MOST MODERN wireless communication applications,

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1 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 60, NO. 4, APRIL One Analog STBC-DCSK Transmission Scheme not Requiring Channel State Information Pingping Chen, Lin Wang, Senior Member, IEEE, and FrancisC.M.Lau, Senior Member, IEEE Abstract Both the inherently wideband differential-chaosshift-keying (DCSK) modulation and the space-time block code (STBC) are techniques that can mitigate the effect of multipath fading. By applying STBC at the chaotic segment level, a novel analog STBC-DCSK scheme is proposed in this paper. The proposed scheme is a simple configuration that combines the advantages of STBC and chaotic modulation. Due to the very low correlation between different analog chaotic signals, the proposed scheme can remarkably suppress the inter-transmit-antenna interference so as to recover the desired information and to achieve the full diversity gain. The theoretical bit-error-rate (BER) performance and the highly consistent simulation results demonstrate that the STBC-DCSK scheme outperforms the conventional single-input-single-output (SISO)-DCSK scheme by about 5 db at a BER of. The performance superiority of the proposed scheme is further demonstrated in a typical UWB channel by simulations. More importantly, the proposed scheme maintains the same low transceiver cost as the SISO-DCSK scheme. Consequently, this proposed scheme is a low-cost alternative for wireless local area network (WLAN) applications. Index Terms Differential chaos shift keying (DCSK), spacetime block code (STBC), STBC-DCSK, ultra-wideband (UWB). I. INTRODUCTION MOST MODERN wireless communication applications, such as mobile or indoor radio, are susceptible to multipath propagation effects. The interaction between the signals from different paths arriving at the receiver results in a degradation in error performance. To overcome the negative effect, the next-generation wireless standards (such as LTE and WiMAX) are likely to use multiple-input-multiple-output (MIMO) technique to improve the system performance and the data throughput [1], [2]. Space-time block code (STBC) [3] [5] is viewed as an attractive technique that can be applied to a number of existing and emerging wireless personal communication standards. It can effectively enhance the signal quality at the receiver by encoding the transmit symbols for different transmit antennas and different time intervals. STBC has been studied for wireless personal area network and wireless local area network (WPAN/ Manuscript received December 07, 2011; revised March 28, 2012; accepted June 26, Date of current version March 23, This work was supported by the National Natural Science Foundation of China under Grant , Grant , and the Fundamental Research Funds for the Central Universities No G017. This paper was recommended by Associate Editor G. Sobelman. P. Chen and L. Wang are with the College of Information Science and Technology, Xiamen University, Fujian, Xiamen, , China ( wanglin@xmu.edu.cn). F. C. M. Lau is with the Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hong Kong, China. Digital Object Identifier /TCSI WLAN) systems over dense-multipath ultra-wideband (UWB) channel environments [6] [12]. It has also been concluded that properly designed analog space-time coding schemes can exhibit a significant performance improvement and can enhance robustness against time jitter in UWB transmissions [8]. Furthermore, over a typical flat-fading channel in a 4G WLAN application, an UWB system based on STBC outperforms the single-input-single-output (SISO) UWB systems with two orders of magnitude in terms of bit error rate (BER) [9]. Another well-known solution against the multipath propagation is the use of spectrum spreading, which performs significantly better than the narrowband systems. Conventional spread-spectrum systems are based on pseudo-random sequences such as -sequence [13]. Recently, chaotic signals have been used for spectrum spreading because they are inherently wideband signals and are readily generated with a relatively simple circuitry [14]. In addition, the nonperiodicity and long-term unpredictability of chaotic signals can provide some sort of security from the view of cryptography [15]. In particular, one chaos-based non-coherent modulation scheme, namely the differential-chaos-shift-keying (DCSK) scheme with a correlation receiver, has been widely studied [16] [23]. Since the DCSK scheme can offer excellent performance under multipath conditions and does not require complex Rake reception or channel estimation, it has attracted considerable interest towards its application to WPAN [24] [27] and WLAN [28] [31]. One recent study has revealed that by using a chaotic carrier with a longer duration, the coverage of the UWB radio devices can be enlarged compared with the impulse-radio (IR) UWB devices using an extremely short pulse duration [32]. In [33], a combined MIMO-DCSK scheme based on STBC and DCSK has been proposed. It has been proved the feasibility of using chaotic communications in MIMO channels, in which the digital Alamouti space code is used to ensure the signals orthogonality. The BER performance of the system has subsequently been analyzed [34]. While the aforementioned scheme achieves a capacity gain compared to the SISO-DCSK scheme (the scheme with one transmit antenna and one receive antenna), it requires extra hardware for estimating the channel state information (CSI) which is essential for carrying out the STBC decoding. In this paper, a novel analog 1 STBC-DCSK scheme that does not requires CSI in the decoding process is proposed. The aim is to reduce the implementation complexity and the cost of the chaos-based STBC scheme. Our contributions in this work are summarized as follows. 1) The proposed STBC-DCSK scheme achieves the same full diversity as the MIMO-DCSK scheme in [33]. Furthermore, it does not require any complicated channel 1 We call the proposed scheme analog STBC-DCSK because the chaotic signals passing into the STBC encoder are of analog nature /$ IEEE

2 1028 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 60, NO. 4, APRIL 2013 estimation, carrier synchronization or Rake reception. Its hardware complexity is similar to that of the SISO-DCSK scheme. 2) Two systems one equipped with two transmit antennas and one receive antenna; and another one equipped with three transmit antennas and one receive antenna are investigated. The theoretical BER performance of the systems has been analyzed. 3) Unlike the MIMO-DCSK scheme in [33] where the DCSK demodulation and the STBC decoding are performed separately, the proposed analog scheme jointly performs DCSK demodulation and STBC decoding with a linear complexity. It should be noted that the inter-transmit-antenna interference (ITI) in the proposed scheme is minimized by the low correlations among different analog chaotic signals [17]. 4) Both the theoretical results and the simulation results show that the proposed scheme accomplishes a notable performance gain (up to 5 db) over the SISO-DCSK scheme at a BER of. To further prove its feasibility in practical applications, this paper also compares the performance between their frequency-modulated (FM) schemes: STBC-FM-DCSK and SISO-FM-DCSK, over a typical UWB CM1 channel. It is observed that the former scheme outperforms the latter one over two orders of magnitude, which suggests that the proposed scheme can extend the coverage of radio devices used in WLAN environments. In summary, the proposed scheme can be viewed as a low-complexity and good error performance solution for multipath channels, such as WLAN/WPAN channels. The paper is organized as follows. Section II gives an explicit description of the proposed analog STBC-DCSK scheme. Section III derives the analytical performance and discusses the characteristics of the proposed scheme. Section IV compares the theoretical BER results and the simulation results. It also compares the error performance of the proposed scheme with the SISO-DCSK scheme. Finally, Section V draws the conclusions. II. SYSTEM MODEL In this section, a brief overview of the DCSK scheme is given before details of the proposed analog STBC-DCSK scheme are presented. A. DCSK Modulation In the DCSK modulation scheme, the binary information to be sent is mapped into a wideband chaotic signal [17]. For each bit, a reference chaotic sequence of length is first obtained by sampling the output of a chaos generator. Then, this reference segment is sent in the first-half of the bit duration. In the second-half of the bit duration, the reference segment is repeated if ;otherwise,if, the reference segment is multiplied by and sent 2. 2 The spreading factor of the DCSK scheme therefore equals. Fig. 1. Cross-correlation of two chaotic signals. The logistic map is used to generate the chaotic signals of length. For the th bit, denote the reference chaotic sequence and the information-bearing sequence, respectively, by and where is a chaos generator. At the receiving side, the reference chaotic sequence correlates with the information-bearing sequence and produces an output. If a positive output is found, the transmitted bit is determined as a ; otherwise, a is decoded. Moreover, it is worth noting that the chaotic sequences into which a bit is mapped are different for different bits and most of them have good auto/cross correlation properties [16] [19]. Suppose the output of the chaos generator has a mean value of zero, i.e., where is the expectation operator. Assume that the logistic map is used. Moreover, two chaotic signals and are generated by and each has a length of. The cross-correlation of and is illustrated in Fig. 1 for different values of. We can observe that for small (8 and 48), the cross-correlation between the chaotic signals is relatively large. When is increased to 128 or 256, the cross-correlation value becomes very small. Thus, for a large spreading factor, the correlation properties of the two chaotic signals and can be approximated by if otherwise (1) where denotes the dot product of the two vectors and,and represents the mean-squared value of the chaotic signal. This good auto/cross correlation is the very characteristic of the analog chaotic signals that we utilize to design the proposed analog scheme. B. Design Principle of the Analog STBC-DCSK System Generally, a transmit diversity scheme is based on the so-called delay diversity scheme [35] in which the same symbol is not transmitted simultaneously from different transmit antennas but is sent with a minimum time delay between the transmissions. We denote the number of transmit antennas by. Moreover, we consider the case in which the transmitter is equipped with two or three transmit antennas (i.e.,, 3) 3. We also assume that the receiver has only one antenna. 1) Transmitter: In the proposed analog scheme, the data stream is firstly serial-to-parallel converted into blocks of 3 Our system can be readily extended to more than 3 transmit antennas.

3 CHEN et al.: ONE ANALOG STBC-DCSK TRANSMISSION SCHEME NOT REQUIRING CHANNEL STATE INFORMATION 1029 TABLE I CHAOTIC SEQUENCE SENT BY THE TH ANTENNA IN THE TH BIT TRANSMISSION PERIOD FOR A TWO-TRANSMIT-ANTENNA SYSTEM Fig. 2. Block diagram of the transmitter in the proposed STBC-DCSK scheme. bits. For example, when there are two transmit antennas, 4 bits are considered at a time. Referring to Fig. 2, the bits are then passed to DCSK modulators. Each of the modulators outputs a reference chaotic segment and a corresponding information-bearing segment. As described in the previous section, the reference chaotic segment and the information-bearing segment for the th bit are denoted by and, respectively. For example, when, the 4 DCSK modulators generate,, and separately. Subsequently, the chaotic segments are fed into the STBC encoder. As shown in (1), the chaotic sequences generated for different data bits have very low cross-correlations. In fact, they are almost orthogonal if the length of the chaotic sequences is long enough [17]. In consequence, with an aim to avoid excessive inter-transmit-antenna interference (ITI) at the receiver, chaotic segments from the same DCSK modulator should not be sent by the different transmit antennas at the same time. For example, if is being sent by the first transmit antenna, neither nor should be sent by other transmit antennas. Based on this rule, the STBC encoder arranges the reference segments and the data segments differently for each of the antennas. The objective is to suppress the ITI effectively by using the inherent low cross-correlation property of the analog chaotic signals. Recall that each antenna has bit durations to complete the transmissions of all the chaotic segments generated by bits of data. Therefore, in order to keep the average bit energy identical for systems with different number of transmit antennas (even for a single-transmit-antenna system), the transmitted signal for each antenna is scaled by a factor of. Let denote the chaotic sequence sent by the th antenna in the th bit transmission period.notethat each comprises two rearranged segments ( or )which are scaled by. By combining all the chaotic sequences together, a chaotic frame sent by the th antenna, denoted by, is formed. For example, when there are transmit antennas,,, and are input to the STBC encoder. To obtain the transmit diversity with little inter-transmit-antenna interference, the chaotic sequences (,2and, 2,3,4) can be designed according to Table I. Then, in the first bit duration, i.e.,,thefirst transmit antenna sends while at the same time the other antenna sends. When there are three transmit antennas, 6 bits are sent in one block. The chaotic segments passing into the STBC encoder are then denoted by,,,, and, which are scaled by and rearranged in the STBC encoder. The STBC then outputs the chaotic sequences (,2,3and )asintableii. 2) Channel Model: We model the channel between each transmit antenna and the receive antenna as a two-ray Rayleigh quasi-static block-faded channel [36]. Denote the gains of the two paths between the th transmit antenna and the receive antenna by and, which are independent Rayleigh distributed random variables. They are assumed to be invariant over at least one STBC frame period, i.e., bit durations. Further, denotes a noise sample following a Gaussian distribution with zero mean and variance (two-sided power spectral density). 3) Receiver: Denote the received signal vector during the th bit duration by.then, can be expressed as where (, 2) denotes the delayed signal (delayed by comparing with the first arriving signal) received during the th bit duration; and is the noise vector containing s as elements. Let and, respectively, denote the first-half and the second-half of,i.e.,. To demodulate the signals, differentially coherent demodulators that process the received signals in a linear manner are designed. For example, for a two-transmit-antenna system, 4 differentially coherent demodulators are constructed based on the transmitted chaotic segments in Table I. In this case, the outputs of the 4 demodulators, denoted by to, should be given by Similarly, when there are transmit antennas, 6 differentially coherent demodulators are designed based on Table II. The outputs of the demodulators are therefore given by From (3) and (4), it can be seen that is obtained by summing the correlations between the first-half and the second-half of for. Moreover, it can readily be (2) (3) (4)

4 1030 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 60, NO. 4, APRIL 2013 TABLE II CHAOTIC SEQUENCE SENT BY THE TH ANTENNA IN THE TH BIT TRANSMISSION PERIOD FOR A THREE-TRANSMIT-ANTENNA SYSTEM shown that contains four components the (required) correlation value between the reference chaotic segment and the information-bearing chaotic segment ( and, respectively, for the 2-transmit-antenna system); the inter-transmit-antenna interference due to the chaotic segments sent by other antennas (e.g., correlation between from the second antenna and from the first antenna for the 2-transmit-antenna system); inter-symbol interference due to the delayed signal from the previous bit duration; and noise-related signal. Finally, the decoded bit is given by Thus, if the required signal, i.e., the correlation value between the reference chaotic segment and the information-bearing chaotic segment, is large compared with the interferences and noise, the information bit will be decoded correctly. It can be seen that the principles for decoding other information bits are the same as those for decoding. Note that is derived from signals received from paths which have different delays and different fading coefficients. Consequently, if the spreading factor is large enough such that the signals from the paths are uncorrelated, the proposed STBC-DCSK system will achieve the full diversity (i.e, )[3]. Remark 1: In the MIMO-DCSK scheme reported in [33], the received signals are first demodulated by a correlator. Afterwards, the information bits are STBC decoded using known channel state information (CSI). In our proposed analog STBC- DCSK scheme, the information bits can be recovered from the demodulator outputs based on correlating the corrupted reference segments with the corrupted information-bearing segments [see (3) and (4)]. In other words, demodulation and STBC decoding are jointly accomplished without the need of any CSI, as in the case of demodulating the SISO-DCSK signal. Thus, the implementation of the complex channel estimation algorithm is not required. A. BER Evaluation if if III. PERFORMANCE ANALYSIS We evaluate the error performance of the STBC-DCSK scheme over a multipath fading channel. Based on Table I, (2), and (3), it can be easily proved that for a two-transmit-antenna system, the statistical properties of (,2,3,4)are identical. Consequently, the probability of an error occurring is identical for any transmitted bit. The same argument holds for the three-transmit-antenna system. (5) Therefore, without loss of generality, we consider the error rate of the bit. According to (3) and (4), the demodulator output that decides the value of the bit can be re-written as where represents the output when the first-half of correlates with its second half, and denotes the th element in.next,we derive the statistic properties of andthenwederivethoseof. It is assumed that the spreading factor is much higher than the multipath time delay, i.e., (,2).Consequently, we can suppose that the first paths from different antennas have a zero delay while the second paths have the same delay denoted by. Thus, we can approximate (2) by where denotes the delayed signal (delayed by in the second path) received during the th bit duration. Substituting (8) into (7) for, is rewritten as where denotes the chaotic signals sent in the previous transmission period and represents the noise sample in the th sampling instant. Since, we assume that the amount of intersymbol interference (ISI) is small compared with other types of interference. Thus, (9) can be approximated as (6) (7) (8) (9)

5 CHEN et al.: ONE ANALOG STBC-DCSK TRANSMISSION SCHEME NOT REQUIRING CHANNEL STATE INFORMATION 1031 where (10) Assume that the logistic map is used as the chaos generators in the DCSK modulators. For a large, it has been proved that the cross-correlation (i.e., )ofany two different chaotic sequences and is very small. As is formed by different chaotic segments and which may be uncorrelated, we substitute those uncorrelated chaotic sequences in (10) with simple symbols and for notational convenience. Furthermore, we denote the chaotic segments in the second path as and, which are the delayed version of and, respectively. We also define and, respectively, as the noise vectors in the first-half and the secondhalf of the bit duration. With all the aforementioned notations, (10) is rewritten as (11), at the bottom of the page Denote and as the variance operator and the covariance of and, respectively. Assume that is transmitted, i.e.,. Then. We further assume that the mean value of the chaotic signal is zero, i.e.,. This assumption is easily justified because a zero mean value minimizes the transmission power. Consequently, the mean and variance of, and in (11) can be readily derived as and (11) (17), we then obtain. Based on (18) (19) Using similar procedures and from (3) and (4), the expectation and variance of given can be readily shown equal to Since, and can be shown equal to (20) (21) (12) (13) (14) (15) (16) (17) (22) (11)

6 1032 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 60, NO. 4, APRIL 2013 Similarly, if a is sent, i.e.,, the mean and variance of can be derived and shown equal to (23) (24) Based on the above formulas and assuming that follows a normal distribution under the given conditions, the conditional BER given the received SNR can be calculated as (25) where is the complementary error function; is given by (26) and is the energy spent to send one bit of information. Since the logistic map is used as the chaos generators in the DCSK modulators, we have, and. Thus, (25) becomes where (27) (28) In addition, when is large, the term in (27) can be neglected. Thus (27) can be approximated as (29) Recall that each transmit antenna has the same average power (refer to Tables I and II). We further assume that the two paths corresponding to each transmit antenna have identical average power gain. Hence, the average received SNR per path, denoted by,isgivenby (30) Moreover, for a multipath Rayleigh fading channel, the probability density function (PDF) of the received SNR is given by (31) Fig. 3. BER performance versus the half-spreading factor parameter for the proposed STBC-DCSK system. and db. where denotes the number of fading paths and equals in our study; and is defined as in (30). Finally, using (29) (31), the average bit error rate BER of can be computed as which is also the BER of the overall STBC-DCSK system. (32) B. Spreading Factor Analysis The performance of the proposed scheme is analyzed in this section. Fig. 3 plots the computed BER (32) [based on (29) (31)] against the half-spreading factor parameter when there are, respectively, two and three transmit antennas (i.e.,,3).thevalueof also varies from 15 db to 30 db. It is illustrated that for each set of (, ), there exists an optimal length of the spreading sequence that minimizes the bit error rate. Two interesting phenomena can be seen in Fig. 3. First, the two-transmit-antenna scheme outperforms slightly the three-transmit-antenna scheme for all values of when db. Second, at db, the two-transmit-antenna scheme outperforms the three-transmit-antenna scheme when but underperforms when. The reasons are as follows. Recall that the total energies for sending one bit are identical for both and (i.e., independent of the number of transmit antennas). Moreover, the average received SNR equals [see (30)]. Thus, decreases with increasing at a fixed. Furthermore, an increasing also increases the inter-transmit-antenna interference (ITI) because there are more interfering signals sent from other transmit antennas. At low-to-moderate (e.g., 15, 20 db), the lost in performance due to the two aforementioned factors overwhelms the diversity-gain improvement when increases from 2 to 3. Thus, there is a net increase in BER. Consider the case when, 30 db. The system still suffers from a net error performance loss when increases from 2 to 3 and is small. However, when reaches a threshold value, the system results in a lower BER when increases from 2 to 3. It is because a larger produces a lower

7 CHEN et al.: ONE ANALOG STBC-DCSK TRANSMISSION SCHEME NOT REQUIRING CHANNEL STATE INFORMATION 1033 correlation among the different chaotic segments and hence a smaller ITI arises. Remark 2: The inspection of Fig. 3 reveals that in the proposed analog STBC-DCSK scheme, it does not always hold true that more transmit antennas will definitely give rise to a better error performance. Therefore, in a practical application, an appropriate number of transmit antennas has to be determined dependingonthespecific requirement on. C. The Analysis of Decoding Delay, Memories, Computations and Rate Through the examination of the proposed scheme, we find that the decoding delays for the two-transmit-antenna system and the three-transmit-antenna system are four-bit periods and six-bit periods, respectively. It is because the demodulation cannot be completed until all the chaotic signals within one frame are received. Therefore, in applications that require a short decoding delay, the two-transmit-antenna is more favorable than the three-transmit-antenna one. Besides, the rate of the space-time block code is defined as the ratio of the number of bits the STBC encoder takes as its input to the number of bit periods required to transmit one block of coded bits. Inspecting Tables I and II indicates that bit periods are required to transmit the input bits. Consequently, the proposed analog STBC-DCSK scheme is a full-rate scheme and hence no bandwidth expansion is required. Such a feature is of much importance for wireless transmissions. Note that in general, the proposed STBC-DCSK scheme with transmit antennas incurs a delay of bits in the demodulation process while the SISO-DCSK scheme has a delay of only one bit. Thus, more hardware such as memories is needed for the proposed STBC-DCSK receiver. In addition, we can deduce from (3) and (4) that correlations have to be performed in order to decode one bit of information in the proposed STBC-DCSK scheme. The complexity therefore increases linearly with the number of transmit antennas. In the case of the SISO-DCSK scheme, only one correlation needs to be done. In summary, the delay and complexity of the proposed STBC-DCSK scheme increases only linearly with the number of transmit antennas. Hence, we can justify the use of the proposed STBC-DCSK scheme which can accomplish a non-trivial gain in the high SNR region (see Section IV). Nonetheless, if the computation complexity and/or the time delay is/are critical, the SISO-DCSK scheme is preferred. IV. SIMULATION RESULTS AND DISCUSSIONS In this section, the numerical and simulated results based on the above-described channel model are provided. It is assumed that the transmit power of each transmit antenna is the same and that the average gain of all paths are identical. Moreover, the total transmit energy per bit are the same in all the simulated systems. Also, the logistic map is used to generate all the analog chaotic signals in the simulations. A. BER Comparison Between the Proposed Scheme and the Conventional SISO Scheme Over Two-Ray Channels In the proposed analog STBC-DCSK scheme, we assume that when, the delays of the two paths in the first transmitreceive channel are 0 and 3 and those in the second transmitreceive channel are 1 and 2. We denote these delays by (0, 3) and (1, 2), respectively. Fig. 4. BER of the STBC-DCSK scheme with and the SISO-DCSK scheme.. Delays of the two paths in the two channels are (0, 3) and (1, 2), respectively. Delays of the two paths in the SISO-DCSK scheme are (0, 1). Fig. 5. BER of the STBC-DCSK scheme with and the SISO-DCSK scheme.. Delays of the two paths in the three channels are (0, 2), (1, 4), and (3, 5), respectively. Delays of the two paths in the SISO-DCSK scheme are (0, 1). With, both the simulated BER and the numerical BER calculated from (29) (32) when are depicted in Fig. 4. We also show the simulated result of the SISO-DCSK scheme when the delays are (0, 1). We can see that the numerical BER is very close to its simulated one, and the performance of the proposed scheme is about 5 db better than its SISO counterpart at. Also, Fig. 5 plots the simulation and numerical results when and. The delays of the two paths in the three channels are (0, 2), (1, 4), and (3, 5), respectively. The results again show about 5 db performance gain of the proposed analog scheme over the SISO scheme at. The simulations over channels with larger delays are also performed and compared. The proposed STBC-DCSK scheme with and the SISO-DCSK scheme are simulated with.inthefirst set of simulations, the delays of the channels in the STBC-DCSK scheme are (0, 20) and (1, 21); while the delays of the channel in the SISO-DCSK scheme are (0, 20). In the second set of simulations, the delays in the STBC-DCSK

8 1034 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 60, NO. 4, APRIL 2013 Fig. 6. Simulated BER of the STBC-DCSK scheme with and the SISO-DCSK scheme.. scheme are (0, 40) and (1, 41); while those in the SISO-DCSK scheme are (0, 40). Through the examination of Fig. 6, the STBC-DCSK scheme achieves a performance gain of about 4 db over the SISO-DCSK scheme at. In other words, compared with the SISO-DCSK scheme, the STBC-DCSK scheme shows an enhanced robustness against the multipath channels with a large delay spread. According to (30) (32), the BER performance of the STBC-DCSK scheme is closely related to the average received SNR and the diversity order when and are fixed. An increase in reduces the average received SNR and increases the diversity order. Thus, at low, the reduction in the average received SNR makes the STBC-DCSK scheme underperform in terms of BER compared with the SISO-DCSK scheme. However, when is larger, the advantage of the diversity order becomes more prominent and hence the STBC-DCSK scheme outperforms the SISO-DCSK scheme. As a result, the BER curves of the two schemes cross each other, as shown in Figs B. BER Performance of the Proposed Scheme With Different Spreading Factors Fig. 7 shows the simulated BER of the proposed STBC-DCSK scheme when and. It is seen that for, the chaotic signals of this length have non-negligible cross-correlations (see Fig. 1), which results in high interference level in the demodulation process. This interference aggravates with an increase in the number of transmit antennas in the STBC-DCSK scheme. As a result, the three-transmit-antenna system performs badly and shows a much deteriorated error performance compared to the two-transmit-antenna system. Moreover, the non-negligible cross-correlations give rise to the error floor in the high region. For a much larger spreading factor, i.e.,,different chaotic segments embedded in the received signals exhibit much lower cross-correlations, which bring low interference levels into the decoding process. Thus, due to its higher diversity order, the three-transmit-antenna system has a 2 db gain over the two-transmit-antenna one at a BER of.these Fig. 7. BER of the STBC-DCSK schemes...delays of the two paths in the two channels are (0, 3) and (1, 2), respectively; delays of the two paths in the three channels are (0, 2), (1, 4), and (3, 5), respectively. observations match with the analyses in Sections III-B and II-B, i.e., in communication systems that require a relatively small spreading factor, the proposed scheme with two transmit antennas is a better choice in terms of error performance and decoding delay. In Fig. 7, we further plot the theoretical BER curve based on Gaussian approximation when and.as expected, we observe that there is a discrepancy between the theoretical BER and the simulated BER. When is small, replacing the Gaussian distribution by the -distribution [37], [38] will provide a more accurate theoretical BER. When is large enough, say 128 and 256, the results in Figs. 4 and 5 show that the Gaussian approximation can give a good estimation of the BER. C. BER Comparison Between the Proposed STBC-DCSK Scheme and the STCB-Aided DCSK Scheme in [33] We compare the proposed analog STBC-DCSK system with the digital STBC-aided DCSK system in [33]. Both systems are equipped with two-transmit antennas and one-receive antenna, and each transmit-receive channel is modeled as a one-ray Rayleigh channel [33]. Perfect channel state information (CSI) is assumed to be known to the receiver in the digital STBC-aided DCSK system while such information is not needed in the proposed analog STBC-DCSK system. In Fig. 8, the BER performance of two systems is plotted for. We can observe that the digital STBC-aided DCSK system outperforms the proposed one by about 3 db. This is not unexpected given that perfect CSI is available at the receiver of the digital STBC-aided DCSK system. However, such a receiver is much more complicated and incurs a higher cost. Comparatively, the proposed analog STBC-DCSK scheme avoids complex channel estimation. In particular, the proposed scheme is suitable for UWB applications which features low-complexity and low-cost. Furthermore, a typical UWB channel consists of a large number of resolvable multipath, i.e., dense multipath environment. Under such conditions, reliable channel estimation is very difficult to obtain and therefore the

9 CHEN et al.: ONE ANALOG STBC-DCSK TRANSMISSION SCHEME NOT REQUIRING CHANNEL STATE INFORMATION 1035 Fig. 10. Structure of the output signal from the FM-DCSK modulator in one bit period. and are frequency-modulated chaotic signals. Fig. 8. BER of the proposed analog STBC-DCSK scheme and the digital STBC-aided DCSK scheme in [33].. Fig. 11. Simulated BER of the STBC-FM-DCSK scheme with and the SISO-FM-DCSK scheme over an UWB CM1 channel. Dashed and solid curves show the UWB systems with and without narrow-band interference, respectively.. Fig. 9. BER of the proposed STBC-DCSK scheme and the STBC- scheme. and. Delays of the two paths in the two channels are (0, 3) and (1, 2). digital STBC-aided DCSK system may not be able to operate effectively. D. BER Comparison Between the STBC-DCSK Scheme and the STBC- Scheme We compare the BER performance of the proposed scheme and that of a STBC system (STBC- system) in which the reference segments are formed by a preferred pair of -sequences [13]. We assume that there are two transmit antennas and one receiver antenna, and that each transmit-receive channel is modeled as a two-ray fading channel. Fig. 9 depicts that the BER of the proposed STBC-DCSK system and the STBC- system when. The results show that the BER performance of the two systems are similar. E. BER Comparison Between STBC-FM-DCSK Scheme and SISO-FM-DCSK Scheme in an UWB Channel In [30], the FM-DCSK-based transmitted-reference (TR) system is proposed for WPAN networks within the IEEE standard. It is also found that the FM-DCSK UWB radio system is a feasible solution for UWB channels. Here we further investigate the performance of our proposed scheme under the framework in [30]. First, we modify the DCSK modulators in Fig. 2 to FM-DCSK modulators, forming the STBC-FM-DCSK scheme. Second, the structure of the output FM-DCSK signal follows that shown in Fig. 10, where represents the reference FM chaotic segment and is the information-bearing FM chaotic segment. If the data bit to be sent equals ;otherwise,.in the figure, denotes the duration of each chaotic segment and gives the guard duration between the two successive chaotic segments. As in the STBC-DCSK scheme, the analog signals in the first-half and the second-half of the bit period in the STBC-FM-DCSK scheme are denoted by and, respectively. These signals are then encoded by the STBC encoder based on Table I for. The simulation parameters are:,, sampling frequency for the simulation,andthe center frequency. The simulated UWB channel is the IEEE a CM1 channel model, which is based on the line-of-sight (LOS) residential environment. We also simulate a SISO-FM-DCSK scheme by converting the DCSK modulator in the SISO-DCSK scheme to a FM-DCSK modulator. Fig. 11 shows that the proposed STBC-FM-DCSK scheme with two transmit antennas outperforms the SISO-FM-DCSK scheme by two orders of magnitude at. We further consider the scenario when the STBC-FM-DCSK scheme or the SISO-FM-DCSK scheme is interfered by a narrow-band signal. Such a scenario is likely to occur because UWB systems are

10 1036 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 60, NO. 4, APRIL 2013 Finally, we plan to extend the proposed STBC-DCSK scheme to two or more receive antennas and we will report our results in future publications. Fig. 12. Frequency spectrum of the combined chaotic signal sent from the two antennas in the STBC-FM-DCSK scheme. designed to co-exist with existing conventional narrow-band communication systems [40]. In Fig. 11, we plot the BER results of the STBC-FM-DCSK scheme and the SISO-FM-DCSK scheme again when the signal-to-interference-ratio (SIR) equals. We observe that both the STBC-FM-DCSK scheme and the SISO-FM-DCSK scheme suffer from a slight degradation in error performance. In addition, under the given parameters, Fig. 12 shows the spectrum of the combined chaotic signal sent by the two antennas in the proposed STBC-FM-DCSK scheme. It can be seen that the signal bandwidth is about 600 MHz and a smooth spectrum meeting the requirement of the Federal Communications Commission (FCC) regulations is achieved. Thus, the superiority and feasibility of the proposed STBC-FM-DCSK scheme in practical applications is demonstrated. V. CONCLUSIONS This paper presents a full rate analog STBC-DCSK scheme. It combines the benefits of STBC and DCSK chaotic modulation and offers better system performance and robustness against multipath fading delay spread. The proposed scheme exploits the low cross-correlation property of different analog chaotic signals to effectively suppress inter-transmit-antenna interference. Moreover, neither complex channel estimation nor Rake reception is required. The proposed system therefore possesses an economical advantage because of its simple hardware implementation. It is also particularly useful in propagation conditions where channel estimation may not be possible. A system with two or three transmit antennas and one receive antenna has been considered in this paper. Based on the system model, the improvements are explicitly discussed and analyzed. Both the simulated BER and the numerical BER have shown that the analog STBC-DCSK scheme achieves a performance gain of about 5 db over the conventional SISO-DCSK scheme at a BER of. The proposed STBC-DCSK scheme has been further extended to form the STBC-FM-DCSK scheme. Over a typical WLAN UWB channel, the STBC-FM-DCSK scheme outperforms the SISO-FM-DCSK by two orders of magnitude, which suggests a gain in the coverage of a radio device. It is further found that for a system requiring a relatively small spreading factor, the proposed scheme with two transmit antennas is superior to the one with three transmit antennas. 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McGillem, A chaotic direct-sequence spread-spectrum communication system, IEEE Trans. Commun., vol. 42, no. 234, pp , [40] 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 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, in From 1984 to 1986, he was 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 and Telecommunication, Chongqing, China. From 1995 to 1996, he spent one year with the Mathematics Department of 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-time Professor and Associate Dean in the School of Information Science and Engineering of Xiamen University, Xiamen, China. Recently he has become the editor of ACTA Electronica Sinica and Guest Associate Editor of the International Journal of Bifurcation and Chaos. He holds 8 patents in the field of physical layer in digital communications and has published over 60 international journal and conference papers. His current research interests are in the area of channel coding, joint source and channel coding, chaos modulation, and their application for wireless communications. Francis C. M. Lau (M 93 SM 03) received the B.Eng. (Hons) degree in electrical and electronic engineering and the Ph.D. degree from King s College London, University of London, U.K., in 1989 and 1993, respectively. He is a Professor and Associate Head at the Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hong Kong, China. He is the coauthor of Chaos-Based Digital Communication Systems (Springer-Verlag, 2003) and Digital Communications with Chaos: Multiple Access Techniques and Performance Evaluation (Elsevier, 2007). He is also a co-holder of three U.S. patents, one pending U.S. patent, and one pending international patent. He has published over 200 papers. His main research interests include channel coding, cooperative networks, wireless sensor networks, chaos-based digital communications, applications of complex-network theories, and wireless communications. He served as an Associate Editor for IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS PART II: EXPRESS BRIEFS in and IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS PART I: REGULAR PAPERS in He was also an Associate Editor of Dynamics of Continuous, Discrete and Impulsive Systems, Series B from 2004 to 2007, a Co-Guest Editor of Circuits, Systems and Signal Processing for the special issue Applications of Chaos in Communications in 2005, and an Associate Editor for IEICE Transactions (Special Section on Recent Progress in Nonlinear Theory and Its Applications) in He has been a Guest Associate Editor of International Journal and Bifurcation and Chaos since 2010 and an Associate Editor of IEEE Circuits and Systems Magazine since Pingping Chen received the B.Sc. degree in computer software engineering from FuZhou University (FZU), Fujian, China, in He is currently working toward the Ph.D. degree in the Department of Electronic Engineering, Xiamen University, Fujian, China. His primary research interests include channel coding, joint source and channel coding, network coding, and UWB communications.