Bayesian Monte Carlo multiuser receiver for space-time coded multicarrier CDMA systems

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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 19, NO. 8, AUGUST 2001

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Bayesian Monte Carlo Multiuser Receiver for Space–Time Coded Multicarrier CDMA Systems Zigang Yang, Ben Lu, and Xiaodong Wang

Abstract—We consider the design of optimal multiuser receivers for space–time block coded (STBC) multicarrier code-division multiple-access (MC-CDMA) systems in unknown frequency-selective fading channels. Under a Bayesian framework, the proposed multiuser receiver is based on the Gibbs sampler, a Markov chain Monte Carlo (MCMC) method for numerically computing the marginal a posteriori probabilities of different users’ data symbols. By exploiting the orthogonality property of the STBC and the multicarrier modulation, the computational complexity of the receiver is significantly reduced. Furthermore, being a soft-input soft-output algorithm, the Bayesian Monte Carlo multiuser detector is capable of exchanging the so-called extrinsic information with the maximum a posteriori (MAP) outer channel code decoders of all users, and successively improving the overall receiver performance. Several practical issues, such as testing the convergence of the Gibbs sampler in fading channel applications, resolving the phase ambiguity as well as the antenna ambiguity, and adapting the proposed receiver to multirate MC-CDMA systems, are also discussed. Finally, the performance of the Bayesian Monte Carlo multiuser receiver is demonstrated through computer simulations. Index Terms—Frequency-selective fading, Gibbs sampler, Monte Carlo signal processing, multicarrier CDMA (MC-CDMA), multiuser detection, space–time code (STC), turbo processing.

I. INTRODUCTION

T

HE INTEREST in code-division multiple-access (CDMA) technology has been increasing over the past decades. Recently, as promising candidates for the third-generation (3G) wideband CDMA (WCDMA) systems, different multicarrier CDMA (MC-CDMA) systems have been proposed and investigated [1], [2]. In particular, in [3], an MC-CDMA system which transmits parallel multiple narrowband direct-sequence (DS) waveforms on different frequency carriers is proposed. In contrast, by combining the orthogonal frequency-division multiplexing (OFDM) technique with CDMA, another type of MC-CDMA system which transmits spread signals on different frequency subcarriers is proposed in [4]. The scheme proposed in [3] is more compatible with the currently deployed second-generation (2G) CDMA cellular systems. On the other hand, without compromising the system performance, the scheme in [4] can further reduce the receiver complexity by taking advantage of the special structure of the OFDM multicarrier modulation. In this paper, we follow the OFDM Manuscript received December 14, 2000; revised May 24, 2001. This work was supported in part by the U.S. National Science Foundation under CAREER Grant CCR-9875314, Grant CCR-9980599, and Grant DMS-0073651. The authors are with the Department of Electrical Engineering, Texas A&M University, College Station, TX 77843-3128 USA (e-mail: [email protected]). Publisher Item Identifier S 0733-8716(01)07229-8.

MC-CDMA scheme in [4] and consider the design of optimal multiuser receivers for this system. To provide high data rate communications and to increase the system capacity are among the key objectives of the 3G WCDMA systems. Recently, employing multiple antennas at both the transmitter and the receiver is shown to be able to significantly increase the system capacity [5], [6]. Moreover, space–time coding (STC) techniques, including space–time trellis coding (STTC) and space–time block coding (STBC), are proposed to exploit both spatial diversity and channel coding in multiple-antenna wireless systems [7], [8]. Due to its simplicity, the STBC has been proposed as part of several standards for 3G WCDMA systems, e.g., [9], [10]. In this paper, we also consider the STBC in MC-CDMA systems. The theme of this paper is on the optimal multiuser receiver design for STBC MC-CDMA systems in unknown frequency-selective fading channels. The problem is formulated under a Bayesian framework, where multiple users’ data symbols are to be estimated from the received signals, in the presence of unknown fading channels. Direct implementation of the optimal solution is computationally prohibitive. Hence, we resort to the Markov chain Monte Carlo (MCMC) approach [11], a powerful numerical Bayesian computation methodology. Recently, in [12], the Gibbs sampler, the most popular MCMC method, is applied to the problem of optimal multiuser detection in synchronous CDMA systems over Gaussian and non-Gaussian noise channels. In this paper, following the framework of [12], we further consider the application of the Gibbs sampler in designing multiuser receivers for STBC MC-CDMA systems over unknown frequency-selective fading channels. The contributions of this paper are as follows. • In [12], simple real-valued synchronous CDMA systems are considered. In this paper, we propose and derive a blind receiver for multicarrier CDMA in the more general frequency-selective fading channels. Moreover, we discuss the problem of resolving ambiguities, i.e., the phase-ambiguity and the antenna-ambiguity, in such systems. • Compared with its applications in additive white Gaussian noise (AWGN) channels [12], the Gibbs sampler is more prone to diverge (after a finite number of iterations) in frequency-selective fading channels; we, therefore, propose a pragmatic approach to monitoring the convergence of the Gibbs sampler, by counting the number of errors corrected by the outer channel decoder. • By exploiting the orthogonality property of the STBC and the OFDM multicarrier modulation, we show that no matrix inversion is needed in the Gibbs-sampler-based

0733–8716/01$10.00 ©2001 IEEE

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Bayesian blind receiver; hence, it has a relatively low computational complexity and is more numerically stable. • We show that with minor modifications, the proposed receiver for single-rate STBC MC-CDMA systems can be easily adapted to multirate STBC MC-CDMA systems. The rest of this paper is organized as follows. In Section II, the system model of an STBC MC-CDMA system is described, and the problem of optimal receiver design is stated; furthermore, the extension of the Bayesian Monte Carlo receiver to multirate STBC MC-CDMA systems is discussed. In Section III, a Bayesian Monte Carlo multiuser receiver is derived for STBC MC-CDMA systems without outer channel coding. For STBC MC-CDMA systems employing outer channel codes, a turbo receiver, which consists of the Bayesian Monte Carlo multiuser detector and a bank of channel decoders of all users, is proposed in Section IV. In Section V, computer simulation results are provided. Section VI contains the conclusion.

Fig. 1. MC-CDMA modulator structure of the k th user, where T represents an interleaver.

Fig. 2. Transmitted signal structure of a particular STBC slot of the k th user in an STBC MC-CDMA system.

II. SYSTEM DESCRIPTION A. Transmitter Signal Model of STBC MC-CDMA Systems As mentioned in the previous section, the MC-CDMA system considered here transmits the spread signals (so-called “chips”) from different frequency subcarriers. Due to the cyclic prefix insertion, the received signals are free of intersymbol interference (ISI) [1], [2]. Hence, the traditional equalizer [13] is not needed and the receiver complexity is significantly reduced, which is very desirable in high data rate applications. Next, we consider a -user MC-CDMA system with subcarriers, where different users transmit information at the same rate. (The MC-CDMA system which supports different information rates for different users will be discussed in Section II-F.) The spreading gain of each user is , an integer that divides ; symconsequently, at each time slot, for each user, bols are spread and transmitted over subcarriers. It is assumed users in the system transmit their signals synchrothat all nously. The structure of the MC-CDMA modulator of the -th user is illustrated in Fig. 1. A stream of BPSK symbols of the -th user are first serial-to-parallel converted, where every symbols are grouped into a vector as (1) in is spread by a spreading sequence . The spreading sequences corresponding to all binary phase-shift keying (BPSK) symbols can be denoted by an -vector as

Next, symbol

(2) and the spread signals can also be denoted by an

-vector as (3)

in (3) will be used We remark that the notation of throughout this paper to represent the spreading operation, instead of the conventional vector multiplication. Finally, in order to avoid the strong correlation among the subcarriers

occupied by a particular symbol, all spread chips in (3) are interleaved before they are transmitted from the subcarriers. in Fig. 1; for The interleaving function is denoted by simplicity, the interleaver is assumed to be the same for all users. Without incurring any power or bandwidth penalty, the space–time block codes (STBC) [14] are employed in the MC-CDMA system considered here to further increase the system capacity. Originally, the STBC was proposed to transmit scalar symbols. In this paper, we extend it to the vector form and apply it in MC-CDMA systems. In vector form, the STBC proposed in [14], as illustrated in Fig. 2, simplest [cf. takes two time slots to transmit two symbol vectors is interleaved, (1)]. At the first time slot, the spread vector are transmitted from the and the interleaved signals first transmitter antenna on subcarriers; meanwhile, another is transmitted from the interleaved spread vector second transmitter antenna. At the second time slot, the symbol is spread by the spreading sequence and the vector is transmitted from the interleaved spread vector is spread first transmitter antenna; while the symbol vector by the spreading sequence and the interleaved spread signals is transmitted from the second transmitter antenna. From Fig. 2, we can see that different spreading sequences and are assigned to different transmitter antennas; this structure is shown to be an efficient way to revolve the so-called “antenna-ambiguity” in blind algorithms, as will be discussed in Section II-E. Finally, the structure of the transmitter for the th user is depicted in Fig. 3. B. Frequency-Selective Fading Channels For notational convenience, in this paper, we only consider the system with one receiver antenna; and the extension to multiple receiver antennas is straightforward. Consider the channel response of the th user between the th transmitter antenna and the receiver antenna. Following [13], at

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fading processes are time-invariant within one signal frame, the received signal model can be written as

Fig. 3. Transmitter structure of the k th user in an STBC MC-CDMA system.

a particular time slot, the time-domain channel impulse response can be modeled by a tap-delay line as (8)

(4) is the Kronecker delta function; , where being the maximum multipath spread of all users with (note that we have assumed the synchronous transmission of being the whole bandwidth of multicarrier all users), and is the complex amplitude of the th tap associated systems; with the th transmitter antenna of the th user, whose relative . delay is For MC-CDMA systems with proper cyclic extensions and sample timing, with tolerable leakage, the channel frequency response of the th user at its th transmitter antenna and at the th subcarrier can be expressed as [15]

(5) where (6) contains the time response of all

taps; and

(7) contains the corresponding discrete Fourier transform (DFT) coefficients.

with diag

diag diag

where Kronecker matrix product; symbol vectors input to the STBC encoder at the th STBC slot; spreading sequence assigned to the th transmitter antenna of the th user; permutation matrix, which acts as chips to their an interleaver mapping the assigned subcarriers; received signal during the th STBC slot; ambient noise, which is circularly symmetric complex Gaussian with covariance matrix . Note that (8) can be used to describe both the slow-fading is large) and the fast-fading (when is small, e.g., (when ) cases. However, the fading channels are assumed to be static during two neighboring time slots (i.e., one STBC slot), this is the only limitation of the signal model in (8). According to the definitions in (8), we have ,

C. Receiver Signal Model of STBC MC-CDMA Systems users, as described in SecThe transmitted signals of all tion II-A, propagate through their respective frequency-selective fading channels and finally reach the receiver antenna. It is assumed that the fading processes associated with different transmitter-receiver antenna pairs are uncorrelated. At the receiver, after matched filtering and chip-rate sampling, the discrete Fourier transform (DFT) is then applied to the received discrete-time signals. Consider all the DFT-ed signals in one STBC slots or equivalently signal frame, which spans time slots. [Recall that each STBC slot consists of two neighboring time slots (see Fig. 2).] Using (5) and assuming that the

(9) where (9) follows from the orthogonality property of the STBC, , and the orthogonality property of the i.e., , as well OFDM multicarrier modulation, i.e., as from the fact that the permutation matrix satisfies . As will be seen in the following sections, the structure of (9) can be exploited to reduce the computational complexity of the optimal receiver for the STBC MC-CDMA system. Note that the interleaver function is the same for all users; hence, the signal model (8) can be written in an alternative

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form, which decouples the signal components corresponding to different symbols

with the correct order will exhibit only a few bit corrections, whereas the one with the incorrect order will exhibit many corrections. E. Bayesian Optimal Blind Receiver

(10) is a submatrix decimated from , where which contains all the rows and columns related to the symbol and ; , and are then the corresponding decimations from , , and . Equation (10) will be used later in deriving the conditional posterior distributions of the unknown symbols [see (26)]. D. Resolving Ambiguity in Blind Approach In this paper, we consider the design of blind optimal multiuser receiver for the STBC MC-CDMA system. Since the channel state information is unknown to the receiver, there are two types of ambiguities inherent in the design of blind receiver for the STBC MC-CDMA system: phase ambiguity and antenna ambiguity. To resolve the phase ambiguity, differential encoding is employed before the STBC encoding. For each signal frame, a block of BPSK bits is input to the differential encoder, and the output is given by .

(11)

are the same set of These differentially encoded bits [cf. (8)] input to the STBC encoder, where bits . Henceforth, they are related as is understood as an implicit function of the the index of of . index Antenna ambiguity is the uncertainty to tell from which transmitter antenna a particular symbol was transmitted, which is inherent to any blind receiver in an STBC system. One possible approach to resolve the antenna ambiguity is to employ the differential space–time modulation as proposed in [16]; however, in that case, the signal constellation will be changed. Fortunately, in the STBC MC-CDMA system, the antenna ambiguity can be resolved by using different spreading sequences on different transmitter antennas. Note that the usage of orthogonal sequences will result in a maximum 50% system loading ) in the STBC MC-CDMA system. (defined as On the other hand, the channel frequency selectivity destroys the orthogonality of the orthogonal sequences. Hence, in this paper, random sequences are employed, which provide sufficient number of spreading sequences and set no constraint on the system loading. When the system employs the outer channel code, it is possible to use the same spreading sequence at different antennas of the same user. In this case, the antenna ambiguity can be resolved by exploiting the coding structure. For example, for the system considered here, due to the antenna ambiguity, we have two possible code bit sequences at the output of the multiuser detector. We can send both of them to the channel decoder, and count the number of bit corrections. Then, the code bit sequence

Denote ; ; ; . The optimal blind receiver estimates the a posteriori probabilities of the multiuser data bits (12) based on the received signals , the signal structure (8), the spreading sequences of all users, and the prior information of , without knowing the channel response and the noise variance . The Bayesian solution to (12) is given by

(13) is a Gaussian density function [see (8)]; where , and are prior distributions of the independent and respectively. Clearly the and unknown quantities , computation in (13) involves a very high-dimensional integral which is certainly infeasible for any practical implementations. Thus, we resort to the Gibbs sampler, a Monte Carlo method, to calculate the a posteriori probabilities of the unknown symbols. F. Multirate STBC MC-CDMA System In the previous sections, we have assumed that the spreading gain is the same among all the users. Here, we extend our discussion to multirate STBC MC-CDMA systems by allowing the to be a user-dependent parameter. That is, each spread gain user can choose the spreading gain according to its own transmission rate—a lower spreading gain corresponds to a higher transmission rate. Consider a -user multirate STBC MC-CDMA system with subcarriers. Assume that for the -th user, the spreading gain ; then at each time slot, symbols are transis subcarriers. The transmitter structure of the mitted over the th user is the same as in Fig. 3. At the receiver, similar to (8), the received signal after DFT is given by

(14) with diag

diag diag

YANG et al.: BAYESIAN MONTE CARLO MULTIUSER RECEIVER FOR STC MC-CDMA SYSTEMS

Similar to (10), the signal model in (14) can be written in an alternative form as follows

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1) For the unknown channel distribution is assumed

, a complex Gaussian prior (16)

(15) -vector decimated from with indexes , where these elements coland ; is a lect all the information about matrix decimated from with all the rows and ; and are then the corcolumns in the index set and . It is easy to show that responding decimations from the orthogonality property of (9) is preserved in multirate systems. Next, we will focus on the design of the Bayesian Monte Carlo multiuser receiver for single-rate STBC MC-CDMA systems, which can be readily extended to multirate systems with minor modifications.

In (15),

is a

corresponds to less informaNote that large value of tive prior. 2) For the noise variance , an inverse chi-square prior distribution is assumed (17) corresponds to the less informative Small value of priors. 3) The data bit sequence is a Markov chain, encoded from . Its prior distribution can be expressed as

III. BLIND BAYESIAN MONTE CARLO MULTIUSER RECEIVER In this section, we consider the problem of computing the a posteriori bit probabilities in (12). The problem is solved under a Bayesian framework, by treating the unknown quantities as realizations of random variables with some prior distributions. The Gibbs sampler [17] is then employed to compute the Bayesian estimates.

(18) where (LLR) of

denotes the a priori log-likelihood ratio , i.e.,

A. Gibbs Sampler

(19)

The Gibbs sampler [17] is a Markov chain Monte Carlo (MCMC) procedure for numerical Bayesian computation. Let be a vector of unknown parameters. Let be the observed data. To generate random samples from , given the samples at the joint posterior distribution th iteration, , at the the th iteration, the Gibbs algorithm iterates as follows to obtain .

samples • For tion

, draw

from the conditional distribu-

Note that in (18), we set . phase ambiguity in

to count for the

C. Conditional Posterior Distributions The following conditional posterior distributions are required by the Bayesian multiuser detector. The derivations can be found in the Appendix. 1) The conditional distribution of the th user’s channel regiven , , , and is [where sponse .] (20)

It is known that under regularity conditions [18]–[21]: , 1) the distribution of converges geometrically to ; as , as , for 2) any integrable function . Hence, the marginal a posteriori distribution of any parameter can be computed easily from the samples drawn by the Gibbs sampler.

with

(21) and

B. Prior Distributions For simplicity, we choose the sampling space, the set of unknown parameters sampled by Gibbs sampler, to be , which are assumed to be independent with each and other. Next, we specify their prior distributions .

(22)

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where (21) follows from the orthogonality property in (9). 2) The conditional distribution of the noise variance given , , and is given by

where

is an indicator such that if if

.

(28)

(23) E. Discussions with (24) given 3) The conditional distribution of the data bit , , , and can be obtained from [where .]

(25) with

(26) On the right side of (25), the first two items correspond to the prior information of the differentially encoded bits , through which the prior information of data bits is incorporated. D. Gibbs Multiuser Detection Given the initial values of the unknown quantities drawn from their prior distributions (16)–(18), at the th iteration, the Gibbs multiuser detector operates as follows. , draw from 1) For given by (20), [where ]; from given by (23); 2) Draw , For , Draw 3) For from given by (25), [where ]. To ensure convergence, the Gibbs iteration is usually carried iterations. The first iterations is called the out for burn-in period and only the samples from the last iterations are used to calculate the Bayesian inference. In particular, the can be obposterior distribution of the multiuser data bits tained by

(27)

Next, we briefly address two notable perspectives in the design of Bayesian blind multiuser receiver in STBC MC-CDMA systems. Choosing the Sampling Space of Data: As we are interested in computing the posterior probabilities of the multiuser data bits, direct sampling can be done on the data bits . Note given , that the conditional posterior distribution of , , and involves large number of received signals, . This long memory in the receiver signal i.e., processing will increase the computational complexity and decrease the convergence speed of the Gibbs procedure. To avoid these disadvantages, the Gibbs procedure designed here samples the differentially encoded bits , and outputs . It is shown in (27) that the a sampling sequence can be computed easily marginal posterior probability of . from the output samples Exploiting the Orthogonality Property: The dominant computations involved in the Gibbs sampler are steps (21), (22). By exploiting the orthogonality property (9) in STBC MC-CDMA in (21) is simply a constant matrix; systems, the matrix , moreover, with no matrix inversion involved in computing the numerical stability is also improved. IV. BLIND TURBO MULTIUSER RECEIVER In this section, we consider employing iterative multiuser detection and decoding to improve the performance of the Bayesian multiuser receiver in a coded STBC MC-CDMA system. Because it utilizes the a priori bit probabilities, and it produces the a posteriori bit probabilities, the Bayesian multiuser detector is well suited for iterative processing, which allows the multiuser detector to refine its processing based on the information from the decoding stage and vice versa. The th user’s transmitter structure is shown in Fig. 4, a are encoded using some block of information bits channel code (e.g., block code, convolutional code, or turbo code). A code-bit interleaver is used to reduce the influence of error bursts at the input of the channel decoder. The interleaved . These code bits are then mapped to BPSK symbols BPSK symbols are differentially encoded to yield the symbol , which are then serial-to-parallel converted and stream to feed into the STBC reorganized in a vector form as encoder followed by the MC-CDMA modulator, and finally transmitted from two antennas. An iterative (turbo) receiver structure is shown in Fig. 5. It consists of two stages: the Bayesian multiuser detector developed in the previous sections, followed by a soft-input softoutput channel decoder. The two stages are separated by a deinis mapped into terleaver and an interleaver. Assume that after deinterleaving. In the first stage, the blind Bayesian multiuser detector in, which is comcorporates the a priori information

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Fig. 4. Transmitter structure of the k th user in an STBC MC-CDMA system employing outer channel code, where 5 represents an interleaver.

Fig. 5.

Turbo Bayesian multiuser receiver structure, where 5 denotes an interleaver and 5

puted by the channel decoder in the previous iteration. At the first iteration, it is assumed that all code bits are equally likely. At the output of the blind Bayesian multiuser detector, the a posteriori LLR is given by (29) According to the “turbo principle” [22]–[24], the a priori inforshould be subtracted from the a posteriori mation to obtain the extrinsic information to deliver to LLR the channel decoders. However, the posterior distribution delivered by the blind Bayesian multiuser detector is a quantized value instead of the true value due to the finite number of samples. Therefore, to ensure numerical stability, the posterior LLR is regarded as the approximated extrinsic information, deinterleaved and fed back to the channel decoder of the th user. The soft-input soft-output channel decoder, using the MAP decoding algorithm [25], computes the a posteriori LLR of each code bit of the th user

(30) It is seen from (30) that the output of the MAP decoder is the , and the extrinsic sum of the prior information delivered by the channel decoder. information After interleaving, the extrinsic information delivered by the is then feedback to the channel decoder blind Bayesian multiuser detector as the refined prior informa[see (19)] for the next iteration. tion Decoder-Assisted Convergence Assessment: Although it is desirable to have the Gibbs sampler to reach convergence within the burn-in period ( iterations), this may not always be the case. Hence, we need some mechanism to detect the convergence. In the coded system considered here, the blind multiuser

denotes the corresponding deinterleaver.

detector is followed by a bank of channel decoders, we can assess convergence by monitoring the number of bit corrections made by the channel decoders [12] (The number of corrections is determined by comparing the the signs of the code-bit LLR at the input and output of the MAP channel decoder). If this number exceeds some predetermined threshold, then we decide convergence is not achieved. In that case, the Gibbs multiuser detector will be applied again to the same data block. V. SIMULATION RESULTS In this section, we provide computer simulation results to illustrate the performance of the proposed Bayesian multiuser receivers in the STBC MC-CDMA system, where there are two subcarriers and transmitter antennas, one receiver antenna, users. All spreading sequences used in simulations are randomly and independently generated for each transmitter antenna of each user. The frequency-selective fading channels are assumed to be uncorrelated and have the same statistics for different transmitter-receiver antenna pairs. For simplicity, all taps of a particular fading channel are assumed to be equal, and have depower, normalized such that , ; and all users in lays as the system are assumed to have equal transmission power. Such a system setup is also the worst case scenario from the interference mitigation point of view [26]. For STBC MC-CDMA systems employing outer channel codes, a four-state, rate-1/2 convolutional code with generator (5,7) in octal notation is chosen for all users. For each block of received signals, samples are drawn by the Gibbs sampler, with the first samples discarded. As discussed in Section IV, at the end of the 100 Gibbs iterations, the convergence of the Gibbs sampler is tested. In very few cases, when the Gibbs sampler is not convergent, it gets restarted for another round of 100 Gibbs iterations. The performance is demonstrated in two forms: one is in terms of the bit-error-rate (BER) and OFDM word-error-rate (WER) versus the number of users at a particular signal-noise[cf. (8)]; the other is in terms ratio (SNR), where

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Fig. 6. BER and OFDM WER of an STBC MC-CDMA system in two-tap frequency-selective fading channels, where

of the BER/WER versus SNR for the system with a particular number of users. A. Example 1: Slow Frequency-Selective Fading In this simulation example, we present the performance of the Bayesian multiuser receiver in an STBC MC-CDMA system transmitting over very slow fading channels, i.e., the fading channel is assumed to be fixed within one signal frame STBC slots. The multicarrier system that consists of subcarriers which can resolve two multipaths with is simulated. The spreading gain of this system is , and each equal to the number of subcarriers, i.e., symbol is spread and transmitted from all subcarriers. First, without employing outer convolutional code, in Fig. 6, the performance of a particular user transmitting at a particdB in systems with different number of users ular SNR is demonstrated in solid curves. Define the . Hence, in this simulation exsystem loading ratio as ample, the system loading ratio ranges from 37.5% to 87.5%. For comparison, in Fig. 6, we also show in dashed curves the performance of a single-user STBC multicarrier system with ideal channel state information (CSI) at the receiver, which is the lower bound to the optimal performance of a particular user in multiuser systems [26]. In order to test the convergence of the Gibbs sampler, the detected data from the Gibbs sampler are compared with the true transmitted data (by the aid of a genie). We observed that the average percentage of restarting the , Gibbs sampler is around 3%–5% at the OFDM WER of which only brings slight receiver complexity increase. Although in practice no genie exists, the performance in Fig. 6 can serve as a reference of the best performance the Gibbs sampler can achieve.

N = 16, G = 16, L = 2, SNR = 10 dB.

Secondly, in systems employing outer convolutional codes, the performance of the turbo multiuser receiver is shown in Fig. 7. For comparison, in the figure, we also include an approximated lower bound in dashed lines, which is obtained by performing the MAP demodulator in a single-user STBC multicarrier system with ideal CSI and iterating sufficient number of turbo iterations (five iterations in our simulations) between the MAP demodulator and the MAP convolutional decoder, similarly as in [27]. As discussed in Section IV, besides providing the coding gain, the outer channel code also helps to assess the convergence of the Gibbs sampler. Moreover, it is seen from Fig. 7 that the performance of the proposed Bayesian Monte Carlo turbo multiuser receiver can perform consistently and close to the approximated lower bound as the system loading ratio varying from 37.5% to 87.5%. B. Example 2: Fast Frequency-Selective Fading In this simulation example, we present the performance of the Bayesian multiuser receivers in medium to fast fading channels, i.e., the fading channel coefficients are assumed to be fixed but are changing from one during each STBC slot STBC slot to another. Simulations are carried out in a multisubcarriers which can resolve as carrier system with multipaths. (It has been assumed that the bandmany as width of each subcarrier is the same as that in simulation ex; hence, ample 1.) The spreading gain of this system is symbols of each user are transmitted at altogether each STBC slot simultaneously. The performance of this system is shown in Figs. 8 and 9, where system loading ratios are the same as in simulation example 1. With ideal CSI at the receiver, the system simulated here should expect better performance, due to its higher diversity

YANG et al.: BAYESIAN MONTE CARLO MULTIUSER RECEIVER FOR STC MC-CDMA SYSTEMS

Fig. 7. where

N

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BER and OFDM WER of an STBC MC-CDMA system employing outer convolutional channel code in two-tap frequency-selective fading channels, = 16, = 16, = 2, SNR = 5 dB.

G

L

Fig. 8. BER and OFDM WER of an STBC MC-CDMA system in five-tap frequency-selective fading channels, where

order , compared with the system considered in the . However, more resolvable multiprevious example paths also lead to the decrease of the SNR at each path, which has negative impact on the performance since the CSI usually needs to be measured in practice. In other words, in a practical system, if not properly handled, a highly frequency-selective fading channel does not necessarily bring performance improvement.

N = 256, G = 16, L = 5, SNR = 6 dB.

The performance of the uncoded and the convolutionally coded system is presented respectively in Figs. 8 and 9. First of all, it is seen that the Bayesian Monte Carlo multiuser receivers can properly exploit the frequency-selective fading diversity resources in fading channels, and needs less SNRs to achieve the same performance as that in Figs. 6 and 7. Second, it is seen that in this highly frequency-selective , the performance degradation due to the channel

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Fig. 9. BER and OFDM WER of an STBC MC-CDMA system employing outer convolutional channel code in five-tap frequency-selective fading channels, = 256, = 16, = 5, SNR = 2 dB. where

N

G

L

Fig. 10. BER and OFDM WER of an STBC MC-CDMA system employing outer convolutional channel code in five-tap frequency-selective fading channels, where = 256, = 16, = 5, = 12.

N

G

L

K

increased number of users is more evident compared with that in the previous example. (Note that this effect is not unexpected, since the number of unknowns increases with respect to the number of users.) However, it is also seen that no abrupt performance degradation occurs as the system

loading increases, which is a favorable property for practical applications. Moreover, the performance is demonstrated through BER/WER versus SNR in Fig. 10 for an STBC MC-CDMA and employing outer system with twelve users

YANG et al.: BAYESIAN MONTE CARLO MULTIUSER RECEIVER FOR STC MC-CDMA SYSTEMS

Fig. 11.

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BER and OFDM WER of an STBC MC-CDMA system employing outer convolutional channel code in frequency-selective fading channels, where

N = 256, G = 16, L = 5, K = 12. The same spreading sequence is used on two transmitter antennas of each user. convolutional code. It is seen that through turbo iterations, the receiver performance is significantly improved and after five iterations is only around 2 dB away from the approximated . lower bound at WER of Finally, for comparison, the performance of a twelve-user system using the same random spreading sequence on two transmitter antennas of each user, as that discussed in Section II-E, is shown in Fig. 11. It is seen that the performance is about 1.5 dB . This is because worse than that in Fig. 10 at the BER of when the same spreading sequence is used at both transmitter antennas, the signals are separable only in the spatial domain but not in the time domain. Hence, for better performance and less complexity, we advocate the use of the different spreading sequences on different transmitter antennas.

as the antenna-ambiguity, and adapting the proposed receiver in multirate MC-CDMA systems, have also been discussed. From computer simulation results, the proposed Bayesian Monte Carlo multiuser receiver exhibits good performance in randomly generated frequency-selective fading channels. APPENDIX A. Derivation of (20) See the first equation at the top of the next page. B. Derivation of (23)

VI. CONCLUSION In this paper, we have considered the design of optimal multiuser receivers for STBC MC-CDMA systems in unknown frequency-selective fading channels. The receiver is based on the Gibbs sampler, an MCMC method for numerically computing the marginal a posterior probabilities of different users’ data symbols. By exploiting the orthogonality property of the STBC and the OFDM multicarrier modulation, the computational complexity of the receiver is significantly reduced. Meanwhile, being a soft-input soft-output algorithm, the Bayesian Monte Carlo multiuser detector is capable of exchanging the so-called extrinsic information with the MAP outer channel code decoders, and successively improving the overall receiver performance. Several practical issues, such as testing the convergence of the Gibbs sampler in fading channel applications, resolving the phase-ambiguity as well

C. Derivation of (25) See the second equation at the top of the next page. As exis also denoted as , and plained in Section II-E,

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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 19, NO. 8, AUGUST 2001

among all the received signal, only is related to can be further simplified into Hence,

.

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[3] S. Kondo and L. B. Milstein, “Performance of multicarrier CDMA systems,” IEEE Trans. Commun., vol. 44, pp. 238–246, Feb. 1996. [4] A. Chouly, A. Brajal, and S. Jourdan, “Orthogonal multicarrier techniques applied to direct sequence spread spectrum cdma systems,” in IEEE Globecom Conf., Houston, TX, Nov. 1993, pp. 1723–1728. [5] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Pers. Commun., vol. 6, pp. 311–335, Mar. 1998. [6] I. E. Telatar, “Capacity of multi-antenna gaussian channels,” Eur. Trans. Telecommun, vol. 10, pp. 585–595, Nov. 1999. [7] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: performance criterion and code construction,” IEEE Trans. Inform. Theory, vol. 44, pp. 744–765, Mar. 1998. [8] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inform. Theory, vol. 45, pp. 1456–1467, Jul. 1999. [9] Space-time block coded transmit antenna diversity for WCDMA, Texas Instruments, Dec. 1998. [10] Downlink diversity improvements through space-time spreading, Lucent Technologies, Aug. 1999.

YANG et al.: BAYESIAN MONTE CARLO MULTIUSER RECEIVER FOR STC MC-CDMA SYSTEMS

[11] M. A. Tanner, Tools for Statistics Inference. New York: Springer–Verlag, 1991. [12] X. Wang and R. Chen, “Adaptive bayesian multiuser detection for synchronous CDMA with gaussian and impulsive noise,” IEEE Trans. Signal Processing, vol. 48, pp. 2013–2028, July 2000. [13] J. G. Proakis, Digital Communications, 3rd ed. New York: McGrawHill, 1995. [14] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Select. Areas Commun., vol. 16, pp. 1451–1458, Oct. 1998. [15] J.-J. van de Beek, O. Edfors, M. Sandell, S. K. Wilson, and P. O. Börjesson, “On channel estimation in OFDM systems,” presented at the IEEE Vehicular Technology Conf., Chicago, IL, July 1995. [16] B. Hughes, “Differential space-time modulation,” in Proc. Wireless Communications and Networking Conf., vol. 1, 1999, pp. 145–149. [17] A. Gelfand and A. Smith, “Sampling-based approaches to calculating marginal densities,” J. Amer. Stat. Assoc., vol. 85, pp. 398–409, 1990. [18] S. Geman and D. Geman, “Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-6, pp. 721–741, Nov. 1984. [19] K. Chan, “Asymptotic behavior of the Gibbs sampler,” J. Amer. Stat. Assoc., vol. 88, pp. 320–326, 1993. [20] J. Liu, W. Wong, and A. Hong, “Covariance structure and convergence rate of the gibbs sampler with various scans,” J. Roy. Statist. Soc. Ser. B, vol. 57, pp. 157–169, 1995. [21] C. Robert and G. Casella, Monte Carlo Statistical Methods. New York: Springer–Verlag, 1999. [22] J. Hagenauer, “The turbo principle: Tutorial introduction and state of the art,” in Int. Symp. Turbo Codes and Related Topics, Brest, France, Sept. 1997. [23] X. Wang and H. V. Poor, “Iterative (Turbo) soft interference cancelation and decoding for coded CDMA,” IEEE Trans. Commun., vol. 47, pp. 1046–1061, July 1999. [24] M. Moher, An iterative multiuser decoder for near-capacity communications, vol. 46, no. 7, pp. 870–880, July 1998. [25] L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inform. Theory, vol. IT-20, pp. 284–287, Mar. 1974. [26] S. Verdú, Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, 1998. [27] G. Bauch, “Concatenation of space-time block codes and ‘Turbo’-TCM,” presented at the Int. Conf. Communications, Vancouver, BC, Canada, June 1999.

Zigang Yang received the B.S. degree in electrical engineering and applied mathematics in 1995 and the M.S. degree in electrical engineering in 1998, both from Shanghai Jiao Tong University (SJTU), Shanghai, China. She is currently working toward the Ph.D. degree in the Department of Electrical Engineering, Texas A&M University, College Station, TX. Her research interests are in the area of statistical signal processing and its applications, primarily in wireless communications.

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Ben Lu received the B.E. and M.S. degrees in electrical engineering in 1994 and 1997, respectively, from Southeast University, Nanjing, China. From 1994 to 1997, he was a Research Assistant with National Mobile Communication Laboratory, Southeast University, Nanjing, China. From 1997 to 1998, he was a Member of the CDMA Research Department, Zhongxing Telecommunication Company, Shanghai, China. Since 1999, he has been a Research Assistant with the Department of Electrical Engineering, Texas A&M University. His general research interests include advanced signal processing and channel coding for wireless communication systems.

Xiaodong Wang received the B.S. degree in electrical engineering and applied mathematics (with the highest honors) from Shanghai Jiao tong University, Shanghai, China, in 1992; the M.S. degree in electrical and Computer engineering from Purdue University, West Layfayette, IN, in 1995; and the Ph.D. degree in electrical engineering from Princeton University, Princeton, NJ, in 1998. In July 1998, he joined the Department of Electrical Engineering, Texas A&M University, as an Assistant Professor. He was also with the AT&T Labs—Research, Red Bank, NJ, during the summer of 1997. He has worked in the areas of digital communications, digital signal processing, parallel and distributed computing, nanoelectronics, and quantum computing. His current research interests include multiuser communications theory and advanced signal processing for wireless communications. Dr. Wang is a member of the American Association for the Advancement of Science. He is currently an Associate Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS and the IEEE TRANSACTIONS ON SIGNAL PROCESSING. He has also received the 1999 NSF CAREER Award.

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