Blind detection in MIMO systems via sequential ... - Semantic Scholar
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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 3, APRIL 2003
Blind Detection in MIMO Systems via Sequential Monte Carlo Dong Guo and Xiaodong Wang, Member, IEEE
Abstract—In this paper, we provide a novel sequential estimation and detection approach for multiple-input–multiple-output (MIMO) systems. The basic idea is to design a probabilistic approximation method for the computation of the maximum a posterior distribution (MAP) via the sequential Monte Carlo methods (SMC). The SMC method has two advantages over the other methods in that it is a blind method and can be computed in parallel. Furthermore, the SMC has characteristics of soft-input and soft-output in nature and, thus, it can be employed as the first stage demodulator in a turbo receiver for a coded MIMO system. Such a turbo receiver successively improves the receiver performance by iteratively exchanging the so-called extrinsic information with the MAP outer channel decoder. Finally, the performance of the proposed sequential Monte Carlo receiver is demonstrated through computer simulations for the MIMO systems over the single-path and multipath fading channels. Index Terms—Multiple-input–multiple-output (MIMO), sequential Monte Carlo (SMC) method, turbo receiver.
I. INTRODUCTION
T
HERE is at present a significant amount of interest in the design of multiple-input–multiple-output (MIMO) communication systems for high date-rate wireless communications. An MIMO system employs multiple antennas at both the transmitter and the receiver, and its capacity increases linearly with the minimum between the numbers of transmit and receive antennas. When the channels of an MIMO systems are known, various detectors can be employed to demodulate the transmitted data symbols, such as the maximum-likelihood sequence detector, the zero-forcing detector, the minimum mean-square error detector, and the detector based on interference cancellation [1], [2]. On the other hand, blind identification and equalization of MIMO systems has also received much attention. Existing work on this topic are primarily based on the exploitation of either the second-order statistics (SOS) [3]–[11], or the higher-order statistics of the received signals [12]–[16]. The sequential Monte Carlo (SMC) methodology [17]–[22] recently emerged in the fields of statistics and engineering has shown a great promise in solving a wide class of sophisticated statistical inference problems. Under a state-space framework, the SMC recursively generate Monte Carlo samples of the state variable or some other latent variables; based on which the posterior distribution of any system parameter of interest Manuscript received May 2, 2002; revised October 31, 2002. The work was supported in part by the U.S. National Science Foundation under Grant CCR9875314, Grant CCR-9980599, and Grant DMS-0073651. The authors are with the Department of Electrical Engineering, Columbia University, New York, NY 10027 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/JSAC.2003.809722
can be approximated. Compared with the classical expectation-maximization (EM) techniques in solving hidden data problems, which are prone to being trapped in locally optimal points, the SMC methods exhibit global convergence and is more robust to the choice of initial conditions [20], [23]. In this paper, we treat the problem of symbol detection in MIMO communication systems with unknown channels under the Bayesian SMC framework. A Bayesian formulation of the problem is first presented, and then the SMC blind adaptive receiver is derived based on the techniques of important sampling and resampling. The new receiver does not require any training or pilot symbols or decision feedback, and it performs well in both flat-fading and frequency-selective fading channels. Another issue treated in this paper is a turbo MIMO receiver structure [24]–[26] that employs the blind SMC receiver as the first-stage soft-input soft-output demodulator. Such a receiver iterates between the Bayesian demodulation stage and the maximum a posterior channel decoding stage; and the so-called extrinsic information of data symbols are iteratively exchanged between these two stages to successively improve the overall receiver performance. The rest of this paper is organized as follows. In Section II, the MIMO systems over flat-fading and frequency-selective fading channels are described. In Section III, some background materials on the SMC techniques are provided. In Section IV, we derive an SMC algorithm for blind symbol detection in MIMO systems. In Section V, a turbo receiver based on the proposed SMC detector is discussed for channel-coded MIMO systems. Simulation results are provided in Section VI and a brief summary is given in Section VII. II. SYSTEM DESCRIPTION A. Flat Fading MIMO Channels We consider an MIMO system with transmit and receive antennas over slow flat-fading channels with additive Gaussian noise. The transmitted symbols are assumed to be independent in time as well as in space. Moreover, they are taken from a finite . At each transmit antenna alphabet set , , the transmitted symbols are differentially modulated from the binary information source symbols with . Such a differential encoding scheme is necessary to resolve the phase ambiguity inherent to any blind receiver method. Note that in blind systems, there is also an antenna ambiguity-namely, the receiver can only recover the symbols transmitted from the transmit antennas up to a permutation ambiguity. In order to resolve this, a different spreading code is used to spread the symbol transmitted at every transmit antenna.
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GUO AND WANG: BLIND DETECTION IN MIMO SYSTEMS VIA SEQUENTIAL MONTE CARLO
Fig. 1.
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An MIMO system employing the SMC receiver.
The block diagram of such a system is shown in Fig. 1. The received signal vector at the th receive antenna and at time is given by
(1) is the transmitted symbol at the th transmit antenna where , is the and at time ; is the spreading code employed at the th transmit antenna; complex fading channel gain between the th transmit antenna is the and the th receive antenna; and represents a circularly symmetric ambient noise, where denotes identity complex Gaussian distribution, and matrix. Note that here, we assume that the channel is block remain fixed fading, such that the fading coefficients symbols. Denote for a block of diag
based on the received signals and the a priori symbol prob, without knowing the channel response abilities of . The exact solution to this problem involves a very high dimensional integration which is infeasible in practice. Here we will employ the sequential Monte Carlo technique to solve the , above inference problem. Denote , and . The basic idea then is to generate random samples from the joint posterior distribu, and then to estimate the marginal distribution of tion the information symbols using these samples. The details of the algorithm will be discussed in Section IV. B. Frequency-Selective Fading MIMO Channels Now assume that the channel between the th transmit antenna and the th receive antenna is subject to frequencyselective fading, which is modeled by a tapped-delay and complex fading gains line with delays [27]. Here, we assume that the delay spread is small compared with the symbol interval, so that the inter-symbol interference is negligible. Then, the received signal vector at the th receive antenna and at time is
and Then, (1) can be rewritten as (8) (2) We further denote
is the delayed version of the spreading code for where the th transmit antenna. Denote
(3) (4) and
(5)
then, we have (6) and . Our goal is to estimate the a posteriori probabilities of the information symbols Let
(7)
diag
Then, (8) can be rewritten as
(9)
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Furthermore, using the same notations as in (3)–(5), we have
to the sample
, we can estimate the quantity of interest, , as
(10) It is seen that the flat-fading MIMO channel (6) and the frequency-selective fading MIMO channel (10) are described by similar equations, only with different forms of the variables . Then involved. Again, we denote our goal is to estimate the a posteriori probabilities of the based on the received signals information symbols and the a priori information symbol probabilities, without knowing the channel response . III. SMC METHODS SMC is a family of probability approximation methods that use Monte Carlo samples to efficiently estimate the posterior distribution of the unknown variables in a dynamic system [21]. A general framework for the SMC methods is briefly explained next. Consider a dynamic system modeled in the following statespace form state equation observation equation
(11)
where , , and are, respectively, the state variable, the observation, the state noise, and the observation noise at time . They can be either scalars or vectors. In the MIMO system described in the previous section [e.g., (6) and (10)], the state , representing both the unobvariable corresponds to served symbols and the unknown fading channel state. and let . Let is of interest; that is, at Suppose an on-line inference of current time we wish to make a timely estimate of a func, based on the currently tion of the state variable , say available observation, . With the Bayes theorem, we realize that the optimal solution to this problem is d . In most cases, an exact evaluation of this expectation is analytically intractable because of the high-complexity of such a dynamic system. Monte Carlo methods provide us with a viable alternative to the required computation. Specifrandom samples from the ically, if we can draw , then we can estimate by distribution (12)
(14)
. The pair , , is with called a properly weighted sample with respect to distribution . To implement Monte Carlo techniques for a dynamic system, a set of random samples properly weighted with respect to are needed for any time . Because the state equation in system (11) possesses a Markovian structure, we can implement a recursive importance sampling strategy, which is the basis of all sequential Monte Carlo techniques [21]. Suppose (with a set of properly weighted samples ) is given at time . Based on respect to these samples at previous time, a SMC algorithm generates a , which is properly weighted at time new one, with respect to . Note that in most applications we are up to a normalizing constant, only able to evaluate which is sufficient for using (14) in Monte Carlo estimation. . The algorithm is described as follows, for from a trial distribution • Draw a sample and let ; • Compute the importance weight
It can be shown that the above algorithm indeed generates properly weighted samples with respect to the distribution [21]. IV. THE SMC DETECTOR FOR MIMO SYSTEMS A. The SMC Algorithm for MIMO Systems Consider the MIMO system described by (1) or (8). De, , note ; and . as the received Denote also signals at the th receive antenna up to time . Then, the aim of the SMC method is to perform an on-line estimate of the a posteriori symbol probability (15)
is often not feasible, but Direct sampling from drawing samples from some trial distribution is easy. In this case, we can use the idea of importance sampling. Suppose a is generated from the trial set of random samples . By associating the weight distribution
(13)
up to time and the a priori based on the received signals , without knowing the channel resymbol probabilities of sponse . , be a sample draw Let . At each by the SMC at time . Denote time , we need to obtain a set of Monte Carlo samples of the , properly weighted with transmitted symbols, . For every symbol respect to the distribution
GUO AND WANG: BLIND DETECTION IN MIMO SYSTEMS VIA SEQUENTIAL MONTE CARLO
, The a posteriori probability of the information symbol can then be estimated as
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where (25)
(26) (16) with where product; and 1
; the operator denotes element-wise is an indicator function defined as if if
.
(27) In (27), channel (1)
is defined as follows: for the flat-fading MIMO
(17)
Based on the general principle of SMC outlined in Section III, we choose the following trial sampling density to draw the samfrom ples (18)
diag
(28)
and for the frequency-selective fading MIMO channel (8) diag
(29)
Hence, the conditional density in (20) is given by
Then, the importance weight can be updated according to
d
(19)
(30)
which is an integral of a Gaussian probability density function (pdf) with respect to another Gaussian pdf. Hence, the resulting pdf is still Gaussian, i.e., (31)
(20) with the mean and the covariance given, respectively, by where
is denoted as (32) Cov
(21) and (19) follows from the assumption that the channels are unis independent of correlated in space; (20) holds because given and is a first-order Markov chain due to the differential encoding rule. Note that the second term in (20) incorporates the a priori probability of the unknown symbol . then (20) becomes Denote
(33) with (34) is similarly defined as (28) and (29) with where placed by . in (20) can be computed by Therefore,
re-
(22) To compute the predictive density in (20), we assign a Gaussian prior to the channel i.e.,
(35) The trial distribution in (18) can be computed as follows:
(23) Then, the conditional distribution of can be computed as
, conditioned on
and (36) (24)
by (19).
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Note that the a posteriori mean and covariance of the channel in (25) and (26) can be updated recursively as follows. At th is drawn, we combine it with step, after a new sample of to form . Let and be the the past samples . It quantities computed by (32) and (33) for the imputed then follows from the matrix inversion lemma that (25) and (26) become
It is seen that at any time , the dominant computation in the one-step prediction of SMC receiver involves the and the one-step up(32) and (33) in computing . Since the samplers date of (37) and (38) for operate independently and in parallel, the SMC detector is well suited for parallel implementations. B. Resampling Procedure
(37) (38) with (39) Note that (37)–(39) can also be obtained from standard Kalman filtering theory. Finally, we summarize the SMC blind detector for MIMO systems as follows. 1) Initialization: Set the initial channel vector values as , for , ; , All importance weights are initialized as . The following steps are implemented at the th recurto update each weighted sample. sion . For and each , compute 2) For and according to (32)–(34); and compute according to (35). for each 3) Compute the trial sampling distribution .
measures the “quality” The importance sampling weight . A relatively of the corresponding imputed signal sequence small weight implies that the sample is drawn far from the main body of the posterior distribution and has a small contribution in the final estimation. Such a sample is said to be ineffective. If there are too many ineffective samples, the Monte Carlo procedure becomes inefficient. To avoid the degeneracy, a useful resampling procedure, which was suggested in [17], and [21], may be used. Roughly speaking, resampling is to multiply the streams with the larger importance weights, while eliminate the ones with small importance weights. A simple but efficient resampling procedure consists of the following two steps. 1) Sample a new set of streams from with probability pro. portional to the importance weights 2) Assign equal weight to each stream in the new samples, . i.e., Resampling can be done at every fixed-length time interval (say, every five steps) or it can be conducted dynamically. The effective sample size 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 [28]
(40)
4) Draw a sample
from the set
with probability (41)
5) Compute the importance weight (42)
and normalize it by (43) 6) Update the a posteriori mean and covariance of channel. in Step 2, then for If the imputed sample , set ; and update and according to (37)–(39). 7) Compute the a posteriori probability of the information according to (16). symbol 8) Perform resampling as described in Section IV-B.
(44) where
, the coefficient of variation, is given by (45)
. In dynamic resampling, a resamwith is pling step is performed once the effective sample size below a certain threshold. Heuristically, resampling can provide chances for good sample streams to amplify themselves and, hence, “rejuvenate” the sampler to produce a better result for future states as system evolves. It can be shown that the samples drawn by the above resampling procedure are also indeed properly weighted with , provided that is sufficiently large. In respect to in practice, when small to modest is used (we use this paper), the resampling 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 estimation. On the other hand, however, resampling significantly reduces the Monte Carlo variance for future samples.
GUO AND WANG: BLIND DETECTION IN MIMO SYSTEMS VIA SEQUENTIAL MONTE CARLO
Fig. 2.
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A single-user coded MIMO systems employing the turbo receiver.
V. TURBO RECEIVERS
block symbol
. The LLR of this code bit is given for by
In this section, we consider employing the SMC detector in a turbo receiver [25], [26] for coded MIMO systems. As shown in the previous section, the SMC detector not only utilizes the a priori symbol probabilities, but also it produces the a posteriori symbol probabilities. Such a soft-in and soft-output nature makes it an ideal candidate to be employed as the demodulator in a turbo receiver. In what follows, we will consider both single-user and multiuser coded systems. The transmitter structure of a single-user coded MIMO system in shown in the upper part of Fig. 2. A block of are encoded into code bits . The code information bits bits are interleaved and then a serial-to-parallel conversion is performed. Each stream of code bits are differentially encoded and then mapped to binary phase-shift keying (BPSK) symbols. Each symbol is then spread by a short signature code to resolve the antenna ambiguity at the receiver end, and transmitted from one of the transmit antennas. Since the combination of BPSK modulation, the differential encoder, as well as the MIMO channel, effectively acts as an inner coder, a turbo receiver can be designed for such a system, as shown in the lower part of Fig. 2. The turbo receiver consists of two stages: the SMC detector developed in the previous section, followed by a MAP channel decoder [29]. The two stages are separated by a deinterleaver and an interleaver. after interleaving. The SMC Assume that is mapped to receiver takes as input the interleaved a priori log-like likefrom the MAP lihood ratios (LLRs) of code bits channel decoder in the previous turbo iteration as well as the received signals . And it computes a posterior LLR of the code is included in a bits as output. Assume that the code bit
(46) Using the Bayes’ rule, (46) can be written as (47)
, represents where the second term in (47), denoted by , which is computed by the a priori LLR of the code bit the channel decoder in the previous iteration, interleaved and then fed back to the SMC receiver. For the first iteration, it is assumed that all code bits are equally likely. The first term in , represents the extrinsic information (47), denoted by delivered by the SMC receiver, based on the received signals , the structure of signal model, and the prior information about , is then all other code bits. The extrinsic information deinterleaved and fed back to the channel decoder as the a priori information for the channel decoder. The MAP decoder employs the standard MAP decoding algorithm—e.g., the BCJR algorithm for the convolutional codes—to compute the a posteriori LLR of each code bit
(48)
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Fig. 3.
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A multiuser coded MIMO system employing the turbo receiver.
It is seen from (48) that the extrinsic information is obtained by subtracting the prior information from . This extrinsic information is the posterior information gleaned from the prior the information about the code bit , based on the information about the other code bits, constraint structure of the code. After interleaving, the extrinsic is fed information delivered by the channel decoder back to the SMC receiver as the prior information. Assume, for example, that the transmitted symbol corresponds to the in the MAP coder output. The prior code bits is then given by symbol probability for each symbol (49) where the code bit probability can be computed from the corresponding LLR as follows [26]:
(50) At the last turbo iteration, the LLRs of information bits are computed followed by hard decisions on them. We next consider a multiuser coded MIMO system. The transmitter structure of such a system is shown in the upper part of Fig. 3, and the corresponding turbo receiver is shown in the lower part of Fig. 3. Here, in contrast to the single-user system, each stream of the source symbols is independently encoded using some channel code, and the code bits are interleaved by independent interleavers. At the receiver, a soft channel decoder is employed for each user to compute the extrinsic information of the code bits of that user. The output
of all channel decoders are fed to the SMC detector as the prior information for all received code bits. The SMC detector computes the extrinsic information of all code bits based on the received signals , and distribute it to the corresponding soft channel decoders. The process iterates to successively improve the receiver performance. VI. SIMULATION RESULTS In this section, we provide computer simulation results to illustrate the performance of the blind SMC detector as well as the SMC-based turbo receiver in both flat-fading and frequency-selective fading MIMO channels. The fading coefficients are generated according to uncorrelated circularly symmetric complex . Hence, the Gaussian distribution, i.e., total received energy of each transmitted symbol is normalized to unity. The channels are assumed to be block fading, i.e., they symbols, but vary remain constant over the entire block of from block to block. We assume that there are two transmit an. The transtennas and two receive antennas, i.e., mitted symbol at each transmit antenna is spread by a short random sequence. In our simulations, the number Monte Carlo . We consider two types of channels, samples is taken as , and frequency-selective fading i.e., flat-fading channel . channel with A. Performance in Uncoded MIMO Systems We first illustrate the performance of the proposed blind SMC detector for one single user in uncoded MIMO systems. The . In Fig. 4, the bit-error-rate (BER) perforblock size is mance of the SMC blind detector in a flat-fading MIMO channel is shown. Also shown in this figure is the performance of a MAP receiver with perfect channel state information, which serves as
GUO AND WANG: BLIND DETECTION IN MIMO SYSTEMS VIA SEQUENTIAL MONTE CARLO
Fig. 4. The BER performance of the blind SMC detector in an uncoded flat-fading MIMO system. There are two transmit antennas and two receive antennas.
Fig. 5. The BER performance of the blind SMC detector in an uncoded frequency-selective fading MIMO system. There are two transmit antennas and two receive antennas.
a lower bound on the achievable performance for any blind receiver. In Fig. 5, we illustrate the BER performance of the SMC detector, as well as the known-channel lower bound, in a 2-tap frequency-selective fading MIMO system. It is seen from these two figures that the performance of the blind SMC detector is within about 1 dB to the known-channel lower bound. B. Performance in Coded MIMO System We next illustrate the performance of the SMC-based blind turbo receiver in coded MIMO systems. We first consider a single-user coded system with two transmit and two receive anconstraint length-5 convolutional code (with tennas. A a rate generators 23 and 35 in octal notation) is employed. The interleaver is randomly generated and is fixed for all simulations. (i.e., 128 information bits). The code bit block size is
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Fig. 6. The BER performance of the blind SMC detector in a coded flat fading MIMO system. There are two transmit antennas and two receive antennas.
Fig. 7. The BER performance of the blind SMC detector in a coded frequency-selective MIMO system. There are two transmit antennas and two receive antennas.
The performance of the SMC-based turbo receiver for such a single-user system in flat-fading and frequency-selective fading MIMO channels is shown in Fig. 6 and Fig. 7. Next, we consider a two-user system where each user’s information bits are independently encoded by the same convolutional encoder, interleaved, and then transmitted from one of the transmit antennas. The turbo receiver employs the SMC detector at the first stage, and two soft convolutional decoders at the second stage. The performance in flat-fading and frequency-selective fading channels of this two-user MIMO system is shown in Fig. 8 and Fig. 9. It is seen from these figures that the proposed SMC-based blind turbo receiver can successively improve the receiver performance in MIMO systems through iterative processing, in both single-user and multiuser systems, and in both flat-fading and frequency-selective fading channels.
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REFERENCES
Fig. 8. The BER performance of the blind SMC detector in a multiuser coded flat-fading MIMO system. There are two transmit antennas and two receive antennas.
Fig. 9. The BER performance of the blind SMC detector in a multiuser coded frequency selective fading MIMO system. There are two transmit antennas and two receive antennas.
VII. CONCLUSION In this paper, we have developed a blind Bayesian receiver for MIMO communication systems over unknown fading channels. This receiver is based on the SMC methods for computing the a posteriori probabilities of the unknown transmitted symbols. The SMC blind detector is seen to perform close to the known channel lower bound in unknown MIMO channels. Moreover, being soft-input and soft-output in nature, the proposed SMC detector is capable of exchanging the so-called extrinsic information with the maximum a posteriori (MAP) outer channel decoder, and successively improving the overall receiver performance. The good performance of such a blind SMC turbo receiver for both single-user and multiuser coded MIMO systems is also demonstrated.
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[28] R. Chen, X. Wang, and J. 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. [29] L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inform. Theory, vol. IT-20, pp. 284–287, Mar. 1974.
Dong Guo received the B.S. degree in geophysics and computer science from China University of Mining and Technology (CUMT), Xuzhou, China, in 1993, the M.S. degree in geophysics from the Graduate School of Research Institute of Petroleum Exploration and Development (RIPED), Beijing, China, in 1996, and the Ph.D. degree in applied mathematics from Beijing University, Beijing, China, in 1999. He is now working toward his second Ph.D. degree in the Department of Electrical Engineering, Columbia University, New York, NY. His research interests are in the area of statistical signal processing and communications.
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Xiaodong Wang (S’98–M’98) 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 in 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 quantum computing, and has published extensively in these areas. His current research interests include multiuser communications theory and advanced signal processing for wireless communications. He worked at the AT&T Labs-Research, Red Bank, NJ, during summer 1997. 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 SIGNAL PROCESSING, and the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS.
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 3, APRIL 2003
Blind Detection in MIMO Systems via Sequential Monte Carlo Dong Guo and Xiaodong Wang, Member, IEEE
Abstract—In this paper, we provide a novel sequential estimation and detection approach for multiple-input–multiple-output (MIMO) systems. The basic idea is to design a probabilistic approximation method for the computation of the maximum a posterior distribution (MAP) via the sequential Monte Carlo methods (SMC). The SMC method has two advantages over the other methods in that it is a blind method and can be computed in parallel. Furthermore, the SMC has characteristics of soft-input and soft-output in nature and, thus, it can be employed as the first stage demodulator in a turbo receiver for a coded MIMO system. Such a turbo receiver successively improves the receiver performance by iteratively exchanging the so-called extrinsic information with the MAP outer channel decoder. Finally, the performance of the proposed sequential Monte Carlo receiver is demonstrated through computer simulations for the MIMO systems over the single-path and multipath fading channels. Index Terms—Multiple-input–multiple-output (MIMO), sequential Monte Carlo (SMC) method, turbo receiver.
I. INTRODUCTION
T
HERE is at present a significant amount of interest in the design of multiple-input–multiple-output (MIMO) communication systems for high date-rate wireless communications. An MIMO system employs multiple antennas at both the transmitter and the receiver, and its capacity increases linearly with the minimum between the numbers of transmit and receive antennas. When the channels of an MIMO systems are known, various detectors can be employed to demodulate the transmitted data symbols, such as the maximum-likelihood sequence detector, the zero-forcing detector, the minimum mean-square error detector, and the detector based on interference cancellation [1], [2]. On the other hand, blind identification and equalization of MIMO systems has also received much attention. Existing work on this topic are primarily based on the exploitation of either the second-order statistics (SOS) [3]–[11], or the higher-order statistics of the received signals [12]–[16]. The sequential Monte Carlo (SMC) methodology [17]–[22] recently emerged in the fields of statistics and engineering has shown a great promise in solving a wide class of sophisticated statistical inference problems. Under a state-space framework, the SMC recursively generate Monte Carlo samples of the state variable or some other latent variables; based on which the posterior distribution of any system parameter of interest Manuscript received May 2, 2002; revised October 31, 2002. The work was supported in part by the U.S. National Science Foundation under Grant CCR9875314, Grant CCR-9980599, and Grant DMS-0073651. The authors are with the Department of Electrical Engineering, Columbia University, New York, NY 10027 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/JSAC.2003.809722
can be approximated. Compared with the classical expectation-maximization (EM) techniques in solving hidden data problems, which are prone to being trapped in locally optimal points, the SMC methods exhibit global convergence and is more robust to the choice of initial conditions [20], [23]. In this paper, we treat the problem of symbol detection in MIMO communication systems with unknown channels under the Bayesian SMC framework. A Bayesian formulation of the problem is first presented, and then the SMC blind adaptive receiver is derived based on the techniques of important sampling and resampling. The new receiver does not require any training or pilot symbols or decision feedback, and it performs well in both flat-fading and frequency-selective fading channels. Another issue treated in this paper is a turbo MIMO receiver structure [24]–[26] that employs the blind SMC receiver as the first-stage soft-input soft-output demodulator. Such a receiver iterates between the Bayesian demodulation stage and the maximum a posterior channel decoding stage; and the so-called extrinsic information of data symbols are iteratively exchanged between these two stages to successively improve the overall receiver performance. The rest of this paper is organized as follows. In Section II, the MIMO systems over flat-fading and frequency-selective fading channels are described. In Section III, some background materials on the SMC techniques are provided. In Section IV, we derive an SMC algorithm for blind symbol detection in MIMO systems. In Section V, a turbo receiver based on the proposed SMC detector is discussed for channel-coded MIMO systems. Simulation results are provided in Section VI and a brief summary is given in Section VII. II. SYSTEM DESCRIPTION A. Flat Fading MIMO Channels We consider an MIMO system with transmit and receive antennas over slow flat-fading channels with additive Gaussian noise. The transmitted symbols are assumed to be independent in time as well as in space. Moreover, they are taken from a finite . At each transmit antenna alphabet set , , the transmitted symbols are differentially modulated from the binary information source symbols with . Such a differential encoding scheme is necessary to resolve the phase ambiguity inherent to any blind receiver method. Note that in blind systems, there is also an antenna ambiguity-namely, the receiver can only recover the symbols transmitted from the transmit antennas up to a permutation ambiguity. In order to resolve this, a different spreading code is used to spread the symbol transmitted at every transmit antenna.
0733-8716/03$17.00 © 2003 IEEE
GUO AND WANG: BLIND DETECTION IN MIMO SYSTEMS VIA SEQUENTIAL MONTE CARLO
Fig. 1.
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An MIMO system employing the SMC receiver.
The block diagram of such a system is shown in Fig. 1. The received signal vector at the th receive antenna and at time is given by
(1) is the transmitted symbol at the th transmit antenna where , is the and at time ; is the spreading code employed at the th transmit antenna; complex fading channel gain between the th transmit antenna is the and the th receive antenna; and represents a circularly symmetric ambient noise, where denotes identity complex Gaussian distribution, and matrix. Note that here, we assume that the channel is block remain fixed fading, such that the fading coefficients symbols. Denote for a block of diag
based on the received signals and the a priori symbol prob, without knowing the channel response abilities of . The exact solution to this problem involves a very high dimensional integration which is infeasible in practice. Here we will employ the sequential Monte Carlo technique to solve the , above inference problem. Denote , and . The basic idea then is to generate random samples from the joint posterior distribu, and then to estimate the marginal distribution of tion the information symbols using these samples. The details of the algorithm will be discussed in Section IV. B. Frequency-Selective Fading MIMO Channels Now assume that the channel between the th transmit antenna and the th receive antenna is subject to frequencyselective fading, which is modeled by a tapped-delay and complex fading gains line with delays [27]. Here, we assume that the delay spread is small compared with the symbol interval, so that the inter-symbol interference is negligible. Then, the received signal vector at the th receive antenna and at time is
and Then, (1) can be rewritten as (8) (2) We further denote
is the delayed version of the spreading code for where the th transmit antenna. Denote
(3) (4) and
(5)
then, we have (6) and . Our goal is to estimate the a posteriori probabilities of the information symbols Let
(7)
diag
Then, (8) can be rewritten as
(9)
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Furthermore, using the same notations as in (3)–(5), we have
to the sample
, we can estimate the quantity of interest, , as
(10) It is seen that the flat-fading MIMO channel (6) and the frequency-selective fading MIMO channel (10) are described by similar equations, only with different forms of the variables . Then involved. Again, we denote our goal is to estimate the a posteriori probabilities of the based on the received signals information symbols and the a priori information symbol probabilities, without knowing the channel response . III. SMC METHODS SMC is a family of probability approximation methods that use Monte Carlo samples to efficiently estimate the posterior distribution of the unknown variables in a dynamic system [21]. A general framework for the SMC methods is briefly explained next. Consider a dynamic system modeled in the following statespace form state equation observation equation
(11)
where , , and are, respectively, the state variable, the observation, the state noise, and the observation noise at time . They can be either scalars or vectors. In the MIMO system described in the previous section [e.g., (6) and (10)], the state , representing both the unobvariable corresponds to served symbols and the unknown fading channel state. and let . Let is of interest; that is, at Suppose an on-line inference of current time we wish to make a timely estimate of a func, based on the currently tion of the state variable , say available observation, . With the Bayes theorem, we realize that the optimal solution to this problem is d . In most cases, an exact evaluation of this expectation is analytically intractable because of the high-complexity of such a dynamic system. Monte Carlo methods provide us with a viable alternative to the required computation. Specifrandom samples from the ically, if we can draw , then we can estimate by distribution (12)
(14)
. The pair , , is with called a properly weighted sample with respect to distribution . To implement Monte Carlo techniques for a dynamic system, a set of random samples properly weighted with respect to are needed for any time . Because the state equation in system (11) possesses a Markovian structure, we can implement a recursive importance sampling strategy, which is the basis of all sequential Monte Carlo techniques [21]. Suppose (with a set of properly weighted samples ) is given at time . Based on respect to these samples at previous time, a SMC algorithm generates a , which is properly weighted at time new one, with respect to . Note that in most applications we are up to a normalizing constant, only able to evaluate which is sufficient for using (14) in Monte Carlo estimation. . The algorithm is described as follows, for from a trial distribution • Draw a sample and let ; • Compute the importance weight
It can be shown that the above algorithm indeed generates properly weighted samples with respect to the distribution [21]. IV. THE SMC DETECTOR FOR MIMO SYSTEMS A. The SMC Algorithm for MIMO Systems Consider the MIMO system described by (1) or (8). De, , note ; and . as the received Denote also signals at the th receive antenna up to time . Then, the aim of the SMC method is to perform an on-line estimate of the a posteriori symbol probability (15)
is often not feasible, but Direct sampling from drawing samples from some trial distribution is easy. In this case, we can use the idea of importance sampling. Suppose a is generated from the trial set of random samples . By associating the weight distribution
(13)
up to time and the a priori based on the received signals , without knowing the channel resymbol probabilities of sponse . , be a sample draw Let . At each by the SMC at time . Denote time , we need to obtain a set of Monte Carlo samples of the , properly weighted with transmitted symbols, . For every symbol respect to the distribution
GUO AND WANG: BLIND DETECTION IN MIMO SYSTEMS VIA SEQUENTIAL MONTE CARLO
, The a posteriori probability of the information symbol can then be estimated as
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where (25)
(26) (16) with where product; and 1
; the operator denotes element-wise is an indicator function defined as if if
.
(27) In (27), channel (1)
is defined as follows: for the flat-fading MIMO
(17)
Based on the general principle of SMC outlined in Section III, we choose the following trial sampling density to draw the samfrom ples (18)
diag
(28)
and for the frequency-selective fading MIMO channel (8) diag
(29)
Hence, the conditional density in (20) is given by
Then, the importance weight can be updated according to
d
(19)
(30)
which is an integral of a Gaussian probability density function (pdf) with respect to another Gaussian pdf. Hence, the resulting pdf is still Gaussian, i.e., (31)
(20) with the mean and the covariance given, respectively, by where
is denoted as (32) Cov
(21) and (19) follows from the assumption that the channels are unis independent of correlated in space; (20) holds because given and is a first-order Markov chain due to the differential encoding rule. Note that the second term in (20) incorporates the a priori probability of the unknown symbol . then (20) becomes Denote
(33) with (34) is similarly defined as (28) and (29) with where placed by . in (20) can be computed by Therefore,
re-
(22) To compute the predictive density in (20), we assign a Gaussian prior to the channel i.e.,
(35) The trial distribution in (18) can be computed as follows:
(23) Then, the conditional distribution of can be computed as
, conditioned on
and (36) (24)
by (19).
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Note that the a posteriori mean and covariance of the channel in (25) and (26) can be updated recursively as follows. At th is drawn, we combine it with step, after a new sample of to form . Let and be the the past samples . It quantities computed by (32) and (33) for the imputed then follows from the matrix inversion lemma that (25) and (26) become
It is seen that at any time , the dominant computation in the one-step prediction of SMC receiver involves the and the one-step up(32) and (33) in computing . Since the samplers date of (37) and (38) for operate independently and in parallel, the SMC detector is well suited for parallel implementations. B. Resampling Procedure
(37) (38) with (39) Note that (37)–(39) can also be obtained from standard Kalman filtering theory. Finally, we summarize the SMC blind detector for MIMO systems as follows. 1) Initialization: Set the initial channel vector values as , for , ; , All importance weights are initialized as . The following steps are implemented at the th recurto update each weighted sample. sion . For and each , compute 2) For and according to (32)–(34); and compute according to (35). for each 3) Compute the trial sampling distribution .
measures the “quality” The importance sampling weight . A relatively of the corresponding imputed signal sequence small weight implies that the sample is drawn far from the main body of the posterior distribution and has a small contribution in the final estimation. Such a sample is said to be ineffective. If there are too many ineffective samples, the Monte Carlo procedure becomes inefficient. To avoid the degeneracy, a useful resampling procedure, which was suggested in [17], and [21], may be used. Roughly speaking, resampling is to multiply the streams with the larger importance weights, while eliminate the ones with small importance weights. A simple but efficient resampling procedure consists of the following two steps. 1) Sample a new set of streams from with probability pro. portional to the importance weights 2) Assign equal weight to each stream in the new samples, . i.e., Resampling can be done at every fixed-length time interval (say, every five steps) or it can be conducted dynamically. The effective sample size 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 [28]
(40)
4) Draw a sample
from the set
with probability (41)
5) Compute the importance weight (42)
and normalize it by (43) 6) Update the a posteriori mean and covariance of channel. in Step 2, then for If the imputed sample , set ; and update and according to (37)–(39). 7) Compute the a posteriori probability of the information according to (16). symbol 8) Perform resampling as described in Section IV-B.
(44) where
, the coefficient of variation, is given by (45)
. In dynamic resampling, a resamwith is pling step is performed once the effective sample size below a certain threshold. Heuristically, resampling can provide chances for good sample streams to amplify themselves and, hence, “rejuvenate” the sampler to produce a better result for future states as system evolves. It can be shown that the samples drawn by the above resampling procedure are also indeed properly weighted with , provided that is sufficiently large. In respect to in practice, when small to modest is used (we use this paper), the resampling 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 estimation. On the other hand, however, resampling significantly reduces the Monte Carlo variance for future samples.
GUO AND WANG: BLIND DETECTION IN MIMO SYSTEMS VIA SEQUENTIAL MONTE CARLO
Fig. 2.
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A single-user coded MIMO systems employing the turbo receiver.
V. TURBO RECEIVERS
block symbol
. The LLR of this code bit is given for by
In this section, we consider employing the SMC detector in a turbo receiver [25], [26] for coded MIMO systems. As shown in the previous section, the SMC detector not only utilizes the a priori symbol probabilities, but also it produces the a posteriori symbol probabilities. Such a soft-in and soft-output nature makes it an ideal candidate to be employed as the demodulator in a turbo receiver. In what follows, we will consider both single-user and multiuser coded systems. The transmitter structure of a single-user coded MIMO system in shown in the upper part of Fig. 2. A block of are encoded into code bits . The code information bits bits are interleaved and then a serial-to-parallel conversion is performed. Each stream of code bits are differentially encoded and then mapped to binary phase-shift keying (BPSK) symbols. Each symbol is then spread by a short signature code to resolve the antenna ambiguity at the receiver end, and transmitted from one of the transmit antennas. Since the combination of BPSK modulation, the differential encoder, as well as the MIMO channel, effectively acts as an inner coder, a turbo receiver can be designed for such a system, as shown in the lower part of Fig. 2. The turbo receiver consists of two stages: the SMC detector developed in the previous section, followed by a MAP channel decoder [29]. The two stages are separated by a deinterleaver and an interleaver. after interleaving. The SMC Assume that is mapped to receiver takes as input the interleaved a priori log-like likefrom the MAP lihood ratios (LLRs) of code bits channel decoder in the previous turbo iteration as well as the received signals . And it computes a posterior LLR of the code is included in a bits as output. Assume that the code bit
(46) Using the Bayes’ rule, (46) can be written as (47)
, represents where the second term in (47), denoted by , which is computed by the a priori LLR of the code bit the channel decoder in the previous iteration, interleaved and then fed back to the SMC receiver. For the first iteration, it is assumed that all code bits are equally likely. The first term in , represents the extrinsic information (47), denoted by delivered by the SMC receiver, based on the received signals , the structure of signal model, and the prior information about , is then all other code bits. The extrinsic information deinterleaved and fed back to the channel decoder as the a priori information for the channel decoder. The MAP decoder employs the standard MAP decoding algorithm—e.g., the BCJR algorithm for the convolutional codes—to compute the a posteriori LLR of each code bit
(48)
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Fig. 3.
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 3, APRIL 2003
A multiuser coded MIMO system employing the turbo receiver.
It is seen from (48) that the extrinsic information is obtained by subtracting the prior information from . This extrinsic information is the posterior information gleaned from the prior the information about the code bit , based on the information about the other code bits, constraint structure of the code. After interleaving, the extrinsic is fed information delivered by the channel decoder back to the SMC receiver as the prior information. Assume, for example, that the transmitted symbol corresponds to the in the MAP coder output. The prior code bits is then given by symbol probability for each symbol (49) where the code bit probability can be computed from the corresponding LLR as follows [26]:
(50) At the last turbo iteration, the LLRs of information bits are computed followed by hard decisions on them. We next consider a multiuser coded MIMO system. The transmitter structure of such a system is shown in the upper part of Fig. 3, and the corresponding turbo receiver is shown in the lower part of Fig. 3. Here, in contrast to the single-user system, each stream of the source symbols is independently encoded using some channel code, and the code bits are interleaved by independent interleavers. At the receiver, a soft channel decoder is employed for each user to compute the extrinsic information of the code bits of that user. The output
of all channel decoders are fed to the SMC detector as the prior information for all received code bits. The SMC detector computes the extrinsic information of all code bits based on the received signals , and distribute it to the corresponding soft channel decoders. The process iterates to successively improve the receiver performance. VI. SIMULATION RESULTS In this section, we provide computer simulation results to illustrate the performance of the blind SMC detector as well as the SMC-based turbo receiver in both flat-fading and frequency-selective fading MIMO channels. The fading coefficients are generated according to uncorrelated circularly symmetric complex . Hence, the Gaussian distribution, i.e., total received energy of each transmitted symbol is normalized to unity. The channels are assumed to be block fading, i.e., they symbols, but vary remain constant over the entire block of from block to block. We assume that there are two transmit an. The transtennas and two receive antennas, i.e., mitted symbol at each transmit antenna is spread by a short random sequence. In our simulations, the number Monte Carlo . We consider two types of channels, samples is taken as , and frequency-selective fading i.e., flat-fading channel . channel with A. Performance in Uncoded MIMO Systems We first illustrate the performance of the proposed blind SMC detector for one single user in uncoded MIMO systems. The . In Fig. 4, the bit-error-rate (BER) perforblock size is mance of the SMC blind detector in a flat-fading MIMO channel is shown. Also shown in this figure is the performance of a MAP receiver with perfect channel state information, which serves as
GUO AND WANG: BLIND DETECTION IN MIMO SYSTEMS VIA SEQUENTIAL MONTE CARLO
Fig. 4. The BER performance of the blind SMC detector in an uncoded flat-fading MIMO system. There are two transmit antennas and two receive antennas.
Fig. 5. The BER performance of the blind SMC detector in an uncoded frequency-selective fading MIMO system. There are two transmit antennas and two receive antennas.
a lower bound on the achievable performance for any blind receiver. In Fig. 5, we illustrate the BER performance of the SMC detector, as well as the known-channel lower bound, in a 2-tap frequency-selective fading MIMO system. It is seen from these two figures that the performance of the blind SMC detector is within about 1 dB to the known-channel lower bound. B. Performance in Coded MIMO System We next illustrate the performance of the SMC-based blind turbo receiver in coded MIMO systems. We first consider a single-user coded system with two transmit and two receive anconstraint length-5 convolutional code (with tennas. A a rate generators 23 and 35 in octal notation) is employed. The interleaver is randomly generated and is fixed for all simulations. (i.e., 128 information bits). The code bit block size is
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Fig. 6. The BER performance of the blind SMC detector in a coded flat fading MIMO system. There are two transmit antennas and two receive antennas.
Fig. 7. The BER performance of the blind SMC detector in a coded frequency-selective MIMO system. There are two transmit antennas and two receive antennas.
The performance of the SMC-based turbo receiver for such a single-user system in flat-fading and frequency-selective fading MIMO channels is shown in Fig. 6 and Fig. 7. Next, we consider a two-user system where each user’s information bits are independently encoded by the same convolutional encoder, interleaved, and then transmitted from one of the transmit antennas. The turbo receiver employs the SMC detector at the first stage, and two soft convolutional decoders at the second stage. The performance in flat-fading and frequency-selective fading channels of this two-user MIMO system is shown in Fig. 8 and Fig. 9. It is seen from these figures that the proposed SMC-based blind turbo receiver can successively improve the receiver performance in MIMO systems through iterative processing, in both single-user and multiuser systems, and in both flat-fading and frequency-selective fading channels.
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REFERENCES
Fig. 8. The BER performance of the blind SMC detector in a multiuser coded flat-fading MIMO system. There are two transmit antennas and two receive antennas.
Fig. 9. The BER performance of the blind SMC detector in a multiuser coded frequency selective fading MIMO system. There are two transmit antennas and two receive antennas.
VII. CONCLUSION In this paper, we have developed a blind Bayesian receiver for MIMO communication systems over unknown fading channels. This receiver is based on the SMC methods for computing the a posteriori probabilities of the unknown transmitted symbols. The SMC blind detector is seen to perform close to the known channel lower bound in unknown MIMO channels. Moreover, being soft-input and soft-output in nature, the proposed SMC detector is capable of exchanging the so-called extrinsic information with the maximum a posteriori (MAP) outer channel decoder, and successively improving the overall receiver performance. The good performance of such a blind SMC turbo receiver for both single-user and multiuser coded MIMO systems is also demonstrated.
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Dong Guo received the B.S. degree in geophysics and computer science from China University of Mining and Technology (CUMT), Xuzhou, China, in 1993, the M.S. degree in geophysics from the Graduate School of Research Institute of Petroleum Exploration and Development (RIPED), Beijing, China, in 1996, and the Ph.D. degree in applied mathematics from Beijing University, Beijing, China, in 1999. He is now working toward his second Ph.D. degree in the Department of Electrical Engineering, Columbia University, New York, NY. His research interests are in the area of statistical signal processing and communications.
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Xiaodong Wang (S’98–M’98) 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 in 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 quantum computing, and has published extensively in these areas. His current research interests include multiuser communications theory and advanced signal processing for wireless communications. He worked at the AT&T Labs-Research, Red Bank, NJ, during summer 1997. 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 SIGNAL PROCESSING, and the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS.
