RECENTLY, multicarrier (MC) direct-sequence (DS)

104 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 24, NO. 1, JANUARY 2006 Transmit Selection Diversity With Maximal-Ratio Combining for Multicarrier DS-CDMA Wireless Networks Over Nakagami-m Fading Channels Jia Tang, Student Member, IEEE, and Xi Zhang, Senior Member, IEEE Abstract We propose the scheme to integrate transmit selection diversity/maximal-ratio combining (TSD/MRC) with multicarrier (MC) direct-sequence code-division multiple access (DS-CDMA) for various wireless networks. Applying this TSD/MRC-based scheme, the transmitter jointly selects the optimal subcarrier-and-antenna pair to significantly decrease the peak-to-average power ratio (PAPR), which is one of the main problems inherently associated with MC DS-CDMA communications. Over the frequency-selective Nakagami- fading channels, we develop the unified analytical framework to analyze the symbol-error rate (SER) of the scheme implemented in different types of wireless networks, while dealing with the perfect and imperfect channel state information (CSI) feedbacks, respectively. The imperfect feedbacks we focus on include delayed feedbacks and erroneous feedbacks. Taking the imperfectness of the feedback into account, the resultant SER is compared with that of both conventional selection diversity (SD)/MRC-based and space time block coding (STBC)/MRC-based schemes. Our analyses show that in a wide variation of the feedback imperfectness, our proposed TSD/MRC-based scheme has significant advantages over the other two schemes for both downlink cellular networks and ad hoc wireless networks. However, our analytical findings indicate that TSD/MRC-based scheme cannot always outperform SD/MRC-based and STBC/MRC-based schemes even when the perfect CSI feedbacks are available. Index Terms Multicarrier (MC) direct-sequence code-division multiple access (DS-CDMA), Nakagami- fading, peak-toaverage power ratio (PAPR), symbol-error rate (SER), transmit selection diversity (TSD), wireless networks. I. INTRODUCTION RECENTLY, multicarrier (MC) direct-sequence (DS) code-division multiple access (CDMA) that integrates the advantages of orthogonal frequency-division multiplexing (OFDM) with DS-CDMA has emerged as a promising technique for the next-generation wireless communications and networks [1] [4]. MC DS-CDMA can significantly alleviate the impacts of frequency-selective fading by mapping the serial data flow into a number of low-rate parallel substreams and transmitting the time-domain spread signals over multiple orthogonal subcarriers. The authors in [1] showed that when appropriately selecting the system parameters and using the Manuscript received October 6, 2004; revised May 29, 2005. This work was supported in part by the U.S. National Science Foundation CAREER Award under Grant ECS-0348694. The authors are with the Networking and Information Systems Laboratory, Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843 USA (e-mail: jtang@ece.tamu.edu; xizhang@ece.tamu.edu). Digital Object Identifier 10.1109/JSAC.2005.858884 antenna diversity, MC DS-CDMA is capable of supporting ubiquitous broadband wireless services over diverse propagation environments. The antenna diversity technique, on the other hand, making use of multiple antennas at transmitter and/or receiver, is another effective approach to combat the time-varying fading channels. Among the large number of antenna diversity schemes, the selection diversity (SD)/maximal-ratio combining (MRC) offers a good tradeoff between complexity and performance and, thus, received a great deal of research attentions [5] [9]. It is natural to consider integrating SD/MRC with MC DS-CDMA to further improve the system performance. However, how to combine these two techniques in the most efficient way, and how the combination can impact the wireless network performance have not been thoroughly studied. A. Previous Related Works The SD/MRC technique can be considered as a special case of the more general hybrid-selection/maximal-ratio combining (H-S/MRC) schemes [10] [16], where a subset of the available antennas are selected at transmitter and/or receiver. The H-S/MRC technique over independent identical fading environment has been studied in [5] [14], followed by some more complicated models such as the unequal fading and power models [15] and the correlated fading models [16]. In [11], the authors developed the novel approach termed as virtual branch to analyze the performance of H-S/MRC scheme, which transforms the mutually dependent order statistics to independent virtual branches and, thus, significantly simplify the analyses. In [14], the authors further studied the impact of channel state information (CSI) estimation error on H-S/MRC scheme. Most previous works in this area mainly focused on H-S/MRC employed at the receiver side only [6], [7], [10], [11], [14] [16] such that the CSI feedback from the receiver to the transmitter is not needed to make the antenna selection decision. In contrast, when the selection is applied at the transmitter side, the CSI feedback is necessary and, thus, the imperfectness of CSI feedback will impair the performance of the H-S/MRC scheme. While the powerful virtual branch technique [11] particularly focuses on the order statistics, this technique is hard to be extended to our imperfect CSI feedback analyses, specifically, for the delayed-csi feedback analysis, because our delayed-feedback analysis involves the induced order statistics [17]. In [5] and [9], the authors investigated the impact of feedback delay on SD/MRC scheme. Also, the authors in [13] studied the impact of feedback 0733-8716/$20.00 2006 IEEE

TANG AND ZHANG: TSD WITH MRC FOR MC DS-CDMA WIRELESS NETWORKS OVER NAKAGAMI- FADING CHANNELS 105 error on space time block coding (STBC)-based antenna-selection scheme proposed in [12]. However, no closed-form symbolerror rate (SER) expressions were obtained in [5], [9], and [13]. B. Our Contributions To make the analyses tractable, in this paper, we focus on the SD-based scheme employed over the independent identical fading channels. We first generalize the SD scheme into multiple-input multiple-output (MIMO) MC DS-CDMA systems. Based on the unique infrastructure of MC DS-CDMA systems, the SD is employed in both spatial and frequency domains, such that the joint optimal subcarrier-and-antenna pair is selected for each substream to transmit data. This two-dimensional selection diversity, referred as transmit selection diversity (TSD), offers not only the higher order of selection diversity, but also the lower peak-to-average power ratio (PAPR), which is the main drawback imposed by MC DS-CDMA communications systems [3]. Taking the feedback delays and errors into considerations, we then develop the unified framework to analyze the SER of the proposed TSD/MRC-based MC DS-CDMA scheme over frequency-selective Nakagami- fading channels. Following the excellent work in [18] [20], we derive the SERs as closed-form expressions. Applying this developed analytical framework, we finally analyze the impact of TSD on MC DS-CDMA systems in various wireless networks. The diverse network architectures we have considered include downlink cellular networks, uplink cellular networks, and ad hoc wireless networks. We also compare the performance of TSD/MRC with conventional SD/MRC and STBC/MRC-based MC DS-CDMA systems, through which we gain the insights about how severe the feedback imperfectness can impair the performance of various schemes. Our analyses show that in a wide variation of CSIfeedback imperfectness, TSD/MRC-based scheme has significant advantages over SD/MRC-based and STBC/MRC-based schemes for both downlink cellular networks and ad hoc wireless networks. However, our analytical findings also indicate that the TSD/MRC-based scheme cannot always outperform SD/MRC and STBC/MRC for uplink cellular networks even when the perfect CSI feedbacks are available. The rest of this paper is organized as follows. Sections II and III derive the SER with perfect and imperfect feedbacks, respectively. Section V generalizes and applies the obtained results to various multiuser MC DS-CDMA wireless networks. numerically evaluates the proposed scheme and compares it with other existing schemes. This paper concludes with Section VII. II. SYSTEM MODEL A. Transmitter Model We first consider a point-to-point wireless link with antennas at the transmitter and antennas at the receiver. We denote the average power by, the total number of subcarriers by, where the parameters and will be detailed below, and the central frequencies of these subcarriers by. First, a block of symbols with each duration of are converted to parallel substreams using the serial-to-parallel (S/P) converter. The signal over the th substream is expressed as, where is the index of substream, is the discrete time-index, denotes the transmitted symbol at time, represents the symbol duration after S/P conversion, and stands for the unit rectangular-pulse function of duration. Each substream is then multiplied by the spread-code, where takes the value of, and denotes the chip duration which follows with the spreading gain. For conventional MC DS-CDMA systems [1], the substreams are then further copied to parallel branches (also known as the identical-bit subcarriers [4]) and transmitted simultaneously. In contrast, in our proposed system, the transmitter jointly selects the optimal branch-and-antenna pair in frequency and spatial domains for each substream. Specifically, for the th substream, the transmitter will select the optimal antenna among antennas and the optimal subcarrier among branches to maximize the received power, where denotes the index of the branch. Finally, the modulation can be implemented by the -points inverse fast Fourier transform (IFFT). B. Channel Model In this paper, we consider the generalized frequency-selective Nakagami- fading channels because this model is very general and often best fits the land-mobile and indoor-mobile multipath propagations [10], [18] [20], [22]. We assume that the system parameters and are designed such that [1], where and denote the minimum and the maximum delay spreads of the channel, respectively. Under such an assumption, on one hand, each subcarrier signal is guaranteed to experience the flat-fading; on the other hand, the -branches of the same substream are ensured to experience the independent fading such that the frequency diversity is achieved. The channel impulse response function, denoted by, between the th transmit antenna and the th receive antenna at the th subcarrier can be expressed as where, is the path delay, is the set of path envelopes, and is the set of path phases. We assume that are independent identically distributed (i.i.d.) random variables (r.v. s) uniformly distributed between, and are i.i.d. r.v. s. The common probability density function (pdf) of, denoted by, follows the Nakagamidistribution specified by where is the average path gain and denotes the Gamma function. Let denote the corresponding path power. When employing TSD, the transmitter will select, for the th substream, the optimal branch-and-antenna pair, indexed by, that maximizes the total received power. We denote the subcarrier central frequency, the path power, the path envelope, and the path phase corresponding (1) (2)

106 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 24, NO. 1, JANUARY 2006 to the selected th substream by,, and, respectively. Thus, the received signal, denoted by, at the th antenna of the receiver is given by (3) where, and denotes the complex additive white Gaussian noise (AWGN) with zero-mean and double-sided power spectral density of per dimension. C. Receiver Model We assume that the receiver has the perfect knowledge of CSI. The received signal first correlates with the referenced waveform, the output of which, denoted by, can be expressed as Fig. 1. PAPR comparisons between our proposed TSD-based MC DS-CDMA scheme and the conventional SD-based and STBC-based MC DS-CDMA schemes using quadrature phase-shift keying (QPSK) modulations. The number of transmit antennas is set equal to N =2. Then, the outputs of the correlators from different receive antennas are combined together, i.e.,. Let the th substream s signal-to-noise ratio (SNR) at the MRC combiner output be denoted by, then, where denotes the average transmission energy per symbol. It is well known that when selection diversity is not employed, the pdf and cumulative distribution function (CDF) of the SNR at MRC combiner output, denoted by and, respectively, are specified by where, denotes the lower incomplete Gamma function [23], and represents the average SNR per receive antenna. D. PAPR Discussions for MC DS-CDMA Systems One of the main problems inherently associated with MC DS-CDMA communications is the high PAPR at the output signal [3]. It is well known that the peak-factor of the multicarrier signal is proportional to the number of used subcarriers [21]. Furthermore, in conventional MC DS-CDMA systems, the frequency-repeated signals transmitted on identical-bit subcarriers are not independent with each other, which will result in the even severer PAPR problem [3]. In our proposed scheme, although it has transmit antennas and subcarriers, the transmitter actually sends the signal by different paths. In average, the number of used subcarriers at each antenna is equal to. In the worst case, the maximum number of used subcarriers at one antenna is, where (4) (5) all substreams are coincidently transmitted by this antenna. Furthermore, by using the frequency-domain selection, the proposed scheme avoids sending duplicated signals on identical-bit subcarriers. In contrast, the conventional MIMO MC DS-CDMA schemes, such as STBC-based transmitter, always send signals by all subcarriers at each antenna, with copies of eachsubstreamsignal[1].ontheotherhand,theconventionalsd 1 schemes cannot avoid the possibility that the same transmit antenna sends several branches of the same substream. In the worst case, one transmit antenna has to send the signals for all subcarriers, which leads to high PAPR. Thus, our proposed TSD scheme can significantly decrease the PAPR imposed by the multicarrier DS-CDMA communications systems. Fig. 1 plots the simulated PAPR comparisons between our proposed TSD/MRC-based scheme and the conventional schemes, including STBC/MRC-based and SD/MRC-based schemes. We can observe that the PAPRs of our proposed scheme are significantly lower than those of the other two conventional schemes. As shown in Fig. 1, the larger the number of subcarriers causes the larger PAPR for both conventional and the proposed systems, which is consistent with the well known results given in [21]. However, the PAPR increasing-rate of our proposed TSD/MRC-based scheme is much lower than those of the two conventional schemes. Furthermore, increasing the frequency-repeating branches from to dramatically degrades the PAPR-performance for the two conventional schemes. In contrast, the higher order of the frequency-diversity guarantees the better PAPR-performance for our proposed scheme. III. SER OF TSD/MRC-BASED MC DS-CDMA SCHEME WITH PERFECT CSI FEEDBACK The performance of the selection diversity under perfect CSI feedbacks has been extensively studied in literatures, e.g., [8] [11]. In this section, we derive the SER within the context 1 In this paper, SD refers to the scheme that employs antenna selection at each subcarrier, with Q identical-bit subcarriers for the same substream.

TANG AND ZHANG: TSD WITH MRC FOR MC DS-CDMA WIRELESS NETWORKS OVER NAKAGAMI- FADING CHANNELS 107 of our proposed TSD/MRC-based scheme. For the integer, the common pdf of the SNRs, denoted by, follows the ordered Gamma distribution, which can be derived by the similar approach in [10] as concomitant) [17] of the original ordered, denoted by, is given by. Thus, the pdf of (10) (6) where denotes the pdf of conditioned on, where and are two correlated Gamma-distribution r.v. s. According to [22], the conditional pdf can be expressed as where and is the multinomial coefficients determined by, with,,,, and. Then, using the alternative representation of the Gaussian -function [18] [20], the SER, denoted by, for a set of commonly used signal constellations, can be expressed as follows: (11) where denotes the modified Bessel function of the first kind with the order of, the correlation coefficient is determined by [6] with denoting the zeroth-order Bessel function of the first kind, and representing the Doppler frequency. Thus, considering CSI feedback delays, we obtain the time-delay impacted SER, denoted by, as follows: (7) where we define In (7),,, and are constellation-dependent parameters (see [20] for details). Solving given in (8) using the approach proposed in [18], we obtain its closed-form expression as follows: (8), de- where we define the time-delayed version of noted by, as follows: (12) (13) As derived in the Appendix, we obtain the closed-form expression of (13) as follows: (9) Then, the closed-form expression for given in (7) can be derived using [20, Appendix 5A], which is omitted here for lack of space. IV. SER OF TSD/MRC-BASED MC DS-CDMA SCHEME WITH IMPERFECT CSI FEEDBACKS A. SER With Time-Delayed CSI Feedbacks The impact of feedback delay on the selection diversity has been studied in [9]. However, the authors in [9] did not obtain any closed-form SER expressions. We assume in this section that the feedback a transmitter receives is error-free, but experiences a time-delay, denoted by. Due to the time-varying nature of the wireless channel, the current optimal SNR may have changed already at the moment when the transmitter receives the feedback after the delay. Let denote the time-delayed version of SNR for the original optimal. According to the order statistics, is called the induced order statistics (or the (14) Similar to Section III, we can also obtain the closed-form expression for the SER by [20, Appendix 5A]. Note that when, derived in (14) reduces to derived in (9), which are expected since corresponds to (i.e., the perfect CSI feedbacks without delay [6]). It is also worth noting that when,wehave (15) which is also expected since if no selection diversity is employed, the feedback delay will not affect the performance of our proposed TSD/MRC-based MC DS-CDMA scheme.

108 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 24, NO. 1, JANUARY 2006 B. SER With Erroneous CSI Feedbacks The impact of CSI feedback error on antenna selection has been studied in [13], where two out of three transmit antennas are selected to employ STBC. However, no closed-form SER expressions are obtained in [13]. In our scheme, for each substream, the indices of the optimal branch-and-antenna pair are decided and fed back to the transmitter by the receiver. In this section, we assume that this feedback is sent to the transmitter without delay, but may be received in error due to the unreliable feedback channel. Let the feedback frame for each substream be carried by a binary vector with bits, representing the index of the optimal subcarrier-and-antenna pair. Thus, a total number of bits of feedback are needed when using our scheme at each subcarrier. Let denote the probability of a single-bit feedback error. Under the assumption that the feedback is uncoded and the feedback bit-errors are independent, the probability of the feedback-frame error, i.e., the transmitter erroneously selects a certain nonoptimal branch-and-antenna pair for data transmission, is determined by. 2 Thus, with the probability of, the th substream s feedback is correct. Then, the transmitter will select the optimal branch-and-antenna pair with the maximal SNR. However, with the probability of, the th substream s feedback is not correct. Then, the transmitter will select an arbitrary branch-and-antenna pair with its SNR being the nonmaximum. The set of those nonmaximum SNRs is the complementary set of the maximal SNR. Using the order statistics [17], the pdf of a random sample within those SNRs, denoted by, can be derived as follows: where we define (19) where is given by (9) and is given by (15). C. SER With Both Time-Delayed and Erroneous CSI Feedbacks Applying the analyses above, we derive the SER when jointly considering feedback delays and feedback errors. The scenario we consider is as follows. With the probability of, the th substream s feedback is correct. Since the feedback is also delayed by, the transmitter will select the branch-and-antenna pair, which has the delayed version of the maximum-snr, i.e., the concomitant to transmit data. On the other hand, with the probability of, the th substream s feedback is received in error. Also, considering the delay, the transmitter will select an arbitrary branch-and-antenna pair with the delayed version of the nonmaximum SNR, to transmit data. Using the induced order statistics [17], we derive the pdf of the delayed version of the nonmaximum SNRs, denoted by, as follows: (20) where is the special case of with. Similar to Section IV-B, the pdf, denoted by, for the combined SNR when jointly taking both feedback errors and delays into account, is given by (21) (16) where is defined in (17). Using (21), we derive their corresponding SER, denoted by as where and are given by (5), and is given by (6). Thus, we obtain the pdf, denoted by, for the combined SNR when taking the erroneous CSI feedbacks into account, as follows: (17) where we define (22) where we further define. Thus, considering the CSI feedback error, we get the corresponding SER, denoted by, as follows: (18) 2 In this paper, we focus on the system with uncoded CSI feedback for derivation convenience. For the system with coded CSI feedback, the analysis in this section also holds except that the expression of frame-error rate " should be changed. (23) where is given by (14) and is given by (15). V. APPLICATIONS TO DIFFERENT MULTIUSER MC DS-CDMA WIRELESS NETWORKS A. Downlink Cellular Networks In this section, we generalize and apply the proposed TSD/MRC-based MC DS-CDMA scheme to multiuser communications over different types of wireless networks. We

TANG AND ZHANG: TSD WITH MRC FOR MC DS-CDMA WIRELESS NETWORKS OVER NAKAGAMI- FADING CHANNELS 109 first consider the synchronous downlink of the cellular networks. We assume that the basestation synchronously transmits signals to mobile users, and the spread-codes assigned to different mobile users are all orthogonal with each other. Since each subcarrier signal experiences the flat-fading, the orthogonality between different spread codes can still be guaranteed. Therefore, the downlink cellular network has the near-single-user performance [1]. As a result, the analytical analyses derived in the previous sections can be directly applied and extended to the downlink cellular networks. 3 Note that certain nonideal factors, such as the nonzero interferences from neighbor cells and the Doppler frequency-drifts impairing the orthogonality between different subcarriers, may degrade the system performance. In regards to the first problem, the neighbor-cell interferences can be modeled as part of AWGN using the standard Gaussian approximation, provided that the number of interference terms is sufficiently large. For the second problem, the performance degradation is neglectable, as long as the mobilities of the users are not too high [1]. B. Uplink Cellular Networks We consider the applications of our proposed scheme to asynchronous uplink of the cellular networks, where mobile users asynchronously transmit signals to the base station. Under the assumption of perfect power control, the received signal at the th antenna at the base station can be expressed as (24) where we use the notations similar to those used in Section II except that the subscript or superscript is added to distinguish different users. We further assume that no multiuser detection (MUD) [24] techniques are employed. By using the framework developed in [2] and the standard Gaussian approximation, the combined SNR for the th mobile user of the th substream, denoted by, can be derived as, where denotes the effective AWGN for each substream, which is defined by (25) where denotes the effective average path gain for each substream at each receive antenna, which is determined by the general expression as follows: (26) where needs to be substituted by the specific pdf expressions [i.e., (6), (30), (17), or (21), respectively], depending on either perfect or imperfect CSI feedback being considered. Substituting the appropriate equation into (26) and solving the integral, we obtain the different effective average path gains s. 3 The results in the previous sections can be considered as the lower bound of SERs for the downlink cellular networks. Fig. 2. SER versus the average SNR with perfect and imperfect CSI feedbacks for our proposed TSD/MRC-based MC DS-CDMA scheme. The number of transmit and receive antennas are set to N =2 and N =1, respectively. The number of branches per substream is Q =2. By substituting the appropriate into given by (25) and letting the new average signal-to-interference-and-noise ratio (SINR) be, the results derived in Sections III and IV can be directly applied to the uplink cellular networks. Applying our proposed scheme over uplink cellular networks, all mobile users select the optimal branch-and-antenna pair transmitting signals to the base station. As a result, at the bases tation, although the received power of the useful signals for each user is enhanced, the strength of interferences is also increased. Noting that the capacity of the cellular networks is interference-limited, the overall uplink performance may degrade when employing the TSD/MRC-based MC DS-CDMA scheme due to the strong interference. C. Ad Hoc Wireless Networks In this section, we consider our proposed scheme applied in asynchronous ad hoc wireless networks, where pairs of mobile users communicate with each other in each pair asynchronously and independently. Without loss of generality, we focus on the first pair of mobile users in the following analyses. We call the th pair s transmitter and receiver by the th transmitter and th receiver, respectively. Then, the received signal at the th antenna of the first receiver, denoted by, can be expressed as (27) where superscript denotes the channel between the th transmitter and the first receiver. Comparing (27) with that at uplink cellular networks in (24), we can find that the useful signal parts and the noise terms have the same structure, while the interference terms are different. Noting that although the channel between the first transmitter and the first receiver (i.e., the useful signal channel) is optimal, the channels between the other transmitters and the first receiver (i.e., the interference channels) are random and independent. Thus, unlike what happening in the

110 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 24, NO. 1, JANUARY 2006 Fig. 3. SER P performance of our proposed TSD/MRC MC DS-CDMA scheme when jointly considering feedback delays and errors. N =2, N =1, and Q =2. (a) SER P versus feedback error rate P and delay f. (b) The projection of SER contour line onto (P ;f ) space. uplink cellular networks, the strong interferences will not be cumulated at the receiver in ad hoc wireless networks. Then, the SNR at the first receiver, denoted by, can be derived as, where denotes the path power between the first transmitter and the first receiver, and also represents the effective AWGN given by (28) which differs from (25) only in that the effective average path gain is in (28), instead of used in (25). VI. NUMERICAL RESULTS FOR PERFORMANCE EVALUATIONS Without loss of generality, we evaluate the performance of our proposed scheme using binary phase-shift keying (BPSK) modulation (with and ) over Rayleighfading channels. The number of substreams is set to and the spreading gain is set equal to. For comparison and illustration purposes, we plot the SERs of MRC-, STBC/MRC-, and SD/MRC-based MC DS-CDMA schemes whenever necessary. Note that both STBC/MRC and MRC are independent of feedbacks imperfectness. Fig. 2 plots the exact SERs and the corresponding Chernoff bounds versus the average SNR, where we also derive the closed-form Chernoff bounds for the SERs, but omit them for lack of space. As shown in Fig. 2, the feedback delay and the feedback error can significantly impact the SER of TSD/MRCbased scheme, especially, for the high SNRs. The plots marked with no CSI feedback means that the transmitter selects the antennas and subcarriers arbitrarily, which corresponds to the worst SER performance of the proposed scheme. Fig. 3(a) plots the SER of TSD/MRC scheme when both feedback delays and feedback errors vary, characterizing s general dynamics. Fig. 3(b) plots the projections of the two intersecting lines, where the SER of TSD/MRC scheme is equal to SERs of SD/MRC and STBC/MRC schemes, respectively. The regions within the projected lines determine the variation region of the tolerable imperfect feedbacks that ensure TSD/MRC outperforming SD/MRC and STBC/MRC, respectively. We can also see from Fig. 3(b) that in a wide range of feedback imperfectness, the TSD/MRC scheme outperforms the SD/MRC and STBC/MRC schemes. The results above can also be considered as the approximations of SERs over the downlink cellular networks. In Fig. 4, we evaluate the performance of TSD/MRC in uplink cellular networks and ad hoc wireless networks, respectively. We assume that we can upper bound the imperfectness of the feedback errors to make the SERs virtually unchanged with, such that only the feedback delay can impact the SER performance. Note that we set the feedback delay equal to, which can significantly deteriorate the SER performance of TSD/MRC scheme. Fig. 4(a) plots the SER against average SNR in uplink cellular networks. We can see from Fig. 4(a) that TSD/MRC-based schemes cannot guarantee the performance superiority over SD/MRC-based and STBC/MRC-based schemes. The feedback delay further degrades SER of TSD/MRC-based and SD/MRC-based schemes. In contrast, for the SER performance over ad hoc wireless networks as shown in Fig. 4(b), TSD/MRC-based schemes always have the better SER performance than the corresponding SD/MRC-based and STBC/MRC-based schemes, respectively, when the feedback is perfect. Even under the large feedback delays, the SER of TSD/MRC still performs the best among the three schemes. The larger the order of the transmit diversity, the higher the superiority of TSD/MRC over the other schemes.

TANG AND ZHANG: TSD WITH MRC FOR MC DS-CDMA WIRELESS NETWORKS OVER NAKAGAMI- FADING CHANNELS 111 Fig. 4. SER performance of TSD/MRC-based MC DS-CDMA scheme over uplink and ad hoc wireless networks. N =2, N =2, and K =60. (a) Uplink wireless networks. (b) Ad hoc wireless networks. VII. CONCLUSION We proposed the scheme that integrates TSD/MRC with MC DS-CDMA for diverse wireless networks. We also developed the analytical framework to analyze the SERs of the proposed scheme over Nakagami- fading channels when taking feedback delays and errors into considerations. The proposed scheme can significantly decrease the PAPR that is inherently associated with MC DS-CDMA communications systems. The resultant SERs are compared with those of SD/MRC-based and STBC/MRCbased MC DS-CDMA schemes in different wireless-network scenarios. Our analyses showed that in a wide variation of feedback imperfectness, the proposed TSD/MRC-based MC DS-CDMA scheme is better applicable to both downlink cellular networks and ad hoc wireless networks. However, the analyses also indicated that TSD/MRC-based MC DS-CDMA scheme cannot always outperform SD/MRC-based and STBC/MRCbased MC DS-CDMA schemes in uplink cellular networks due to the imposed stronger interference. APPENDIX DERIVATION OF EQ. (14) Expanding the Bessel function in (11) by an infinite series [23], and then solving the integral of (10), we can derive the closed-form expression of the pdf as follows: (29) where denotes the confluent (Kummer) hypergeometric function [23], which can be expressed as a finite series expansion by [25]. Thus, we obtain a more explicit closed-form expression for as follows: (30) Substituting (30) into (13) and using the approach proposed in [18], we obtain the closed-form expression of as shown in (14). ACKNOWLEDGMENT The authors would like to thank Prof. H.-H. Chen and the anonymous reviewers for their valuable comments. REFERENCES [1] L.-L. Yang and L. Hanzo, Multicarrier DS-CDMA: A multiple access scheme for ubiquitous broadband wireless communications, IEEE Commun. Mag., pp. 116 124, Oct. 2003. [2], Performance of generalized multicarrier DS-CDMA over Nakagami-m fading channels, IEEE Trans. Commun., vol. 50, no. 6, pp. 956 966, Jun. 2002. [3] X.-K. Zhao and X.-D. Zhang, Peak-to-average power ratio analysis in Multicarrier DS-CDMA, IEEE Trans. Veh. Technol., vol. 52, no. 3, pp. 561 568, May 2003. [4] E. A. Sourour and M. Nakagawa, Performance of orthogonal multicarrier CDMA in a multipath fading channel, IEEE Trans. Commun., vol. 44, no. 3, pp. 356 367, Mar. 1996. [5] T. Skinner and J. Cavers, Selective diversity for Rayleigh fading channels with a feedback link, IEEE Trans. Commun., vol. 21, no. 2, pp. 117 126, Feb. 1973.

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New York: Dover, 1965. [24] S. Verdu, Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, 1998. [25] [Online]. Available: http://functions.wolfram.com/07.20.03.0025.01 Xi Zhang (S 89 SM 98) received the B.S. and M.S. degrees from Xidian University, Xi an, China, the M.S. degree from Lehigh University, Bethlehem, PA, all in electrical engineering and computer science, and the Ph.D. degree in electrical engineering and computer science (Electrical Engineering Systems) from The University of Michigan, Ann Arbor. He is currently an Assistant Professor and the Founding Director of the Networking and Information Systems Laboratory, Department of Electrical and Computer Engineering, Texas A&M University, College Station. He was an Assistant Professor and the Founding Director of the Division of Computer Systems Engineering, Department of Electrical Engineering and Computer Science, Beijing Information Technology Engineering Institute, Beijing, China, from 1984 to 1989. He was a Research Fellow with the School of Electrical Engineering, University of Technology, Sydney, Australia, and the Department of Electrical and Computer Engineering, James Cook University, Queensland, Australia, under a Fellowship from the Chinese National Commission of Education. He worked as a Summer Intern with the Networks and Distributed Systems Research Department, Bell Laboratories, Murray Hills, NJ, and with AT&T Laboratories Research, Florham Park, NJ, in 1997. He has published more than 70 technical papers. His current research interests focus on the areas of wireless networks and communications, mobile computing, cross-layer designs and optimizations for QoS guarantees over mobile wireless networks, wireless sensor and Ad Hoc networks, wireless and wireline network security, network protocols design and modeling for QoS guarantees over multicast (or unicast) wireless (or wireline) networks, statistical communications theory, random signal processing, and distributed computer-control systems. Dr. Zhang received the U.S. National Science Foundation CAREER Award in 2004 for his research in the areas of mobile wireless and multicast networking and systems. He served or is serving as the Panelist on the U.S. National Science Foundation Research-Proposal Review Panel in 2004, the WiFi-Hotspots and QoS Panelist at the IEEE QShine 2004, as the Symposium Chair for the IEEE International Cross-Layer Designs and Protocols Symposium within the IEEE International Wireless Communications and Mobile Computing Conference (IWCMC) 2006, the Technical Program Committee Co-Chair for the IEEE IWCMC 2006, the Poster Chair for the IEEE QShine 2006, the Publicity Co-Chair for the IEEE WirelessCom 2005, and as the Technical Program Committee members for IEEE GLOBECOM, IEEE ICC, IEEE VTC, IEEE QShine, IEEE WoWMoM, IEEE WirelessCom, and IEEE EIT. He is a member of the Association for Computing Machinery (ACM). Jia Tang (S 03) received the B.S. degree in electrical engineering from Xi an Jiaotong University, Xi an, China, in 2001. He is currently a Research Assistant working towards the Ph.D. degree in the Networking and Information Systems Laboratory, Department of Electrical and Computer Engineering, Texas A&M University, College Station. His research interests include mobile wireless communications and networks, with emphasis on cross-layer design and optimizations, wireless quality-of-service (QoS) provisioning for mobile multimedia networks, wireless diversity techniques, and wireless resource allocation. Mr. Tang received the Fouraker Graduate Research Fellowship Award from the Department of Electrical Engineering, Texas A&M University in 2005.