of publication November 11, 2010; date of current version December 30, This work was supported in part by the City University of Hong Kong under SRG Grant and in part by the National Natural Science Foundation of China under Grant and Grant This paper was recommended by Associate Editor M. Laddomada. W. Xu and L. Wang are with the Department of Communication Engineering, Xiamen University, Fujian , China ( xweikai@xmu.edu.cn; wanglin@xmu.edu.cn). G. Chen is with the Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China ( eegchen@cityu.edu.hk). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TCSI with autocorrelation receiver (AcR) does not need code acquisition and synchronization, nor channel estimation, but only requires frame or symbol rate sampling, which is also desirable in applications. Based on these advantages, some ultrawideband systems based on DCSK or FM-DCSK modulations have been proposed recently, e.g., for wireless personal area networks (WPAN) [17] [21]. In a wireless network, system performance degrading is mainly attributed to signal fading and intersymbol interference (ISI) arising under multipath propagation environments. In general, signal fading can be mitigated by using a diversity technique by which redundant signals over independent channel realizations are transmitted so as to obtain anti-fading performance with suitable receivers. Multiple-antenna (spatial) diversity techniques are particularly attractive as they can be easily combined with other forms of diversity, meanwhile offering significant performance gains even if other forms of diversity are unavailable. Over the past two decades, multiple-antenna diversity has been unprecedentedly developed [22] [26]. In [27], a single-input multiple-output (SIMO) system based on FM-DCSK was proposed. Notice, however, that the multiple-antenna transmission diversity suggested in [22], [23] is hardly deployed in DCSK due to the variation of its carrier in different bit periods. Moreover, in wireless sensor networks (WSN), multiple-antenna diversity is impractical due to the cost and size of the sensors. In order to overcome this limit, a new form of spatial diversity user cooperative diversity was proposed by emulating the transmitting antenna diversity in [28] and [29]. In this paper, the two-user cooperative diversity technique is introduced into the DCSK system. The Walsh code sequences with excellent cross-correlation characteristics are adopted as user multiple access [30]. Compared with the cooperative diversity system based on CDMA [28], [29], the proposed system obtains both multipath diversity gain and cooperative diversity gain using a simpler AcR. The remainder of this paper is organized as follows. The principle of DCSK and the two-user cooperative system based on DCSK are presented in Section II. The bit error probability (BEP) lower bound of the proposed system is analyzed in Section III. Then, simulation results and discussions are presented in Section IV. Finally, conclusions are drawn in Section V. II. SYSTEM MODEL This section reviews the related system models, based on which the proposed system is developed /$ IEEE

2 XU et al.: PERFORMANCE OF DCSK COOPERATIVE COMMUNICATION SYSTEMS OVER MULTIPATH FADING CHANNELS 197 Fig. 2. DCSK demodulator using GML detection. For example, for,, second- and fourth-order Walsh code sequences are, respectively, Fig. 1. Block diagram of DCSK transceiver. (a) Transmitter. (b) Receiver. A. Principle of DCSK DCSK uses a chaotic signal as the carrier, with a differential shift keying modulator, for transmission. The chaotic signal is generated by a chaotic mapping method [31], and the simple Logistic chaotic map is chosen here for implementation. Fig. 1 shows the block diagram of the binary DCSK communication system. The binary DCSK modulation unit transmits a reference segment of the chaotic signal in the first half of the symbol duration, and repeats or reverses the segment in the last half of the symbol duration, according to the digital information 1 or 0, respectively. The modulated signal is represented by two orthogonal basis functions, and, as follows During the demodulation process, the generalized maximum likelihood (GML) detection rule may be applied [32]. As a universal method, second-order Walsh code is used for binary DCSK and higher order Walsh code is employed for multiuser or nonbinary DCSK. The binary DCSK demodulator using GML detection is illustrated in Fig. 2. As shown in Fig. 2, the weighted energy received is (5) (1) Here, is modulated signal for transmission and is the bit energy. The two orthogonal basis functions are where is the bit duration and is the chaotic signal. In DCSK, each basis function consists of a reference and an information-bearing segment. The receiver can be implemented using a suboptimum AcR. In AcR, the observation signal is given by where is the received signal into the AcR. B. Multiple-User System Based on Walsh Codes For a multiple user or nonbinary DCSK communication system, the orthogonal Walsh code sequences were adopted for implementation [30]. Let be a -order Walsh code sequence,,. The -order Walsh code sequences are recursively constructed as follows:, (2) (3) (4) where is the bit duration, is the received signal, and are the corresponding elements in the secondorder Walsh code, which are being multiplied as the weights during the modulation. The demodulator needs to find the index that maximizes, and then make a decision on the bit, either 1 or 0. The above-described Walsh code-based DCSK modulation scheme is now extended to a multiple-access system. Consider a system with users. Let denote the length of each carrier segment, and denote the global spread-spectrum factor. One may use -order Walsh code to accommodate the users. During the modulation, one forces so as to make sure that the global spread-spectrum factor f is kept constant. The transmitted signal of the user can be expressed by the -order orthogonal Walsh code, as follows: where, is a row vector of the -order Walsh code. The demodulation process of multiple users based on the Walsh code can be readily extended further from binary DCSK (6) (7)

3 198 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 58, NO. 1, JANUARY 2011 Fig. 3. Block diagram of the u -user receiver. Fig. 4. The cooperative transmission scheme (a) Odd period, transmit self data (b) Even period, relay the data of partner s. with GML detection rule. The block diagram of the demodulator of the user is shown in Fig. 3, and the weighted energy combined of the -user bit is expressed as where ( or 1) denotes the decision statistic of the user. C. Two-User Cooperation Model For exposition, consider a cellular system in which two mobiles are communicating with a base station. The channels between each user and the base station (the uplink channel) are independent, so are the channels between the two users (the interuser channel). Due to the advantages of DCSK over frequency-selective channels, all channels are assumed to subject to static block frequency-selective fading; that is, the channel state remains constant during each cooperative period. It is assumed that a cooperative period is divided into broadcast phase and cooperative phase, denoted as odd period and even period, respectively. The transceiver model used is illustrated by Fig. 4. The signals at the receiver can be expressed as (9) and (10) at the bottom of the page. Here, is the convolution operator,, and are the baseband model of signals at the base station, user 1 and user 2, respectively, during a frame period. Moreover, is the signal transmitted by user,, and is a white Gaussian noise random process with zero mean and two-sided power spectral density,. The channel multipath impulse response are modeled as a linear time-invariant process,, where (8) Fig. 5. Cooperation protocol of the two-user DCSK-CC system (a) Conventional cooperation (b) Space-time cooperation. and denote the attenuation and delay of the -path, respectively, and is the number of the multipath components. It is typically assumed that the channel multipath impulse responses are same for both odd periods and even periods. User 1 reconstructs the partner s information into through in odd periods, user 2 uses in a similar fashion to reconstruct the partner s information into. Then, the two users cooperate to both send their messages to the base station in even periods. In general, user-cooperative algorithms at relay, such as amplify-and-forward (AF) or decode-and-forward (DF), were applied [33]. In the case of AF, the user simply amplifies its partner s signal and then forwards it to the destination, whereas in the case of DF, the partner s signal is decoded, reencoded, and then forwarded to the destination. Although both full-duplex and half-duplex relay channels can be implemented for user cooperative schemes [33], this paper only discusses half-duplex transmission. Unlike the conventional time-division or frequency-division relay channels, a pair of orthogonal sequences of fourth-order Walsh code is allocated to each user in the DCSK-CC system. Fig. 5 shows conventional cooperation and space-time cooperation, respectively. III. PERFORMANCE LOWER BOUND As a lower bound, BEP of the DCSK-CC system is analyzed under the ideal condition, namely, error-free DF at the relay user. During the whole cooperative period, there is a distinction between odd and even periods. During the odd period, each user sends only its own data, which are received and detected by the base station and by its partner, respectively. The (9) (10)

4 XU et al.: PERFORMANCE OF DCSK COOPERATIVE COMMUNICATION SYSTEMS OVER MULTIPATH FADING CHANNELS 199 signal transmitted by user 1 is as seen in (9). It is received by its partner according to, and by the base station according to, respectively. Adopting the GML demodulation algorithm based on Walsh codes, the partner uses to form a hard estimate of of user 1, whereas the base station depends on its received signal to make a soft decision of. The BEP of the partner s hard decision estimate of is equal to (11) where is the conditional BEP over a fading multipath channel, and is the instantaneous signal-to-noise ratio (SNR) of the fading multipath channel. Assuming that bit 1 is transmitted, can be expressed as (12) According to the GML algorithm, where and are weighted energies of bit 0 and bit 1 of user 1, respectively. With DCSK modulation, for a large spread-spectrum factor and if the logistic map is used, the corresponding can be simplified as [34] For, the function can be approximately expressed by (16) Substituting (13), (15), and (16) into (11), the BEP of user 1 at the partner is approximately expressed as (17) at the bottom of the page. So, the BEP can be evaluated through numerical integral of (17). On the other hand, due to the orthogonality of Walsh code sequences, the base station makes a soft decision in the odd period by calculating (18) During the even period, the two users both send to the base station a cooperative signal consisting of what each user estimates from its partner s bit in the odd period. The transmitted signals of the two partners are (13) Denoting, and, and assuming is the independent and identically-distributed(i.i.d.) Rayleigh distribution random variables, one can verify that is an i.i.d. exponential distribution random variable, with probability density function (PDF). The BEP of the partner s hard decision estimate of is calculated by (11) and (13). The PDF of can be obtained, as (14) where is the PDF of the instantaneous signal-to-noise ratio (SNR) of the path component. To simplify the analysis, a three-path channel is assumed, whose power delay profile is. From (14), the PDF of is expressed by (15) (19) where and are the estimated bits of user 1 and user 2 at their partners in the odd period, respectively. The base station receives signal and extracts a soft decision statistic by calculating, the prob- At the base station, assuming transmitted bit ability of bit error of user 1 is calculated by (20) (21) where denotes the weighted energy of user 1, which is a combination of the odd period and the even period. Denote as a combined weight for the relay link decision statistics at the base station, (21) is expressed in more detail as (22) at the bottom of the page. When, the combination (17)

5 200 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 58, NO. 1, JANUARY 2011 manner collapses to combined equal gains; that is, the even period has the same combination weight as the odd period. More specifically, if the interuser channel is perfect, i.e.,, then the lower bound of BEP is approximately estimated as (23) where and are SNR conditioned channels between user 1 to the base station and between user 2 to the base station. It is assumed that the power delay profile of channels between user 1 and user 2 and between the users and the base station are all the same. Thus, the conditioned SNRs ( and ) are i.i.d. random variables. The distribution of SNR at the base station is given as Substituting (15) into (24), the conditioned SNR (25) at the bottom of the page, where (24) becomes Substituting (13) and (25) into (11) and exploiting (16), the approximate BEP lower bound can be easily evaluated by numerical computation. IV. RESULTS AND DISCUSSIONS A. BEP Lower Bound of the Two-User Cooperative System The BEP lower bound given above is a benchmark of the DCSK-CC system in wireless multipath environment. Herein, numerical and simulated results are provided for evaluating this BEP lower bound. In Fig. 6, the numerical BEP lower bound and Fig. 6. BEP curves of the DCSK-CC system with error-free or error forwarding. simulated results of the two-user DCSK-CC system are plotted for and, respectively. The numerical results are obtained by numerical computation using (11), (13), and (25), while the simulated results are obtained by the Monte Carlo method assuming error-free DF at the relay user. It can be seen that the approximately BEP lower bound well agrees with the simulation results. The effect of the spread spectrum factor on numerical results of the BEP lower bound is shown in Fig. 6. Here, when is 32, the numerical results do not agree well with the simulated ones, since in the derivation of the BEP lower bound, the spread spectrum factor is assumed to be much higher than the multipath time delay so that ISI could be neglected. Under this assumption, for a given fixed multipath time delay, the greater the spread spectrum factor is, the better the agreement of numerical results and simulation results is. In addition, the BEP curves of the DCSK-CC System with error DF in an equal user distance situation are provided in Fig. 6. It shows that there is a large gap between BEP lower bounds of the DCSK-CC and BEP of DCSK-CC with error DF; (22) (25)

6 XU et al.: PERFORMANCE OF DCSK COOPERATIVE COMMUNICATION SYSTEMS OVER MULTIPATH FADING CHANNELS 201 namely, between the two different groups of curves. This is because there are errors when relay user forwards information in the actual cooperative system. How can the gap between the lower bound and a real system be reduced? One method is to use the adaptive decode-forward protocol that the cyclic redundancy check (CRC) are adopted for parity check at the relay user. The relay user does not relay partner s data if the parity check is not satisfied. Correspondingly, the complexity of the relay user and the delay of data transmission will be increased. B. Performance Comparison Between the DCSK-CC System and the CDMA-CC System The user-cooperative scheme was proposed as an efficient wireless diversity technique and introduced into CDMA systems in [28] and [29], where performance of two-user cooperative diversity was studied over flat Rayleigh fading channels. For convenient comparison with DCSK-CC, the CDMA-CC system is extended over to frequency-selective Rayleigh fading channels. The Gold-sequences are adopted as spread-spectrum sequences. Notice that the length of a Gold-sequence is always odd, whereas the global spread-spectrum factor of the DCSK-CC system is always even. Therefore, similarly to the standard IMT-2000 [35], an extra 0 is added at the end of each Gold-sequence to match the spread-spectrum factor, thereby ensuring the same bandwidth efficiency of the two systems. Thus, the comparison between the DCSK-CC and the CDMA-CC can be easily carried out and clearly shown. At the receiver, the conventional receive algorithm, i.e., the single path correlation receiver, is adopted. The near-far effect is a common phenomenon in wireless communication systems (in particular, CDMA). To mitigate the near-far effect, power control is an effective strategy for CDMA systems. Assume that the path loss of all links is proportional to, where d denotes the link distance between two users. The near-far effect corresponds to the effect of relay s location in a user cooperative communication system. Assume that source, relay, and destination terminals are located in a two-dimensional plane, with,, and denoting the link distances of source to destination, source to relay and relay to destination, respectively. All distances are normalized by the distance, so the distance ratio represents the geometric positions of all users. So, except for the distance ratio of 1:1:1, there are near-far effects. The performance comparison between DCSK-CC and CDMA-CC based on conventional cooperation schemes [cooperation pattern of Fig. 5(a)] without power control is considered here. The error performance of the proposed two-user DCSK-CC and CDMA-CC have been simulated by the Monte Carlo method over three-path multipath quasi-static block-faded channels. The path gains of the channels are the same with those assumed in Section III, and the delays are, where is the sampling period of the chaotic signal. Figs. 7 and 8 show the BEP performance of the DCSK-CC system and the CDMA-CC system in various scenarios with different user-distance ratios and with spread-spectrum factors of 32 and 64, respectively. From these figures, one can see that the BEP curves of the DCSK-CC system become lower than that of the CDMA-CC system Fig. 7. BEP performance comparisons of the DCSK-CC system and the CDMA-CC system, f =32. Fig. 8. BEP performance comparisons of the DCSK-CC system and the CDMA-CC system, f =64. in high-snr range, except for the case with distance ratios 1:1:1. The slopes of the BEP curves of the DCSK-CC system become steeper than that of the CDMA-CC system at BEP levels below for the scenarios with near-far effects. The difference of the slopes aforementioned indicates that the performance of the DCSK-CC system with a steeper slope is more sensitive to noise than the CDMA-CC system at high values of SNR. The DCSK-CC system is more effective at high SNR for the reason given below. Autocorrelation receiver of DCSK obtains the gain of multipath diversity as well as the user-cooperative diversity gain in the proposed scheme. On the contrary, the conventional single path correlation receiver of CDMA cannot get multipath diversity gain unless the RAKE receiver is applied. However, compared with AcR of DCSK, the RAKE receiver requires complete knowledge about the channel state information, which is usually obtained by the complex channel estimation algorithm. So, it is unsuitable for low-power and low-complexity applications, for example the WSN. Consequently, the DCSK-CC system has performance advantages at the BEP of to meet the low-cost

7 202 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 58, NO. 1, JANUARY 2011 Fig. 9. BEP performance comparisons of the DCSK-CC system and the CDMA-CC system over COST 207 RA channel, f =64. Fig. 10. BEP performance of conventional cooperative system and space-time cooperative system, f = 32. data transmission demands. In particular, it achieves greater performance advantages in near-far effect scenarios. Thus, compared with the CDMA-CC network, the energy consumption and complexity of the proposed network are reduced, without using the power control techniques at the transmitter. To further justify the advantages of the proposed scheme, its performances on COST 207 Rural Area multipath channel [36], whose power delay profile is, are evaluated and compared. Assume that the path gains have Rayleigh distributions, and the delay vector is. Fig. 9 shows the BEP curves of the DCSK-CC system and CDMA-CC system in various scenes of user-distance ratios, with a spread-spectrum factor of 64. One can see that the advantages of DCSK-CC system are held in the far-near scenarios as well. C. Comparison With the Space-Time Cooperative System To further improve the error performance of the user-cooperative systems, space-time cooperation was proposed in [37]. The cooperation framework is illustrated in Fig. 5(b) where it could be noticed that cooperative users transmit their partner s data as well as their own bits in even frame compared with the conventional cooperative pattern shown in Fig. 5(a). Figs. 10 and 11 show the BEP performances of the conventional cooperative and space-time cooperative systems with spread-spectrum factors 32 and 64, respectively. It is worth noting that the performance of the space-time cooperative system is not better than that of the conventional cooperative system in the DCSK-CC system at all times. On the contrary, the performance of the conventional cooperative system outperforms that of the space-time cooperative system at higher SNR with spread-spectrum factor 32, except for the case with 1:1:1. Unlike the user-cooperative communication systems based on traditional digital modulations, it implies that the conventional cooperation is a better cooperation protocol than the space-time cooperation in DCSK-CC system. D. Performances of Multiple Relays In this section, the performance of multiuser DCSK-CC system is evaluated in a more general situation. The four-user Fig. 11. BEP performance of conventional cooperative system and space-time cooperative system, f =64. DCSK-CC systems, with two relays and three relays, respectively, are considered and compared with that of the situation with only one relay. Assume that the link distances between relay users are negligible; thereby, the same distance ratio criterion of the two-user cooperation system is adopted. Fig. 12 shows the BEP performances of the DCSK-CC with one relay, two relays, and three relays in the scene where the distance ratios of source to destination, source to relays, and relays to destination are of 1:0.8:0.4. It shows that the performance gain is remarkable when the number of relays is increasing from one to two. However, this gain decreases greatly when the number of relays increases from two to three. Thus, it can be concluded that the performance contribution of increasing the number of relays is negligible when it is greater than two in a multiuser DCSK-CC system. E. Performances of Different Chaotic Maps Finally, the performances of various DCSK-CC systems are investigated with the following chaotic maps: logistic map, cubic map and Bernoulli-shift map, defined respectively by

8 XU et al.: PERFORMANCE OF DCSK COOPERATIVE COMMUNICATION SYSTEMS OVER MULTIPATH FADING CHANNELS 203 Fig. 12. BEP performance of the DCSK-CC system with different relays in a multiuser environment, f =64. V. CONCLUSION In this paper, the performance of a user-cooperative system has been investigated based on DCSK modulation, which employs the orthogonal Walsh code to achieve orthogonal subchannels for cooperation. Over multipath Rayleigh fading channels, the performance comparisons between the DCSK-CC systems and the CDMA-CC systems subject to the same bandwidth efficiency have been evaluated, which show that the former outperforms the latter at the BEP of. The advantages are prominent in near-far scenes. Thus, the common power control device which mitigates near-far effects is not needed in the proposed system comparing to CDMA-CC systems. Therefore, the DCSK-CC has great potential for improving link performances by only increasing overheads of the relay protocol while keeping all terminal conditions unchanged. This implies that the former has a simpler algorithmic design with lower transceiver cost than the latter. It has also been found that space-time cooperation in the proposed system cannot improve the performance, different from the traditional cooperative communication systems. In other words, conventional cooperation is a good choice for the proposed system. As a performance benchmark, the BEP performance lower bound of the proposed system with a conventional cooperation protocol has been derived over three-path Rayleigh fading channels, which well agree with its counterparts in simulations. Thus, the proposed system can be expected applicable to energy-constrained and low-cost wireless networks with simple cooperation protocols. Fig. 13. BEP performances of DCSK-CC system when logistic map, cubic map, and Bernoulli-shift map are used, respectively, f =32. when when. The BEP curves for the two-user DCSK-CC system, corresponding to different chaotic maps with distance ratio 1:0.8:0.4, are plotted in Fig. 13. It is found that the chaotic sequences generated by the Bernoulli-shift map produce higher BEP, while the BEPs for the system using the cubic map and the logistic map are the same. There are similar results in conventional multiuser DCSK systems [6]. This observation can be explained in terms of the variances of chaotic sequences; that is, the variance of Bernoulli-shift map sequence is larger than that of the logistic map and the cubic map sequences [6]. REFERENCES [1] L. Kocarev, K. S. Halle, K. Eckert, L. O. Chua, and U. Parlitz, Experimental demonstration of secure communications via chaotic synchronization, Int. J. Bifurcation Chaos, vol. 2, pp , [2] H. Dedieu, M. P. Kennedy, and M. Hasler, Chaos shift keying: Modulation and demodulation of a chaotic carrier using self-synchronizing chua s circuit, IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 40, pp , Oct [3] G. Kolumbán, B. Vizvari, W. Schwarz, and A. Abel, Differential chaos shift keying: A robust coding for chaos communications, in Proc.,4th Int. Workshop Nonlinear Dyn. Electron. Syst. (NDES 96), Seville, Spain, Jun. 1996, pp [4] G. Kolumbán, M. P. Kennedy, G. Kis, and Z. Jákó, FM-DCSK: A novel method for chaotic communications, in Proc. IEEE ISCAS, May 1998, vol. 4, pp [5] G. Kolumbán, Theoretical noise performance of Correlator-based chaotic communications schemes, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 47, pp , Dec [6] W. M. Tam, F. C. M. Lau, and C. K. Tse, Analysis of bit error rates for multiple access CSK and DCSK communication systems, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 50, pp , May [7] F. C. M. Lau, C. K. Tse, Y. Ming, and S. F. Hau, Coexistence of chaos-based and conventional digital communication systems of equal bit rate, IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 51, no. 2, pp , Feb [8] L. F. Ye, G. R. Chen, and L. Wang, Essence and advantages of FM-DCSK technique versus conventional spreading spectrum communication method, Circuits, Syst. Signal Process., vol. 24, pp , Oct [9] S. Kozic, T. Schimming, and M. Hasler, Controlled one-and multidimensional modulations using chaotic maps, IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 53, no. 9, pp , Sep [10] W. M. Tam, F. C. M. Lau, and C. K. Tse, Generalized correlationdelay-shift-keying scheme for noncoherent chaos-based communication systems, IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 53, no. 3, pp , Mar [11] A. B. Salberg and A. Hanssen, A subspace theory for differential chaos-shift keying, IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 53, no. 1, pp , Jan [12] F. J. Escribano, L. López, and M. A. F. Sanjuán, Chaos coded modulations over Rayleigh and Rician flat fading channels, IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 55, no. 6, pp , Jun

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Lin Wang (S 99 M 03 SM 09) received the B.Sc. degree in mathematics (with first class honors) from the Chongqing Normal University, Chongqing, China, in 1984, the M.Sc. degree in applied mathematics from the Kunming University of Technology, Kunming, China, in 1988, and the Ph.D. degree in electronics engineering from the University of Electronic Science and Technology of China, Chengdu, China, in From 1984 to 1986, he was a Teaching Assistant in the Mathematics Department of Chongqing Normal University. From 1989 to 2002, he was Teaching Assistant, Lecturer, and then Associate Professor in Applied Mathematics and Communication Engineering in the Chongqing University of Post & Telecommunications, Chongqing. From 1995 to 1996, he spent one year with the Mathematics Department at the University of New England, Australia. In 2003, he spent three months as visiting researcher in the Center for Chaos and Complexity Networks at the City University of Hong Kong. Since 2002, he has been Full Professor and Associate Dean in the School of Information Science and Technology, Xiamen University, Xiamen, China. He holds five patents in the field of physical-layer digital communications and published over 60 journal and conference papers. His current research interests are in the areas of channel coding and chaos modulation, and their applications to wireless communications and storage systems. Guanrong (Ron) Chen (M 89 SM 92 F 97) received the M.Sc. degree in computer science from the Sun Yat-sen (Zhongshan) University, Guangzhou, China, in 1981 and the Ph.D. degree in applied mathematics from Texas A&M University, College Station, in Currently he is a Chair Professor and the Director of the Center for Chaos and Complex Networks at the City University of Hong Kong, Hong Kong, China. Prof. Chen serves as Editor at different ranks for several IEEE Transactions and international journals. He is an ISI highly cited researcher in engineering, and has received four best journal paper awards in 1998, 2001, 2002, and 2005, and the 2008 State Natural Science Award of China. He is Honorary Professor at different ranks at more than 30 universities worldwide.