A new class of soft mimo demodulation algorithms - Semantic Scholar
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A New Class of Soft MIMO Demodulation Algorithms Bin Dong, Xiaodong Wang, and Arnaud Doucet
Abstract—We propose a new class of soft-input soft-output demodulation schemes for multiple-input multiple-output (MIMO) channels, based on the sequential Monte Carlo (SMC) framework under both stochastic and deterministic settings. The stochastic SMC sampler generates MIMO symbol samples based on importance sampling and resampling techniques, whereas the deterministic SMC approach recursively performs exploration and selection steps in a greedy manner. By exploiting the artificial sequential structure of the existing simple Bell-Labs layered space-time (BLAST) detection method based on nulling and cancellation, the proposed algorithms achieve an error probability performance that is orders of magnitude better than the traditional BLAST detection schemes while maintaining a low computational complexity. In fact, the new methods offer performance comparable with that of the sphere decoding algorithm without attendant increase in complexity. More importantly, being soft-input soft-output in nature, both the stochastic and deterministic SMC detectors can be employed as the first-stage demodulator in a turbo receiver in coded MIMO systems. Such a turbo receiver successively improves the receiver performance by iteratively exchanging the so-called extrinsic information between the soft outer channel decoder and the inner soft MIMO demodulator under both known channel state and unknown channel state scenarios. Computer simulation results are provided to demonstrate the performance of the proposed algorithms. Index Terms—MIMO, sequential Monte Carlo, soft-input softoutput, turbo receiver.
I. INTRODUCTION
T
HE ever increasing demand for high-speed wireless data transmission has posed great challenges for wireless system designers to achieve high-throughput wireless communications in radio channels with limited bandwidth. Multiple transmit and receive antennas are most likely to be the dominant solution in future broadband wireless communication systems, as the capacity of such a multiple-input multipleoutput (MIMO) channel increases linearly with the minimum between the numbers of transmit and receive antennas in a rich-scattering environment, without increasing the bandwidth or transmitted power [1]–[3]. Because of the extremely high spectrum efficiency, MIMO techniques will most probably be incorporated into various future high-speed wireless applications including wireless LAN and wireless cellular systems. Manuscript received December 12, 2002; revised April 9, 2003. This work was supported in part by the U.S. National Science Foundation under Grants DMS-0225692 and CCR-0225826 and by the U.S. Office of Naval Research under Grant N00014-03-1-0039. The associate editor coordinating the review of this paper and approving it for publication was Prof. Brian Hughes. B. Dong and X. Wang are with Department of Electrical Engineering, Columbia University, New York, NY 10027-4712, USA. A. Doucet is with Department of Engineering, Cambridge University, Cambridge, CB2 1PZ, U.K. Digital Object Identifier 10.1109/TSP.2003.818155
The Bell-Labs layered space-time (BLAST) architecture is an example of uncoded MIMO systems now under implementation. In the literature [4]–[6], different BLAST detection schemes have been proposed based on nulling and interference cancellation (IC), such as the method of zero-forcing (ZF) nulling and IC with ordering, and the method based on MMSE nulling and IC with ordering. The performance of these simple detection strategies is significantly inferior to that of the maximum likelihood (ML) detection, whose complexity grows exponential in terms of the number of transmit antennas. In [7]–[9], sphere decoding is proposed as a near-optimal BLAST detection method, whose complexity is cubic in terms of the number of transmit antennas. As hard decision algorithms, the above schemes suffer performance losses when concatenated with outer channel decoder in coded MIMO systems. In [8], a list sphere decoding algorithm is proposed to yield soft-decision output by storing a list of symbol sequence candidates. However, the complexity is significantly increased compared with the original sphere decoding algorithm. In this paper, we develop a new family of soft MIMO demodulation algorithms based on sequential Monte Carlo methods. The new algorithms achieve near-optimal performance with low complexities. The sequential Monte Carlo (SMC) methodology [10]–[15] originally emerged in the field of statistics and engineering has provided a promising new paradigm for the design of low-complexity signal processing algorithms with performance approaching the theoretical optimum for fast and reliable communication in highly severe and dynamic wireless environments. The SMC can be loosely defined as a class of methods for solving online estimation problems in dynamic systems by recursively generating Monte Carlo samples of the state variables or some other latent variables. In [10]–[12] and [15], SMC has been successfully applied to a number of problems in wireless communications including channel equalization, joint data detection and channel tracking in fading channels, and adaptive OFDM receiver in time dispersive channels. Because the SMC detectors not only utilize the a priori symbol probabilities but also produce the a posteriori symbol probabilities, they are ideal candidates to serve as the soft-input soft-output demodulator in a turbo receiver. In this paper, we propose a new class of soft-input soft-output MIMO detection schemes for quasistatic MIMO fading channels. This novel class of receivers are based on the SMC framework and can be in an either stochastic or deterministic setting. The stochastic SMC demodulator employs the techniques of importance sampling and resampling, where the trial sampling distribution is formed by exploiting the artificial sequential structure of the existing simple nulling and cancellation BLAST detection scheme.
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Fig. 1. Transmitter structure of a coded MIMO system.
Fig. 2. Turbo receiver structure for a coded MIMO system, where 5 denotes the interleaver and 5
The deterministic SMC demodulator recursively performs exploration and selection steps to detect the final survivor signal paths with the corresponding importance weights. Being soft-input and soft-output in nature, these SMC detectors can serve as the soft demodulator in a turbo receiver for coded MIMO systems, where the extrinsic information is iteratively exchanged between the soft demodulator and the soft outer channel decoder to successively improve the receiver performance. Moreover, when the channel is unknown and is estimated by using pilot symbols, the decoded code bits with high reliability can act as pilots for channel re-estimation. By performing iterative channel estimation and data detection, the receiver performance can be further improved. Note that in [16], an SMC MIMO receiver was developed whose complexity is exponential in terms of the number of transmit antennas, whereas the SMC receivers proposed in this paper has linear complexities in terms of the number of transmit antennas. The remainder of this paper is organized as follows. Section II describes the system under consideration. Background materials on the SMC methodology are provided in Section III. In Section IV, we derive the new class of soft MIMO demodulation algorithms using the stochastic SMC and the deterministic SMC, based on the simple nulling and cancellations BLAST detection scheme. Computer simulation results are provided in Section V, and conclusions are drawn in Section VI.
denotes the deinterleaver.
II. SYSTEM DESCRIPTIONS In this section, we consider a generic coded MIMO system with a turbo receiver. The transmitter and receiver structures are shown in Figs. 1 and 2, respectively. A. Baseband MIMO Signal Model are At the transmitter, a block of information bits . The code bits are then randomly encoded into code bits interleaved and mapped to an M-PSK or M-QAM modulation symbol stream, taking values from a finite alphabet set . Each symbol is serial-to-parallel substreams via demultiplexing, and each converted to substream is associated with a transmit antenna. At each time instance, one symbol from each substream is transmitted from symbols transmitted its corresponding antenna, resulting in simultaneously in the same frequency band. Such a space-time bit interleaved coded modulation (BICM) allows for a better exploitation of the spatial, temporal, and frequency diversity resources available in the wireless MIMO systems [17], [18]. Following the common assumption for BLAST systems [1], transmit and re[5], we consider an MIMO system with . The wireless channel is assumed ceive antennas with to have rich-scattering and flat fading. The fading between each transmit and receive antenna pair is assumed to be independent.
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The channel is also assumed quasistatic, i.e., it is static over a data burst and changes from burst to burst. At the receiver, after matched filtering and symbol rate samreceive antennas is pling, the received signal vector from all denoted as
for symbol
. The a priori symbol probability for each is then given by
(4) where the code bit probability can be computed from the corresponding LLR as follows [21]:
In complex baseband representation, the received signal can be expressed as the linear combination of the transmitted signal (5)
(1) is the total signal energy at the transis the complex fading channel matrix, is the spatially and temporally white Gaussian noise, and is the data burst length. . The received signal is matched-filtered Note that and whitened to obtain (here for simplicity, we drop the time index )
where mitter,
,
(2)
Note that initially, the extrinsic information is set to be zero so that the QPSK symbols are equally probable. At the output of the soft demodulator are the a posteriori symbol probabilities . Note that
The exact expression of
is given by
. Based on (2), the maxwhere imum likelihood (ML) MIMO detector is given by (3) (6) Subsequently, the complexity of the ML receiver grows exponentially in the number of transmit antennas. Alternative suboptimal approaches, such as the method based on zero-forcing nulling and cancellation with ordering and the method based on MMSE nulling and cancellation with ordering [1], [6], have lower complexities but also incur substantial performance loss. The sphere decoding method has recently emerged as a near-optimal BLAST detection algorithm. However, its complexity is cubic in term of the number of transmit antennas [7], [8]. B. Turbo Receiver Since the combination of M-PSK or M-QAM modulation and symbol-antenna mapping effectively acts as an inner encoder in the MIMO transmitter, the whole system can be perceived as a serial concatenated system, and an iterative (turbo) receiver [19], [20] can be designed for such a system as shown in Fig. 2. The turbo receiver consists of two stages: the soft-input soft-output SMC demodulator to be developed in Section IV followed by a soft channel decoder. The two stages are separated by a deinterleaver and an interleaver. The soft MIMO demodulator takes as the input the extrinsic delivered by the channel decoder in the information previous turbo iteration as well as the received signals given by (2). First, the symbol a priori probability is calculated as follows. Assuming QPSK modulation is employed, the is mapped to symbol . Additionally, bit pair assume that in the MIMO systems, the transmitted symbol corresponds to the interleaved code bit pair ,
possible Since the summations in (6) are over all the , its complexity is exponential vectors in in the number of transmit antennas and impractical for systems with high spatial-multiplexing gain. In this paper, we will develop some low complexity algorithms for approximating (6). Based on the symbol a posteriori probabilities computed by the soft demodulator, the bit LLR computer calculates the a posteriori log-likelihood ratios (LLRs) of the interleaved code bits . Assuming that the code bit is included in a QPSK , the LLR of this code bit is given by symbol
(7) Using Bayes’ rule, (7) can be written as (8)
, repwhere the second term in (8), which is denoted by , which is computed resents the a priori LLR of the code bit by the channel decoder in the previous iteration, interleaved, and
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then fed back to the soft MIMO demodulator. For the first iteration, it is assumed that all code bits are equally likely. The , represents the first term in (8), which is denoted by extrinsic information delivered by the soft MIMO demodulator, based on the received signals , the MIMO signal structure, and the a priori information about all other code bits. The extrinsic is then deinterleaved and fed back to the information channel decoder as the a priori information for the channel decoder. The soft decoder takes as input the a priori LLRs of the code bits and delivers as output an update of the LLRs of the coded bits, as well as the LLRs of the information bits based on the code constraints. The soft channel decoding algorithm [22] computes the a posteriori LLR of each code bit code constraints code constraints
One way to produce such a training sequence is to use the complex Walsh codes generated by the Hadamard matrix. As shown in Fig. 2, at the end of each turbo iteration, the a posteriori LLRs of the code bits are fed back from the soft channel decoder through the interleaver to the MIMO channel estimator for channel re-estimation. If the LLRs of all code bits corresponding to one QPSK MIMO symbol are above a predetermined threshold, the corresponding decoded MIMO symbol is considered to be with high reliability and will act as a training symbol for channel re-estimation at the beginning of the next turbo iteration. Intuitively, as the turbo receiver iterates, such a scheme obtains increasingly more accurate channel estimation since it makes use of more and more training symbols. Accordingly, the overall receiver performance will be improved compared with the scheme where the channel is estimated only once using the pilot symbols before the first turbo iteration.
(9) where the factorization (9) is shown in [21]. It is seen from (9) that the output of the soft decoder is the sum of the a priori inand the extrinsic information delivered formation by the channel decoder. This extrinsic information is the information about the code bit gleaned from the a priori informabased on the constraint tion about the other code bits structure of the code. Note that at the first iteration, the extrinsic and is statistically independent, information but subsequently, since it uses the same information indirectly, it will become increasingly more correlated, and finally, the improvement through the iterations will diminish. C. MIMO Channel Estimation Since the receiver has no knowledge of channel state information in realistic MIMO systems, pilot symbols embedded in the data stream are required to estimate the channel impulse time slots are used at the beginresponse. Assume ning of each data burst to transmit known pilot symbols . Denote the corresponding received signal as . Then, we have
III. BACKGROUND ON SEQUENTIAL MONTE CARLO Sequential Monte Carlo (SMC) is a class of efficient methods that obtains random samples from a sequence of probability distributions known only up to a normalizing constant. A general framework for the SMC methods is briefly explained next. Consider the following generic sequence of , the a posteriori probability distributions is the set of unobserved pawhere is the set rameters to be estimated, and of available observations at index . We emphasize the fact that is not necessarily a time index. Typically, a posteriori distributions are only known up to a normalizing constant as is not available in closed form, and thus, SMC is of great interest in this context. Assume one wants to compute the minimum mean-square error , which is given by (MMSE) estimate of (14) Given
(10)
random samples
distributed according to
, this expectation can be approximated numerically through
. In [23], two forms of channel estimawhere tors are given, namely, the maximum likelihood (ML) estimator, hwich is given by (11) and the minimum mean-square error (MMSE) channel estimator, which is given by (12) It is also shown in [23] and [24] that the optimal training sequence in the sense of minimizing the channel estimation error should satisfy the following orthogonality condition: (13)
(15) is often not feasible, and it is Sampling directly from necessary to derive approximate methods. A. Sequential Importance Sampling is often not feasible or Sampling directly from too computationally expensive, but drawing samples from some trial density “close” to the distribution of interest is often easy. In this case, we can use the idea of importance sampling. Supdistributed according pose a set of random samples is available. We can still come up with valid to estimates of (14) by correcting for the discrepancy between
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and
. By associating the importance
weight
to the sample
with
, we can indeed estimate (14) as
. The pair
is called
a properly weighted sample with respect to the distribution . It is crucial to notice that one only needs to compute the weights up to a normalizing constant so that the is not necessary. In the “optimal” case knowledge of , then all the weights are equal where and have zero variance. Roughly speaking, the performance of the method typically deteriorates when the variance of the weights increases. The importance sampling method is a generic method that is not sequential. However, it is easy to come up with a sequential version of it. Suppose a set of properly weighted samples with respect to is available . Based on these samples, we want to generate a at index that are properly weighted new set of samples . The sequential importance sampling with respect to (SIS) algorithm proceeds as follows. from a trial distribution • Draw a sample , and let • Update the importance weight
(16)
One can check that the above algorithm indeed generates properly weighted samples with respect to the distribution . It is indeed simply an importance sampling method with trial distribution satisfying
This procedure is very general but it is crucial to design a “good” trial distribution to obtain good performance. It is easy to establish that the optimal importance distribution—optimal as it minimizes the conditional variance of the weights [10]—is given by
B. Resampling measures the “quality” of the The importance weight . A relatively small corresponding imputed signal sequence weight implies that the sample is drawn far from the main body of the a posteriori distribution and has a small contribution in the final estimation. Such a sample is said to be ineffective. Whatever the importance distribution is used, the SIS algorithm is getting inefficient as increases. This is because the discrepancy and can only increase with . between In practice, after a few steps of the procedure, only one stream dominates all the others, i.e., its importance weight is close to 1, whereas the others are close to 0. To make the SIS procedure efficient in practice, it is necessary to use a resampling procedure, as suggested in [25]. Roughly speaking, the aim of resampling is to duplicate the streams with large importance weights while eliminating the streams with small ones, i.e., one focuses the computational efforts in the promising zones of the space. Each is copied times with . Many resample sampling procedures have been proposed in the literature [11], [14], [15], [25], [26]. We use here the systematic resampling procedure proposed in [26]. , • Compute the cumulative distribution for • Draw a random number uniformly in . Compute with • Set • Assign uniform weights to the new set of streams This algorithm is very computationally efficient. In the class , this of unbiased resampling schemes, i.e., and displays better algorithm minimizes the variance performance than other algorithms. Note that if resampling is used in conjunction with the optimal importance distribution (17), one should perform the resampling step before the sampling step at time as, in this very specific case, the importance weight at time is independent of . the samples Resampling can be done at every fixed-length time interval (say, every five time steps), or it can be conducted dynamically. The effective sample size is a criterion that can be used to monitor the variation of the importance weights of the sample streams and to decide when to resample as the system evolves. The effective sample size is defined as
where
, which is the coefficient of variation, is given by
(17) In this case, the importance weight recursion is
with . Roughly speaking, the effective sample size is a measure of variation over the weights. In the optimal case where all the weights are equal, then it is maximum
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and equal to . When one weight dominates all the others, then it is very small. In dynamic resampling, a resampling step is is below a certain performed once the effective sample size threshold. An alternative criterion such as the entropy of the weights could also be used. Heuristically, resampling can provide chances for good sample streams to amplify themselves and, hence, “rejuvenate” the samples to produce better results in the next steps. It can be shown that if is sufficiently large, the resampled streams drawn by the above resampling scheme are also properly . In practice, when small to weighted with respect to in this paper), the resampling modest is used (we use procedure can be seen as a tradeoff between the bias and the variance. That is, the new samples with their weights resulting from the resampling procedure are only approximately proper, which introduces small bias in the Monte Carlo estimates. On the other hand, resampling significantly reduces the Monte Carlo variance for the future samples. C. Alternative Deterministic Procedure SMC is a very general set of methods to sample probability distributions on any space one wants. We restrict ourselves to a case of great interest in telecommunications, namely, the case can only take values in a finite set say . In this case, where could be computed exthe a posteriori distribution actly, but it is typically far too computationally expensive when is large, as can take possible values, where is the number of elements in . In this very specific case, the SIS procedure with the optimal importance distribution (17) takes the following form. • Draw a sample
from
and let
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streams. Indeed, for each
, one needs to compute
for . By sampling , this information is somehow discarded. An alternative deterministic “candidate” approach consists of keeping among the , which are the trajectories at time , trajectories with the highest a posteriori distribution. Clearly, the local approximation error one commits at the selection step using this deterministic strategy is lower than using the randomized strategy. Nevertheless, it does not guarantee performance to be better overall. with a posteriori distributions To sum up, given (known up to a normalizing constant)
, the algo-
rithm proceeds as follows at time . • Compute (19) “best” distinct streams • Select and preserve only the hypotheses. amongst the The drawback of this approach is that clearly, the past observations are as important as the current observation in the calculation of the weight. If an error is committed, then it has a significant impact on further decisions. The randomized algorithm is less sensitive to this problem as when the streams are . resampled, their weights are set to It is worth noticing that both the stochastic and deterministic algorithms are well suited for parallel implementation. IV. SOFT MIMO DEMODULATION ALGORITHMS In this section, we will derive soft-input soft-output MIMO demodulation algorithms based on the sequential Monte Carlo principle. The importance sampling density is obtained by utilizing the artificial sequential structure of the existing simple nulling and cancellation BLAST detection scheme. First, the channel parameters are assumed to have been estimated perfectly at the receiver through the short training sequence before detection. Later, we will examine the effect of channel estimation error on systems performance with pilot symbols along with high-quality symbol decision feedback.
. • Update the importance weight
The importance weight is computed using
(18)
A. Simple Nulling and Cancellation BLAST Detection Algorithm Consider the signal model (2), we denote the QR-decompoas sition of
Two points can be criticized about this algorithm. First, assume is equal to with . that the number of streams ; then, one could compute exactly!). (If possibilities The algorithm should enumerate the first and compute their a posteriori probabilities, i.e., one should exactly. This can be easily incorporated in compute the initialization phase of an SMC algorithm. Second, one can see that the calculation of the weights (18) involves computing the a posteriori distribution up to a normalizing constant of
(20) where is a unitary matrix, and is an upper triangular matrix. The nulling operation is a coordinate rotation that left multiplies to produce a sufficient statistic the vector on (2) by (21)
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where . Since is unitary, there is no noise enhancement, and the noise whitening characteristic is maintained by . We rewrite (21) as nulling, i.e.,
.. .
..
.
.. .
.. .
.. .
(22)
We can detect data symbols directly by nulling operation, i.e., . However, in [5], it is shown that by utimultiply by lizing the upper triangular structure of , significant improvement over zero-forcing can be obtained with the following successive interference cancellation method:
Fig. 3. BER performance of various MIMO demodulation algorithms in an uncoded MIMO system with n = n = 8. QPSK modulation.
probability distributions
. This
sequence of “artificial” distributions is defined by (25) The aim of SMC MIMO detection is to compute an estimate of the a posteriori symbol probability
.. .
(26) (23) after nulling. Let be a sample drawn by the SMC at each symbol interval, where is the number of samples. In order to implement the SMC, we need to obtain a set of Monte Carlo samples of the transmitted symbols properly weighted with respect to the distribution of . In in the application of MIMO demodulation, the function as (15) is specified by the indicator function based on the received signal
. Although the above where simple nulling and cancellation scheme has a very low complexity, its performance is much worse than that of the methods based on zero-forcing or MMSE nulling and interference cancellation with ordering as well as that of the sphere decoding algorithm (see Fig. 3). In what follows, we develop SMC-based MIMO demodulation algorithms using the above simple nulling and cancellation method as the kernel. As will be seen in Section V, these new algorithms provide near-optimal performance in both uncoded and coded MIMO systems. B. Stochastic SMC MIMO Demodulator As from (22), the artificial sequential structure of the simple nulling and interference cancellation scheme due to the uppertriangular structure is well suited for applying the SMC method to MIMO data detection with the particularity of operating on to antenna 1. Indeed, spatial domain starting from antenna one has (24)
, and . We where can thus use SMC methods to simulate from the sequence of
if if
.
(27)
Hence, the a posteriori probability of the information symbol can then be estimated as
(28)
where Following (17) in Section III-A, we choose the trial distribution as (29)
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For this trial distribution, the importance weight is updated according to
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3) Compute the importance weight (38) 4) Compute the a posteriori probability according to of the information symbol (28). 5) Perform resampling as described in Section III-B.
(30) We next specify the computation of the predictive distribution in (29) and (30). First, we consider the trial distributions in (29) (31) in (21) is white Gaussian, i.e., Since the noise , we have (32) where the mean
is given by
C. Deterministic SMC MIMO Demodulator The deterministic method for estimating the sequence of probability distributions proceeds as follows. Similarly to the stochastic case, we samples are drawn at each iteration, where assume that with . Note that if , it is equivalent to the maximum likelihood detection and involves calculations. In this step, we compute exactly the distribution by enumerating all particles for down to antenna . As a result, we have antenna distinct QPSK sequences with a set of weights
satisfying
(33) Note that (34) (35) (39)
Hence, the predictive distribution in (30) is given by where
(40) (36) Finally, we summarize the stochastic SMC MIMO demodulator as follows: 0) Initialization: All importance weights . The are initialized as following steps are implemented at the th recursion to update : each weighted sample. For 1) Compute the trial sampling density for each according to (35) with the a priori symbol probability obtained from the last turbo iteration calculated using (4). from the set with 2) Draw a sample probability (37)
(41) is given by (33). and The second step is algorithm described in Section III-C, to antenna 1 (i.e., which is performed from antenna ). We need to update the importance weight according to
(42) where the distinct QPSK symbol sequences with highest importance weights are selected as the survivor paths over hypotheses. The deterministic SMC MIMO demodulation algorithm is summarized as follows. , 0) Initialization: For calculate the exact expression of the probability distribution via (41) by enuparticles. merating all
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The following steps are implemented at the th recursion to up: date each weight sample. For 1) Compute the importance weight (43) is given by (41). where 2) Select and preserve only the “best”distinct streams with the highest weights amongst the hypotheses with weights set . 3) Compute the a posteriori probability according to of the information symbol (28). D. Combined Deterministic/Stochastic SMC MIMO Demodulator We next propose a combined deterministic and stochastic SMC (CDSMC) demodulator. The new method has an adto antenna ditional parameter such that from antenna (first stages), the demodulator generates the QPSK symbol samples in the same manner as the deterministic SMC, whereas in the remainder of the process, the demodulator generates samples in the same manner as the stochastic SMC demodulator. We summarize the combined deterministic and stochastic SMC MIMO demodulation algorithm as follows. , 0) Initialization: For calculate the exact expression of the probability distribution via (41) by enuparticles. merating all The following steps are implemented at the th recursion to update each weight sample. For : 1) Compute the importance weight (44) is given by (41). where 2) Select and preserve only the “best”distinct streams with the highest weights amongst the hypotheses with weights set . The following steps are implemented at the th recursion ( ) to update : each weight sample. For 3) Compute the trial sampling density for each according to (35) with the a priori symbol probability obtained from the last turbo iteration calculated using (4). from the set with 4) Draw a sample probability (45)
5) Compute the importance weight (46) 6) Compute the a posteriori probability according to of the information symbol (28). 7. Perform resampling as described in Section III-B. V. SIMULATION RESULTS In this section, we provide computer simulation results to illustrate the performance of the proposed turbo receivers in . The fading coflat-fading MIMO channels with . Channels are efficients are generated according to assumed to be quasistatic, i.e., they remain constant over the entire frame of symbols but vary from frame to frame. Simulation results are obtained by averaging over 500 channel realizations. The number of samples drawn in the stochastic SMC . The number of samples drawn in the dealgorithm is (i.e., ) in the uncoded case terministic algorithm is (i.e., ) in the coded case. and A. Performance in Uncoded MIMO Systems We first illustrate the performance of the proposed stochastic and deterministic SMC MIMO demodulation algorithms in an uncoded MIMO system. The BER performance of these two new algorithms, together with some existing detection algorithms (including the sphere decoding algorithm, the method based on MMSE nulling and cancellation with ordering, the method based on zero-forcing nulling and cancellation with ordering, as well as the simple nulling and cancellation method) are shown in Fig. 3. Several observations are in order. First of all, the simple nulling and cancellation method displays very poor performance. However, when it is combined with SMC (deterministic or stochastic), we obtain a much more powerful MIMO detector. Second, the deterministic detector outperforms the stochastic SMC detector, and it actually slightly outperforms the sphere decoding algorithms, which is the best suboptimal MIMO detector known so far. This is because the sphere decoding minimizes the probability of sequence , whereas the sequential Monte Carlo error, i.e., calculates the a posteriori symbol probabilities and minimizes . In addition, it is well the symbol error rate, known that the MAP decoding performs slightly better than the ML decoding in terms of symbol error rate. Fig. 4 illustrates the BER performance of the combined deterministic and stochastic SMC (CDSMC) detector with different combination of deterministic SMC and stochastic SMC. The number of samples drawn in all three schemes (i.e., ). It is seen that the performance of is the CDSMC demodulator is better than that of the stochastic SMC demodulator, whereas it is still worse than that of the deterministic SMC demodulator. As the parameter increases, the performance of the CDSMC approaches that of the deterministic SMC. Note that the CDSMC demodulator with
DONG et al.: NEW CLASS OF SOFT MIMO DEMODULATION ALGORITHMS
Fig. 4. BER performance of various combined D&SMC demodulation = n = 8. QPSK algorithms in an uncoded MIMO system with n modulation.
Fig. 5. BER performance of a turbo MIMO receiver employing the stochastic SMC MIMO demodulator. Channel estimation is based on pilots only. n = n = 8. Rate-1/2 constraint length-5 convolutional code. QPSK modulation.
is equivalent to the deterministic SMC demodulator in MIMO . systems with B. Performance in Coded MIMO Systems For the coded MIMO system, a rate-1/2 constraint length-5 convolutional code (with generators 23 and 35 in octal notation) is employed at the transmitter. The bit interleaver is randomly generated and fixed throughout the simulations. The code bit block size is 512, corresponding to 256 information bits and 32 QPSK MIMO symbols. The channel estimator employed here is the MMSE MIMO channel estimator given by (12), where , and QPSK orthogonal MIMO pilot symbols are used. The number of turbo iterations in each simulation run is four.
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Fig. 6. BER performance of a turbo MIMO receiver employing the stochastic SMC MIMO demodulator. Iterative channel estimation is based on pilots and the decoded symbols with high reliability. n = n = 8. Rate-1/2 constraint length-5 convolutional code. QPSK modulation.
Fig. 7. BER performance of a turbo MIMO receiver employing the stochastic SMC MIMO demodulator. Genie-aided channel estimation is based on pilots and all information symbols. n = n = 8. Rate-1/2 constraint length-5 convolutional code. QPSK modulation.
In Figs. 5–7, the BER performance of the turbo receiver employing the stochastic SMC demodulator is plotted. The solid curves in these figures correspond to the BER performance with perfectly known channel, whereas the dashed curves correspond to the BER performance with different channel estimation schemes. In Fig. 5, the channel is estimated only once based on the pilots and used throughout all turbo iterations. In Fig. 6, the channel is re-estimated at the beginning of each turbo iteration using both the pilots and the decoded symbols with high-reliability, as discussed in Section II-C, and in Fig. 7, we assume that the channel is estimated by a genie that knows both the pilot symbols and all information symbols.
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Fig. 8. BER performance of a turbo MIMO receiver employing the deterministic SMC MIMO demodulator. Channel estimation is based on pilots only. n = n = 8. Rate-1/2 constraint length-5 convolutional code. QPSK modulation.
Fig. 10. BER performance of a turbo MIMO receiver employing the deterministic SMC MIMO demodulator. Genie-aided channel estimation is based on pilots and all information symbols. n = n = 8. Rate- 1=2 constraint length-5 convolutional code. QPSK modulation.
VI. CONCLUSIONS
Fig. 9. BER performance of a turbo MIMO receiver employing the deterministic SMC MIMO demodulator. Iterative channel estimation is based on pilots and the decoded symbols with high reliability. n = n = 8. Rate-1/2 constraint length-5 convolutional code. QPSK modulation.
Such a genie-aided channel estimate is used throughout all turbo iterations. Obviously, the scenario for Fig. 7 provides an upper bound on the achievable performance for the turbo receivers with different channel estimation schemes. It is seen that by performing iterative channel estimation and symbol detection, the turbo receiver offers a performance approaching the genie-aided bound. The corresponding BER performance of turbo receiver employing the deterministic SMC demodulator is shown in Figs. 8–10. It is seen that the deterministic SMC algorithm offers improved and more stable performance than its stochastic counterpart.
In this paper, we have developed a new family of soft-input soft-output MIMO demodulation algorithms. These new techniques use the conventional BLAST detection based on simple nulling and cancellation as the kernel and are based on the sequential Monte Carlo (SMC) method for Bayesian inference. Two versions of such SMC MIMO demodulation algorithms are developed, based on, respectively, stochastic and deterministic sampling. As hard MIMO detection algorithms, the proposed SMC demodulation algorithms significantly outperform all existing BLAST detection methods. Moreover, the deterministic SMC MIMO detector slightly outperforms the sphere decoding algorithm while maintaining the lower complexity. In fact, the complexity of the deterministic SMC demodulator , the complexity of the stochastic is , and the complexity of the SMC demodulator is . Furthermore, in coded MIMO syssphere decoder is tems, the proposed SMC algorithms can naturally serve as the soft MIMO demodulator in a turbo receiver. Simulation results indicate that overall, the deterministic SMC MIMO demodulator offers better and more stable performance compared with its stochastic counterpart in both the uncoded and the coded MIMO systems. It remains a challenging open problem to theoretically justify the superior performance of the deterministic SMC algorithm over that of the stochastic SMC algorithm. REFERENCES [1] G. J. Foschini and M. J. Gans, “On the limits of wireless communications in a fading environment when using multiple antennas,” Wireless Pers. Commun., vol. 6, no. 3, pp. 311–335, 1998. [2] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans. Telecommun., vol. 10, no. 6, pp. 585–595, Nov. 1999. [3] C. N. Chuah, D. N. C. Tse, J. M. Kahn, and R. A. Valenzuela, “Capacity scaling in MIMO wireless systems under correlated fading,” IEEE Trans. Inform. Theory, vol. 48, pp. 637–650, Mar. 2002.
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[4] P. W. Wolniansky, G. J. Roschini, G. D. Golden, and R. A. VAlenzuela, “V-BLAST: An architecture for realizing very high data rates over the rich-scattering wireless channel,” in Proc. Int. Symp. Signals, Systems, Electron., Pisa, Italy, Sept. 1998. [5] G. J. Foschini, “Layed space-time architecture for wireless communication in a fading environment when using multi-element antennas,” Bell Labs. Tech. J., vol. 1, no. 2, pp. 41–59, 1996. [6] G. D. Golden, G. J. Foschini, R. A. Valenzuela, and P. W. Wolniansky, “Detection algorithm and initial laboratory results using V-BLAST space-time communication architecture,” Electron. Lett., vol. 35, pp. 14–16, Jan. 1999. [7] O. Damen, A. Chkeif, and J. Belfiore, “Lattice code design for space-time codes,” IEEE Commun. Lett., vol. 4, pp. 161–163, May 2000. [8] B. M. Hochwald and S. T. Brink, “Achieving near-capacity on a multiple-antenna channel,” IEEE Trans. Commun., submitted for publication. [9] A. M. Chan and I. Lee, “A new reduced sphere decoder for multiple antenna systems,” in Proc. 2002 Int. Commun. Conf., New York, Apr. 2002. [10] A. Doucet, S. J. Godsill, and C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Statist. Comput., vol. 10, no. 3, pp. 197–208, 2001. [11] X. Wang, R. Chen, and J. S. Liu, “Monte Carlo Bayesian signal processing for wireless communications,” J. VLSI Signal Process., vol. 30, no. 1–3, pp. 89–105, Jan.–Mar. 2002. [12] R. Chen, X. Wang, and J. S. Liu, “Adaptive joint detection and decoding in flat-fading channels via mixture Kalman filtering,” IEEE Trans. Inform. Theory, vol. 46, pp. 2079–2094, Sept. 2000. [13] R. Chen and J. S. Liu, “Sequential Monte Carlo methods for dynamic systems,” J. Amer. Stat. Assoc., vol. 93, pp. 1302–1044, 1998. , “Mixture Kalman filters,” J. Amer. Stat. Assoc. (B), vol. 62, pp. [14] 493–509, 2000. [15] Z. Yang and X. Wang, “A sequential Monte Carlo blind receiver for OFDM systems in frequency-selective fading channels,” IEEE Trans. Signal Processing, vol. 50, pp. 271–280, Feb. 2002. [16] D. Guo and X. Wang, “Blind detection in MIMO systems via sequential Monte Carlo,” IEEE J. Select. Areas. Commun., vol. 21, pp. 453–464, Apr. 2003. [17] A. M. Tonello, “On turbo equalization of interleaved space-time codes,” in Proc. Fall Vehicular Technology Conf., Oct. 2001. [18] J. J. Boutros, F. Boixadera, and C. Lamy, “Bit-interleaved coded modulations for multiple-input multiple-output channels,” in Proc. IEEE 6th Int. Symp. Spread-Spectrum Technol. Applications, Sept. 2000. [19] H. Dai and A. F. Molisch, “Multiuser detection for interference-limited MIMO systems,” in Proc. Spring Vehicular Technology Conf., May 2002. [20] M. Sellathurai and S. Haykin, “TURBO-BLAST for wireless communications: Theory and experiments,” IEEE Trans. Signal Processing, vol. 50, pp. 2538–2546, Oct. 2002. [21] X. Wang and H. V. Poor, “Iterative (Turbo) soft interference cancellation and CDMA,” IEEE Trans. Commun., vol. 47, pp. 1046–1061, Jul. 1999. [22] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inform. Theory, vol. 20, pp. 284–287, Mar. 1974. [23] T. L. Marzetta, “BLAST training: Estimation channel characteristics for high-capacity space-time wireless,” in Proc. 37th Annu. Allerton Conf. Commun., Comput., Control, Sept. 1999. [24] Q. Sun, D. C. Cox, A. Lozano, and H. C. Huang, “Training-based channel estimation for continuous flat fading BLAST,” in Proc. Int. Conf. Commun., New York, Apr. 2002. [25] N. J. Gordon, D. J. Salmond, and A. F. M. Smith, “Novel approach to nonlinear non-Gaussian Bayesian state estimation,” Proc., Inst. Elect. Eng. F. Radar Sonar Navigat., vol. 140, no. 2, Apr. 1993. [26] G. Kitagawa, “Monte Carlo filter and non-Gaussian nonlinear state space models,” J. Comput. Graph. Statist., vol. 5, no. 1, pp. 1–25, 1996.
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Bin Dong received the B.S. degree in electrical engineering from Shanghai Jiao Tong University, Shanghai, China, in 2000 and the M.S. degree in electrical and computer engineering from Queen’s University, Kingston, ON, Canada, in 2002. He is currently pursuing the Ph.D. degree with the Department of Electrical Engineering, Columbia University, New York, NY. His research are in the area of statistical signal processing, wireless communications, and wireless networks with the application in wireless MIMO, OFDM, and CDMA systems. Mr. Dong reecived the W. W. King Fellowship in 2001.
Xiaodong Wang received the B.S. degree in electrical engineering and applied mathematics (with the highest honor) from Shanghai Jiao Tong University, Shanghai, China, in 1992, the M.S. degree in electrical and computer engineering from Purdue University, West Lafayette, IN, in 1995, and the Ph.D. degree in electrical engineering from Princeton University, Princeton, NJ, in 1998. From July 1998 to December 2001, he was an Assistant Professor with the Department of Electrical Engineering, Texas A&M University, College Station. In January 2002, he joined the Department of Electrical Engineering, Columbia University, New York, NY, as an Assistant Professor. His research interests fall in the general areas of computing, signal processing, and communications. He has worked in the areas of digital communications, digital signal processing, parallel and distributed computing, nanoelectronics, and bioinformatics and has published extensively in these areas. His current research interests include wireless communications, Monte Carlo-based statistical signal processing, and genomic signal processing. Dr. Wang received the 1999 NSF CAREER Award and the 2001 IEEE Communications Society and Information Theory Society Joint Paper Award. He currently serves as an Associate Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS, the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, the IEEE TRANSACTIONS ON SIGNAL PROCESSING, and for the EURASIP Journal of Applied Signal Processing.
Arnaud Doucet was born in France on November 2, 1970. He graduated from Institut National des Telecommunications in June 1993 and received the Ph.D. degree from Universite Paris-Sud, Orsay, France, in December 1997. From January 1998 to February 2001, he was a research associate with Cambridge University, Cambridge, U.K. From March 2001 to August 2002, he was a Senior Lecturer with the Department of Electrical Engineering, Melbourne University, Parkville, Australia. Since September 2002, he has been a University Lecturer with the Engineering Department, Cambridge University. His research interests include sequential Monte Carlo methods, Markov chain Monte Carlo methods, optimal filtering, and control and reinforcement learning.
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 11, NOVEMBER 2003
A New Class of Soft MIMO Demodulation Algorithms Bin Dong, Xiaodong Wang, and Arnaud Doucet
Abstract—We propose a new class of soft-input soft-output demodulation schemes for multiple-input multiple-output (MIMO) channels, based on the sequential Monte Carlo (SMC) framework under both stochastic and deterministic settings. The stochastic SMC sampler generates MIMO symbol samples based on importance sampling and resampling techniques, whereas the deterministic SMC approach recursively performs exploration and selection steps in a greedy manner. By exploiting the artificial sequential structure of the existing simple Bell-Labs layered space-time (BLAST) detection method based on nulling and cancellation, the proposed algorithms achieve an error probability performance that is orders of magnitude better than the traditional BLAST detection schemes while maintaining a low computational complexity. In fact, the new methods offer performance comparable with that of the sphere decoding algorithm without attendant increase in complexity. More importantly, being soft-input soft-output in nature, both the stochastic and deterministic SMC detectors can be employed as the first-stage demodulator in a turbo receiver in coded MIMO systems. Such a turbo receiver successively improves the receiver performance by iteratively exchanging the so-called extrinsic information between the soft outer channel decoder and the inner soft MIMO demodulator under both known channel state and unknown channel state scenarios. Computer simulation results are provided to demonstrate the performance of the proposed algorithms. Index Terms—MIMO, sequential Monte Carlo, soft-input softoutput, turbo receiver.
I. INTRODUCTION
T
HE ever increasing demand for high-speed wireless data transmission has posed great challenges for wireless system designers to achieve high-throughput wireless communications in radio channels with limited bandwidth. Multiple transmit and receive antennas are most likely to be the dominant solution in future broadband wireless communication systems, as the capacity of such a multiple-input multipleoutput (MIMO) channel increases linearly with the minimum between the numbers of transmit and receive antennas in a rich-scattering environment, without increasing the bandwidth or transmitted power [1]–[3]. Because of the extremely high spectrum efficiency, MIMO techniques will most probably be incorporated into various future high-speed wireless applications including wireless LAN and wireless cellular systems. Manuscript received December 12, 2002; revised April 9, 2003. This work was supported in part by the U.S. National Science Foundation under Grants DMS-0225692 and CCR-0225826 and by the U.S. Office of Naval Research under Grant N00014-03-1-0039. The associate editor coordinating the review of this paper and approving it for publication was Prof. Brian Hughes. B. Dong and X. Wang are with Department of Electrical Engineering, Columbia University, New York, NY 10027-4712, USA. A. Doucet is with Department of Engineering, Cambridge University, Cambridge, CB2 1PZ, U.K. Digital Object Identifier 10.1109/TSP.2003.818155
The Bell-Labs layered space-time (BLAST) architecture is an example of uncoded MIMO systems now under implementation. In the literature [4]–[6], different BLAST detection schemes have been proposed based on nulling and interference cancellation (IC), such as the method of zero-forcing (ZF) nulling and IC with ordering, and the method based on MMSE nulling and IC with ordering. The performance of these simple detection strategies is significantly inferior to that of the maximum likelihood (ML) detection, whose complexity grows exponential in terms of the number of transmit antennas. In [7]–[9], sphere decoding is proposed as a near-optimal BLAST detection method, whose complexity is cubic in terms of the number of transmit antennas. As hard decision algorithms, the above schemes suffer performance losses when concatenated with outer channel decoder in coded MIMO systems. In [8], a list sphere decoding algorithm is proposed to yield soft-decision output by storing a list of symbol sequence candidates. However, the complexity is significantly increased compared with the original sphere decoding algorithm. In this paper, we develop a new family of soft MIMO demodulation algorithms based on sequential Monte Carlo methods. The new algorithms achieve near-optimal performance with low complexities. The sequential Monte Carlo (SMC) methodology [10]–[15] originally emerged in the field of statistics and engineering has provided a promising new paradigm for the design of low-complexity signal processing algorithms with performance approaching the theoretical optimum for fast and reliable communication in highly severe and dynamic wireless environments. The SMC can be loosely defined as a class of methods for solving online estimation problems in dynamic systems by recursively generating Monte Carlo samples of the state variables or some other latent variables. In [10]–[12] and [15], SMC has been successfully applied to a number of problems in wireless communications including channel equalization, joint data detection and channel tracking in fading channels, and adaptive OFDM receiver in time dispersive channels. Because the SMC detectors not only utilize the a priori symbol probabilities but also produce the a posteriori symbol probabilities, they are ideal candidates to serve as the soft-input soft-output demodulator in a turbo receiver. In this paper, we propose a new class of soft-input soft-output MIMO detection schemes for quasistatic MIMO fading channels. This novel class of receivers are based on the SMC framework and can be in an either stochastic or deterministic setting. The stochastic SMC demodulator employs the techniques of importance sampling and resampling, where the trial sampling distribution is formed by exploiting the artificial sequential structure of the existing simple nulling and cancellation BLAST detection scheme.
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Fig. 1. Transmitter structure of a coded MIMO system.
Fig. 2. Turbo receiver structure for a coded MIMO system, where 5 denotes the interleaver and 5
The deterministic SMC demodulator recursively performs exploration and selection steps to detect the final survivor signal paths with the corresponding importance weights. Being soft-input and soft-output in nature, these SMC detectors can serve as the soft demodulator in a turbo receiver for coded MIMO systems, where the extrinsic information is iteratively exchanged between the soft demodulator and the soft outer channel decoder to successively improve the receiver performance. Moreover, when the channel is unknown and is estimated by using pilot symbols, the decoded code bits with high reliability can act as pilots for channel re-estimation. By performing iterative channel estimation and data detection, the receiver performance can be further improved. Note that in [16], an SMC MIMO receiver was developed whose complexity is exponential in terms of the number of transmit antennas, whereas the SMC receivers proposed in this paper has linear complexities in terms of the number of transmit antennas. The remainder of this paper is organized as follows. Section II describes the system under consideration. Background materials on the SMC methodology are provided in Section III. In Section IV, we derive the new class of soft MIMO demodulation algorithms using the stochastic SMC and the deterministic SMC, based on the simple nulling and cancellations BLAST detection scheme. Computer simulation results are provided in Section V, and conclusions are drawn in Section VI.
denotes the deinterleaver.
II. SYSTEM DESCRIPTIONS In this section, we consider a generic coded MIMO system with a turbo receiver. The transmitter and receiver structures are shown in Figs. 1 and 2, respectively. A. Baseband MIMO Signal Model are At the transmitter, a block of information bits . The code bits are then randomly encoded into code bits interleaved and mapped to an M-PSK or M-QAM modulation symbol stream, taking values from a finite alphabet set . Each symbol is serial-to-parallel substreams via demultiplexing, and each converted to substream is associated with a transmit antenna. At each time instance, one symbol from each substream is transmitted from symbols transmitted its corresponding antenna, resulting in simultaneously in the same frequency band. Such a space-time bit interleaved coded modulation (BICM) allows for a better exploitation of the spatial, temporal, and frequency diversity resources available in the wireless MIMO systems [17], [18]. Following the common assumption for BLAST systems [1], transmit and re[5], we consider an MIMO system with . The wireless channel is assumed ceive antennas with to have rich-scattering and flat fading. The fading between each transmit and receive antenna pair is assumed to be independent.
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The channel is also assumed quasistatic, i.e., it is static over a data burst and changes from burst to burst. At the receiver, after matched filtering and symbol rate samreceive antennas is pling, the received signal vector from all denoted as
for symbol
. The a priori symbol probability for each is then given by
(4) where the code bit probability can be computed from the corresponding LLR as follows [21]:
In complex baseband representation, the received signal can be expressed as the linear combination of the transmitted signal (5)
(1) is the total signal energy at the transis the complex fading channel matrix, is the spatially and temporally white Gaussian noise, and is the data burst length. . The received signal is matched-filtered Note that and whitened to obtain (here for simplicity, we drop the time index )
where mitter,
,
(2)
Note that initially, the extrinsic information is set to be zero so that the QPSK symbols are equally probable. At the output of the soft demodulator are the a posteriori symbol probabilities . Note that
The exact expression of
is given by
. Based on (2), the maxwhere imum likelihood (ML) MIMO detector is given by (3) (6) Subsequently, the complexity of the ML receiver grows exponentially in the number of transmit antennas. Alternative suboptimal approaches, such as the method based on zero-forcing nulling and cancellation with ordering and the method based on MMSE nulling and cancellation with ordering [1], [6], have lower complexities but also incur substantial performance loss. The sphere decoding method has recently emerged as a near-optimal BLAST detection algorithm. However, its complexity is cubic in term of the number of transmit antennas [7], [8]. B. Turbo Receiver Since the combination of M-PSK or M-QAM modulation and symbol-antenna mapping effectively acts as an inner encoder in the MIMO transmitter, the whole system can be perceived as a serial concatenated system, and an iterative (turbo) receiver [19], [20] can be designed for such a system as shown in Fig. 2. The turbo receiver consists of two stages: the soft-input soft-output SMC demodulator to be developed in Section IV followed by a soft channel decoder. The two stages are separated by a deinterleaver and an interleaver. The soft MIMO demodulator takes as the input the extrinsic delivered by the channel decoder in the information previous turbo iteration as well as the received signals given by (2). First, the symbol a priori probability is calculated as follows. Assuming QPSK modulation is employed, the is mapped to symbol . Additionally, bit pair assume that in the MIMO systems, the transmitted symbol corresponds to the interleaved code bit pair ,
possible Since the summations in (6) are over all the , its complexity is exponential vectors in in the number of transmit antennas and impractical for systems with high spatial-multiplexing gain. In this paper, we will develop some low complexity algorithms for approximating (6). Based on the symbol a posteriori probabilities computed by the soft demodulator, the bit LLR computer calculates the a posteriori log-likelihood ratios (LLRs) of the interleaved code bits . Assuming that the code bit is included in a QPSK , the LLR of this code bit is given by symbol
(7) Using Bayes’ rule, (7) can be written as (8)
, repwhere the second term in (8), which is denoted by , which is computed resents the a priori LLR of the code bit by the channel decoder in the previous iteration, interleaved, and
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then fed back to the soft MIMO demodulator. For the first iteration, it is assumed that all code bits are equally likely. The , represents the first term in (8), which is denoted by extrinsic information delivered by the soft MIMO demodulator, based on the received signals , the MIMO signal structure, and the a priori information about all other code bits. The extrinsic is then deinterleaved and fed back to the information channel decoder as the a priori information for the channel decoder. The soft decoder takes as input the a priori LLRs of the code bits and delivers as output an update of the LLRs of the coded bits, as well as the LLRs of the information bits based on the code constraints. The soft channel decoding algorithm [22] computes the a posteriori LLR of each code bit code constraints code constraints
One way to produce such a training sequence is to use the complex Walsh codes generated by the Hadamard matrix. As shown in Fig. 2, at the end of each turbo iteration, the a posteriori LLRs of the code bits are fed back from the soft channel decoder through the interleaver to the MIMO channel estimator for channel re-estimation. If the LLRs of all code bits corresponding to one QPSK MIMO symbol are above a predetermined threshold, the corresponding decoded MIMO symbol is considered to be with high reliability and will act as a training symbol for channel re-estimation at the beginning of the next turbo iteration. Intuitively, as the turbo receiver iterates, such a scheme obtains increasingly more accurate channel estimation since it makes use of more and more training symbols. Accordingly, the overall receiver performance will be improved compared with the scheme where the channel is estimated only once using the pilot symbols before the first turbo iteration.
(9) where the factorization (9) is shown in [21]. It is seen from (9) that the output of the soft decoder is the sum of the a priori inand the extrinsic information delivered formation by the channel decoder. This extrinsic information is the information about the code bit gleaned from the a priori informabased on the constraint tion about the other code bits structure of the code. Note that at the first iteration, the extrinsic and is statistically independent, information but subsequently, since it uses the same information indirectly, it will become increasingly more correlated, and finally, the improvement through the iterations will diminish. C. MIMO Channel Estimation Since the receiver has no knowledge of channel state information in realistic MIMO systems, pilot symbols embedded in the data stream are required to estimate the channel impulse time slots are used at the beginresponse. Assume ning of each data burst to transmit known pilot symbols . Denote the corresponding received signal as . Then, we have
III. BACKGROUND ON SEQUENTIAL MONTE CARLO Sequential Monte Carlo (SMC) is a class of efficient methods that obtains random samples from a sequence of probability distributions known only up to a normalizing constant. A general framework for the SMC methods is briefly explained next. Consider the following generic sequence of , the a posteriori probability distributions is the set of unobserved pawhere is the set rameters to be estimated, and of available observations at index . We emphasize the fact that is not necessarily a time index. Typically, a posteriori distributions are only known up to a normalizing constant as is not available in closed form, and thus, SMC is of great interest in this context. Assume one wants to compute the minimum mean-square error , which is given by (MMSE) estimate of (14) Given
(10)
random samples
distributed according to
, this expectation can be approximated numerically through
. In [23], two forms of channel estimawhere tors are given, namely, the maximum likelihood (ML) estimator, hwich is given by (11) and the minimum mean-square error (MMSE) channel estimator, which is given by (12) It is also shown in [23] and [24] that the optimal training sequence in the sense of minimizing the channel estimation error should satisfy the following orthogonality condition: (13)
(15) is often not feasible, and it is Sampling directly from necessary to derive approximate methods. A. Sequential Importance Sampling is often not feasible or Sampling directly from too computationally expensive, but drawing samples from some trial density “close” to the distribution of interest is often easy. In this case, we can use the idea of importance sampling. Supdistributed according pose a set of random samples is available. We can still come up with valid to estimates of (14) by correcting for the discrepancy between
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and
. By associating the importance
weight
to the sample
with
, we can indeed estimate (14) as
. The pair
is called
a properly weighted sample with respect to the distribution . It is crucial to notice that one only needs to compute the weights up to a normalizing constant so that the is not necessary. In the “optimal” case knowledge of , then all the weights are equal where and have zero variance. Roughly speaking, the performance of the method typically deteriorates when the variance of the weights increases. The importance sampling method is a generic method that is not sequential. However, it is easy to come up with a sequential version of it. Suppose a set of properly weighted samples with respect to is available . Based on these samples, we want to generate a at index that are properly weighted new set of samples . The sequential importance sampling with respect to (SIS) algorithm proceeds as follows. from a trial distribution • Draw a sample , and let • Update the importance weight
(16)
One can check that the above algorithm indeed generates properly weighted samples with respect to the distribution . It is indeed simply an importance sampling method with trial distribution satisfying
This procedure is very general but it is crucial to design a “good” trial distribution to obtain good performance. It is easy to establish that the optimal importance distribution—optimal as it minimizes the conditional variance of the weights [10]—is given by
B. Resampling measures the “quality” of the The importance weight . A relatively small corresponding imputed signal sequence weight implies that the sample is drawn far from the main body of the a posteriori distribution and has a small contribution in the final estimation. Such a sample is said to be ineffective. Whatever the importance distribution is used, the SIS algorithm is getting inefficient as increases. This is because the discrepancy and can only increase with . between In practice, after a few steps of the procedure, only one stream dominates all the others, i.e., its importance weight is close to 1, whereas the others are close to 0. To make the SIS procedure efficient in practice, it is necessary to use a resampling procedure, as suggested in [25]. Roughly speaking, the aim of resampling is to duplicate the streams with large importance weights while eliminating the streams with small ones, i.e., one focuses the computational efforts in the promising zones of the space. Each is copied times with . Many resample sampling procedures have been proposed in the literature [11], [14], [15], [25], [26]. We use here the systematic resampling procedure proposed in [26]. , • Compute the cumulative distribution for • Draw a random number uniformly in . Compute with • Set • Assign uniform weights to the new set of streams This algorithm is very computationally efficient. In the class , this of unbiased resampling schemes, i.e., and displays better algorithm minimizes the variance performance than other algorithms. Note that if resampling is used in conjunction with the optimal importance distribution (17), one should perform the resampling step before the sampling step at time as, in this very specific case, the importance weight at time is independent of . the samples Resampling can be done at every fixed-length time interval (say, every five time steps), or it can be conducted dynamically. The effective sample size is a criterion that can be used to monitor the variation of the importance weights of the sample streams and to decide when to resample as the system evolves. The effective sample size is defined as
where
, which is the coefficient of variation, is given by
(17) In this case, the importance weight recursion is
with . Roughly speaking, the effective sample size is a measure of variation over the weights. In the optimal case where all the weights are equal, then it is maximum
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and equal to . When one weight dominates all the others, then it is very small. In dynamic resampling, a resampling step is is below a certain performed once the effective sample size threshold. An alternative criterion such as the entropy of the weights could also be used. Heuristically, resampling can provide chances for good sample streams to amplify themselves and, hence, “rejuvenate” the samples to produce better results in the next steps. It can be shown that if is sufficiently large, the resampled streams drawn by the above resampling scheme are also properly . In practice, when small to weighted with respect to in this paper), the resampling modest is used (we use procedure can be seen as a tradeoff between the bias and the variance. That is, the new samples with their weights resulting from the resampling procedure are only approximately proper, which introduces small bias in the Monte Carlo estimates. On the other hand, resampling significantly reduces the Monte Carlo variance for the future samples. C. Alternative Deterministic Procedure SMC is a very general set of methods to sample probability distributions on any space one wants. We restrict ourselves to a case of great interest in telecommunications, namely, the case can only take values in a finite set say . In this case, where could be computed exthe a posteriori distribution actly, but it is typically far too computationally expensive when is large, as can take possible values, where is the number of elements in . In this very specific case, the SIS procedure with the optimal importance distribution (17) takes the following form. • Draw a sample
from
and let
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streams. Indeed, for each
, one needs to compute
for . By sampling , this information is somehow discarded. An alternative deterministic “candidate” approach consists of keeping among the , which are the trajectories at time , trajectories with the highest a posteriori distribution. Clearly, the local approximation error one commits at the selection step using this deterministic strategy is lower than using the randomized strategy. Nevertheless, it does not guarantee performance to be better overall. with a posteriori distributions To sum up, given (known up to a normalizing constant)
, the algo-
rithm proceeds as follows at time . • Compute (19) “best” distinct streams • Select and preserve only the hypotheses. amongst the The drawback of this approach is that clearly, the past observations are as important as the current observation in the calculation of the weight. If an error is committed, then it has a significant impact on further decisions. The randomized algorithm is less sensitive to this problem as when the streams are . resampled, their weights are set to It is worth noticing that both the stochastic and deterministic algorithms are well suited for parallel implementation. IV. SOFT MIMO DEMODULATION ALGORITHMS In this section, we will derive soft-input soft-output MIMO demodulation algorithms based on the sequential Monte Carlo principle. The importance sampling density is obtained by utilizing the artificial sequential structure of the existing simple nulling and cancellation BLAST detection scheme. First, the channel parameters are assumed to have been estimated perfectly at the receiver through the short training sequence before detection. Later, we will examine the effect of channel estimation error on systems performance with pilot symbols along with high-quality symbol decision feedback.
. • Update the importance weight
The importance weight is computed using
(18)
A. Simple Nulling and Cancellation BLAST Detection Algorithm Consider the signal model (2), we denote the QR-decompoas sition of
Two points can be criticized about this algorithm. First, assume is equal to with . that the number of streams ; then, one could compute exactly!). (If possibilities The algorithm should enumerate the first and compute their a posteriori probabilities, i.e., one should exactly. This can be easily incorporated in compute the initialization phase of an SMC algorithm. Second, one can see that the calculation of the weights (18) involves computing the a posteriori distribution up to a normalizing constant of
(20) where is a unitary matrix, and is an upper triangular matrix. The nulling operation is a coordinate rotation that left multiplies to produce a sufficient statistic the vector on (2) by (21)
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where . Since is unitary, there is no noise enhancement, and the noise whitening characteristic is maintained by . We rewrite (21) as nulling, i.e.,
.. .
..
.
.. .
.. .
.. .
(22)
We can detect data symbols directly by nulling operation, i.e., . However, in [5], it is shown that by utimultiply by lizing the upper triangular structure of , significant improvement over zero-forcing can be obtained with the following successive interference cancellation method:
Fig. 3. BER performance of various MIMO demodulation algorithms in an uncoded MIMO system with n = n = 8. QPSK modulation.
probability distributions
. This
sequence of “artificial” distributions is defined by (25) The aim of SMC MIMO detection is to compute an estimate of the a posteriori symbol probability
.. .
(26) (23) after nulling. Let be a sample drawn by the SMC at each symbol interval, where is the number of samples. In order to implement the SMC, we need to obtain a set of Monte Carlo samples of the transmitted symbols properly weighted with respect to the distribution of . In in the application of MIMO demodulation, the function as (15) is specified by the indicator function based on the received signal
. Although the above where simple nulling and cancellation scheme has a very low complexity, its performance is much worse than that of the methods based on zero-forcing or MMSE nulling and interference cancellation with ordering as well as that of the sphere decoding algorithm (see Fig. 3). In what follows, we develop SMC-based MIMO demodulation algorithms using the above simple nulling and cancellation method as the kernel. As will be seen in Section V, these new algorithms provide near-optimal performance in both uncoded and coded MIMO systems. B. Stochastic SMC MIMO Demodulator As from (22), the artificial sequential structure of the simple nulling and interference cancellation scheme due to the uppertriangular structure is well suited for applying the SMC method to MIMO data detection with the particularity of operating on to antenna 1. Indeed, spatial domain starting from antenna one has (24)
, and . We where can thus use SMC methods to simulate from the sequence of
if if
.
(27)
Hence, the a posteriori probability of the information symbol can then be estimated as
(28)
where Following (17) in Section III-A, we choose the trial distribution as (29)
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For this trial distribution, the importance weight is updated according to
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3) Compute the importance weight (38) 4) Compute the a posteriori probability according to of the information symbol (28). 5) Perform resampling as described in Section III-B.
(30) We next specify the computation of the predictive distribution in (29) and (30). First, we consider the trial distributions in (29) (31) in (21) is white Gaussian, i.e., Since the noise , we have (32) where the mean
is given by
C. Deterministic SMC MIMO Demodulator The deterministic method for estimating the sequence of probability distributions proceeds as follows. Similarly to the stochastic case, we samples are drawn at each iteration, where assume that with . Note that if , it is equivalent to the maximum likelihood detection and involves calculations. In this step, we compute exactly the distribution by enumerating all particles for down to antenna . As a result, we have antenna distinct QPSK sequences with a set of weights
satisfying
(33) Note that (34) (35) (39)
Hence, the predictive distribution in (30) is given by where
(40) (36) Finally, we summarize the stochastic SMC MIMO demodulator as follows: 0) Initialization: All importance weights . The are initialized as following steps are implemented at the th recursion to update : each weighted sample. For 1) Compute the trial sampling density for each according to (35) with the a priori symbol probability obtained from the last turbo iteration calculated using (4). from the set with 2) Draw a sample probability (37)
(41) is given by (33). and The second step is algorithm described in Section III-C, to antenna 1 (i.e., which is performed from antenna ). We need to update the importance weight according to
(42) where the distinct QPSK symbol sequences with highest importance weights are selected as the survivor paths over hypotheses. The deterministic SMC MIMO demodulation algorithm is summarized as follows. , 0) Initialization: For calculate the exact expression of the probability distribution via (41) by enuparticles. merating all
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The following steps are implemented at the th recursion to up: date each weight sample. For 1) Compute the importance weight (43) is given by (41). where 2) Select and preserve only the “best”distinct streams with the highest weights amongst the hypotheses with weights set . 3) Compute the a posteriori probability according to of the information symbol (28). D. Combined Deterministic/Stochastic SMC MIMO Demodulator We next propose a combined deterministic and stochastic SMC (CDSMC) demodulator. The new method has an adto antenna ditional parameter such that from antenna (first stages), the demodulator generates the QPSK symbol samples in the same manner as the deterministic SMC, whereas in the remainder of the process, the demodulator generates samples in the same manner as the stochastic SMC demodulator. We summarize the combined deterministic and stochastic SMC MIMO demodulation algorithm as follows. , 0) Initialization: For calculate the exact expression of the probability distribution via (41) by enuparticles. merating all The following steps are implemented at the th recursion to update each weight sample. For : 1) Compute the importance weight (44) is given by (41). where 2) Select and preserve only the “best”distinct streams with the highest weights amongst the hypotheses with weights set . The following steps are implemented at the th recursion ( ) to update : each weight sample. For 3) Compute the trial sampling density for each according to (35) with the a priori symbol probability obtained from the last turbo iteration calculated using (4). from the set with 4) Draw a sample probability (45)
5) Compute the importance weight (46) 6) Compute the a posteriori probability according to of the information symbol (28). 7. Perform resampling as described in Section III-B. V. SIMULATION RESULTS In this section, we provide computer simulation results to illustrate the performance of the proposed turbo receivers in . The fading coflat-fading MIMO channels with . Channels are efficients are generated according to assumed to be quasistatic, i.e., they remain constant over the entire frame of symbols but vary from frame to frame. Simulation results are obtained by averaging over 500 channel realizations. The number of samples drawn in the stochastic SMC . The number of samples drawn in the dealgorithm is (i.e., ) in the uncoded case terministic algorithm is (i.e., ) in the coded case. and A. Performance in Uncoded MIMO Systems We first illustrate the performance of the proposed stochastic and deterministic SMC MIMO demodulation algorithms in an uncoded MIMO system. The BER performance of these two new algorithms, together with some existing detection algorithms (including the sphere decoding algorithm, the method based on MMSE nulling and cancellation with ordering, the method based on zero-forcing nulling and cancellation with ordering, as well as the simple nulling and cancellation method) are shown in Fig. 3. Several observations are in order. First of all, the simple nulling and cancellation method displays very poor performance. However, when it is combined with SMC (deterministic or stochastic), we obtain a much more powerful MIMO detector. Second, the deterministic detector outperforms the stochastic SMC detector, and it actually slightly outperforms the sphere decoding algorithms, which is the best suboptimal MIMO detector known so far. This is because the sphere decoding minimizes the probability of sequence , whereas the sequential Monte Carlo error, i.e., calculates the a posteriori symbol probabilities and minimizes . In addition, it is well the symbol error rate, known that the MAP decoding performs slightly better than the ML decoding in terms of symbol error rate. Fig. 4 illustrates the BER performance of the combined deterministic and stochastic SMC (CDSMC) detector with different combination of deterministic SMC and stochastic SMC. The number of samples drawn in all three schemes (i.e., ). It is seen that the performance of is the CDSMC demodulator is better than that of the stochastic SMC demodulator, whereas it is still worse than that of the deterministic SMC demodulator. As the parameter increases, the performance of the CDSMC approaches that of the deterministic SMC. Note that the CDSMC demodulator with
DONG et al.: NEW CLASS OF SOFT MIMO DEMODULATION ALGORITHMS
Fig. 4. BER performance of various combined D&SMC demodulation = n = 8. QPSK algorithms in an uncoded MIMO system with n modulation.
Fig. 5. BER performance of a turbo MIMO receiver employing the stochastic SMC MIMO demodulator. Channel estimation is based on pilots only. n = n = 8. Rate-1/2 constraint length-5 convolutional code. QPSK modulation.
is equivalent to the deterministic SMC demodulator in MIMO . systems with B. Performance in Coded MIMO Systems For the coded MIMO system, a rate-1/2 constraint length-5 convolutional code (with generators 23 and 35 in octal notation) is employed at the transmitter. The bit interleaver is randomly generated and fixed throughout the simulations. The code bit block size is 512, corresponding to 256 information bits and 32 QPSK MIMO symbols. The channel estimator employed here is the MMSE MIMO channel estimator given by (12), where , and QPSK orthogonal MIMO pilot symbols are used. The number of turbo iterations in each simulation run is four.
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Fig. 6. BER performance of a turbo MIMO receiver employing the stochastic SMC MIMO demodulator. Iterative channel estimation is based on pilots and the decoded symbols with high reliability. n = n = 8. Rate-1/2 constraint length-5 convolutional code. QPSK modulation.
Fig. 7. BER performance of a turbo MIMO receiver employing the stochastic SMC MIMO demodulator. Genie-aided channel estimation is based on pilots and all information symbols. n = n = 8. Rate-1/2 constraint length-5 convolutional code. QPSK modulation.
In Figs. 5–7, the BER performance of the turbo receiver employing the stochastic SMC demodulator is plotted. The solid curves in these figures correspond to the BER performance with perfectly known channel, whereas the dashed curves correspond to the BER performance with different channel estimation schemes. In Fig. 5, the channel is estimated only once based on the pilots and used throughout all turbo iterations. In Fig. 6, the channel is re-estimated at the beginning of each turbo iteration using both the pilots and the decoded symbols with high-reliability, as discussed in Section II-C, and in Fig. 7, we assume that the channel is estimated by a genie that knows both the pilot symbols and all information symbols.
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Fig. 8. BER performance of a turbo MIMO receiver employing the deterministic SMC MIMO demodulator. Channel estimation is based on pilots only. n = n = 8. Rate-1/2 constraint length-5 convolutional code. QPSK modulation.
Fig. 10. BER performance of a turbo MIMO receiver employing the deterministic SMC MIMO demodulator. Genie-aided channel estimation is based on pilots and all information symbols. n = n = 8. Rate- 1=2 constraint length-5 convolutional code. QPSK modulation.
VI. CONCLUSIONS
Fig. 9. BER performance of a turbo MIMO receiver employing the deterministic SMC MIMO demodulator. Iterative channel estimation is based on pilots and the decoded symbols with high reliability. n = n = 8. Rate-1/2 constraint length-5 convolutional code. QPSK modulation.
Such a genie-aided channel estimate is used throughout all turbo iterations. Obviously, the scenario for Fig. 7 provides an upper bound on the achievable performance for the turbo receivers with different channel estimation schemes. It is seen that by performing iterative channel estimation and symbol detection, the turbo receiver offers a performance approaching the genie-aided bound. The corresponding BER performance of turbo receiver employing the deterministic SMC demodulator is shown in Figs. 8–10. It is seen that the deterministic SMC algorithm offers improved and more stable performance than its stochastic counterpart.
In this paper, we have developed a new family of soft-input soft-output MIMO demodulation algorithms. These new techniques use the conventional BLAST detection based on simple nulling and cancellation as the kernel and are based on the sequential Monte Carlo (SMC) method for Bayesian inference. Two versions of such SMC MIMO demodulation algorithms are developed, based on, respectively, stochastic and deterministic sampling. As hard MIMO detection algorithms, the proposed SMC demodulation algorithms significantly outperform all existing BLAST detection methods. Moreover, the deterministic SMC MIMO detector slightly outperforms the sphere decoding algorithm while maintaining the lower complexity. In fact, the complexity of the deterministic SMC demodulator , the complexity of the stochastic is , and the complexity of the SMC demodulator is . Furthermore, in coded MIMO syssphere decoder is tems, the proposed SMC algorithms can naturally serve as the soft MIMO demodulator in a turbo receiver. Simulation results indicate that overall, the deterministic SMC MIMO demodulator offers better and more stable performance compared with its stochastic counterpart in both the uncoded and the coded MIMO systems. It remains a challenging open problem to theoretically justify the superior performance of the deterministic SMC algorithm over that of the stochastic SMC algorithm. REFERENCES [1] G. J. Foschini and M. J. Gans, “On the limits of wireless communications in a fading environment when using multiple antennas,” Wireless Pers. Commun., vol. 6, no. 3, pp. 311–335, 1998. [2] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans. Telecommun., vol. 10, no. 6, pp. 585–595, Nov. 1999. [3] C. N. Chuah, D. N. C. Tse, J. M. Kahn, and R. A. Valenzuela, “Capacity scaling in MIMO wireless systems under correlated fading,” IEEE Trans. Inform. Theory, vol. 48, pp. 637–650, Mar. 2002.
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[4] P. W. Wolniansky, G. J. Roschini, G. D. Golden, and R. A. VAlenzuela, “V-BLAST: An architecture for realizing very high data rates over the rich-scattering wireless channel,” in Proc. Int. Symp. Signals, Systems, Electron., Pisa, Italy, Sept. 1998. [5] G. J. Foschini, “Layed space-time architecture for wireless communication in a fading environment when using multi-element antennas,” Bell Labs. Tech. J., vol. 1, no. 2, pp. 41–59, 1996. [6] G. D. Golden, G. J. Foschini, R. A. Valenzuela, and P. W. Wolniansky, “Detection algorithm and initial laboratory results using V-BLAST space-time communication architecture,” Electron. Lett., vol. 35, pp. 14–16, Jan. 1999. [7] O. Damen, A. Chkeif, and J. Belfiore, “Lattice code design for space-time codes,” IEEE Commun. Lett., vol. 4, pp. 161–163, May 2000. [8] B. M. Hochwald and S. T. Brink, “Achieving near-capacity on a multiple-antenna channel,” IEEE Trans. Commun., submitted for publication. [9] A. M. Chan and I. Lee, “A new reduced sphere decoder for multiple antenna systems,” in Proc. 2002 Int. Commun. Conf., New York, Apr. 2002. [10] A. Doucet, S. J. Godsill, and C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Statist. Comput., vol. 10, no. 3, pp. 197–208, 2001. [11] X. Wang, R. Chen, and J. S. Liu, “Monte Carlo Bayesian signal processing for wireless communications,” J. VLSI Signal Process., vol. 30, no. 1–3, pp. 89–105, Jan.–Mar. 2002. [12] R. Chen, X. Wang, and J. S. Liu, “Adaptive joint detection and decoding in flat-fading channels via mixture Kalman filtering,” IEEE Trans. Inform. Theory, vol. 46, pp. 2079–2094, Sept. 2000. [13] R. Chen and J. S. Liu, “Sequential Monte Carlo methods for dynamic systems,” J. Amer. Stat. Assoc., vol. 93, pp. 1302–1044, 1998. , “Mixture Kalman filters,” J. Amer. Stat. Assoc. (B), vol. 62, pp. [14] 493–509, 2000. [15] Z. Yang and X. Wang, “A sequential Monte Carlo blind receiver for OFDM systems in frequency-selective fading channels,” IEEE Trans. Signal Processing, vol. 50, pp. 271–280, Feb. 2002. [16] D. Guo and X. Wang, “Blind detection in MIMO systems via sequential Monte Carlo,” IEEE J. Select. Areas. Commun., vol. 21, pp. 453–464, Apr. 2003. [17] A. M. Tonello, “On turbo equalization of interleaved space-time codes,” in Proc. Fall Vehicular Technology Conf., Oct. 2001. [18] J. J. Boutros, F. Boixadera, and C. Lamy, “Bit-interleaved coded modulations for multiple-input multiple-output channels,” in Proc. IEEE 6th Int. Symp. Spread-Spectrum Technol. Applications, Sept. 2000. [19] H. Dai and A. F. Molisch, “Multiuser detection for interference-limited MIMO systems,” in Proc. Spring Vehicular Technology Conf., May 2002. [20] M. Sellathurai and S. Haykin, “TURBO-BLAST for wireless communications: Theory and experiments,” IEEE Trans. Signal Processing, vol. 50, pp. 2538–2546, Oct. 2002. [21] X. Wang and H. V. Poor, “Iterative (Turbo) soft interference cancellation and CDMA,” IEEE Trans. Commun., vol. 47, pp. 1046–1061, Jul. 1999. [22] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inform. Theory, vol. 20, pp. 284–287, Mar. 1974. [23] T. L. Marzetta, “BLAST training: Estimation channel characteristics for high-capacity space-time wireless,” in Proc. 37th Annu. Allerton Conf. Commun., Comput., Control, Sept. 1999. [24] Q. Sun, D. C. Cox, A. Lozano, and H. C. Huang, “Training-based channel estimation for continuous flat fading BLAST,” in Proc. Int. Conf. Commun., New York, Apr. 2002. [25] N. J. Gordon, D. J. Salmond, and A. F. M. Smith, “Novel approach to nonlinear non-Gaussian Bayesian state estimation,” Proc., Inst. Elect. Eng. F. Radar Sonar Navigat., vol. 140, no. 2, Apr. 1993. [26] G. Kitagawa, “Monte Carlo filter and non-Gaussian nonlinear state space models,” J. Comput. Graph. Statist., vol. 5, no. 1, pp. 1–25, 1996.
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Bin Dong received the B.S. degree in electrical engineering from Shanghai Jiao Tong University, Shanghai, China, in 2000 and the M.S. degree in electrical and computer engineering from Queen’s University, Kingston, ON, Canada, in 2002. He is currently pursuing the Ph.D. degree with the Department of Electrical Engineering, Columbia University, New York, NY. His research are in the area of statistical signal processing, wireless communications, and wireless networks with the application in wireless MIMO, OFDM, and CDMA systems. Mr. Dong reecived the W. W. King Fellowship in 2001.
Xiaodong Wang received the B.S. degree in electrical engineering and applied mathematics (with the highest honor) from Shanghai Jiao Tong University, Shanghai, China, in 1992, the M.S. degree in electrical and computer engineering from Purdue University, West Lafayette, IN, in 1995, and the Ph.D. degree in electrical engineering from Princeton University, Princeton, NJ, in 1998. From July 1998 to December 2001, he was an Assistant Professor with the Department of Electrical Engineering, Texas A&M University, College Station. In January 2002, he joined the Department of Electrical Engineering, Columbia University, New York, NY, as an Assistant Professor. His research interests fall in the general areas of computing, signal processing, and communications. He has worked in the areas of digital communications, digital signal processing, parallel and distributed computing, nanoelectronics, and bioinformatics and has published extensively in these areas. His current research interests include wireless communications, Monte Carlo-based statistical signal processing, and genomic signal processing. Dr. Wang received the 1999 NSF CAREER Award and the 2001 IEEE Communications Society and Information Theory Society Joint Paper Award. He currently serves as an Associate Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS, the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, the IEEE TRANSACTIONS ON SIGNAL PROCESSING, and for the EURASIP Journal of Applied Signal Processing.
Arnaud Doucet was born in France on November 2, 1970. He graduated from Institut National des Telecommunications in June 1993 and received the Ph.D. degree from Universite Paris-Sud, Orsay, France, in December 1997. From January 1998 to February 2001, he was a research associate with Cambridge University, Cambridge, U.K. From March 2001 to August 2002, he was a Senior Lecturer with the Department of Electrical Engineering, Melbourne University, Parkville, Australia. Since September 2002, he has been a University Lecturer with the Engineering Department, Cambridge University. His research interests include sequential Monte Carlo methods, Markov chain Monte Carlo methods, optimal filtering, and control and reinforcement learning.
