Reduced-Complexity Detection Algorithms for Systems Using Multi-Element Arrays

Transcription

1 Reduced-Complexity Detection Algorithms for Systems Using Multi-Element Arrays Xiaodong Li, Howard C. Huang, Angel Lozano and Gerard J. Foschini Bell Laboratories (Lucent Technologies) 791 Holmdel-Keyport Road Holmdel, NJ 07733, USA Abstmct- In BLAST (Bell-laboratories LAyered Space Time) systems, multiple transmit and receive antennas are employed to achieve very high spectral efflciencies [l]. The ideal detection method for such systems is the maximumlikelihood (ML) algorithm. However, the ML complexity increases exponentially with the number of transmit antennas and the number of bits per modulation symbol. A reduced-complexity detection method has been suggested, using ordered successive interference cancellation [1][2]. In this paper, we consider two other suboptimum techniques: channel-based adaptive group detection and multistep reduced-constellation detection. The goal is to reduce the two forementioned complexity exponentials. The algorithms efficiently combine linear processing with local ML search. We limit the complexity by maintaining small ML searching areas, while maximizing the performance under the complexity constraint by optimizing the front-end linear processing and the selection of the search areas. I. INTRODUCTION We consider the signal detection problem at the receiver for systems with transmit and receive antenna arrays [I]. Suppose there are Nt antennas at the transmitter and NT antennas at the receiver. Represent the baseband signal vector transmitted during a symbol period as X=[ZO, 21,..., XN,-~]~, where each element zi, i=o,l,...,nt-l is a complex signal drawn from a modulation constellation and T indicates vector/matrix transpose. For narrowband transmission, the channel between any pair of transmit and receive antennas is assumed to be flat fading. Then the overall channel can be represented as an N, x Nt complex matrix where hij is the fading coefficient of the channel from the jth transmit antenna to the ith receive antenna, and the column vector hj, j = 0,1,..., Nt - 1 corresponds to the channel fading coefficients from the jth transmit antenna to each receive antenna. In this paper, we assume that the system operates in a rich scattering environment and all the fading coefficients are independent and complex Gaussian. The received signal vector is where n is the N,xl vector of complex additive white Gaussian noise with noise spectral density NO per element. We assume channel information is known at the receiver, possibly through additional channel estimation using training signals known to both ends. Then the detection problem is to recover x from r, given H. The problem is similar to multi-user detection for CDMA [4], if we consider each hj as the spreading code for the ith transmitter. The energies of the spreading codes, llhj1i2,j = 0,1,-.., Nt - 1, are also not normalized for a particular realization of the channel, even though we are normalized. The optimal solution to this detection problem is of course the maximum-likelihood (ML) rule for equallylikely input x. But the ML complexity grows exponentially with Nt and therefore it is impractical for even moderate system configurations. The focus is then to seek suboptimal solutions which yield a reasonable tradeoff between performance and complexity. Linear operations using the principles of zero-forcing (ZF) or minimum mean square error (MMSE) are obvious candidates. Another more effective solution is the algorithm originally proposed for BLAST by Foschini, using ordered successive interference cancellation (OSIC) [1][3]. In this paper, we consider two other suboptimal techniques, which directly aim to reduce the two forementioned complexity exponentials. Our first algorithm, using channel-based adaptive group detection, builds upon and improves over the group detection method originally suggested for CDMA multi-user detection [5]. At the receiver, the transmit signals to be detected are partitioned into small groups. Linear subspace projection is used to null/suppress the interference outside the group and local ML search is performed within the group. By maintaining small groups, the complexity of the ML search is limited. But the performance is significantly improved compared with those using nulling/suppression alone. We propose adaptive and generalized grouping methods, which utilize the channel information observed at the receiver and assume the ensemble averages of llhj!i /00/$ Zoo0 IEEE 1072

2 allow overlapped grouping, to further improve the performance and to provide flexibility. Our second algorithm is called multi-step reducedconstellation detection, which aims to reduce the searches required in the ML detection for high-order modulations. The first step of our detector generates an initial data estimate for each user using a suboptimal detector. In the second step, a localized ML search is performed over the combination of reduced constellations (neighbor lists) surrounding the initial estimation. In the optional following steps, new localized ML search can be performed over newly generated neighbor lists based on previous search results. The algorithm can continue until the most likely data combination is stabilized or a prescribed computation limit is met. 11. BACKGROUND A. Maximum-Likelihood us. Zero Forcing Methods Given the channel matrix H, the ML solution for detecting x from r is 12 = argmin X Ilr - Hx1I2. (4) The ML detction considers all possible inputs x, and chooses the input which minimizes the squared Euclidean distance. It is easy to see that the computation complexity of ML detection increases exponentially with the number of transmit antennas, Nt, and the number of informations bits per modulation symbol, m, assuming the constellation size (M = 2m) for each transmit antenna is the same. The ZF method contains two steps. First we calculate T Y= [!/O,yl,...,yNt-l] =Vr, (5) with v = (H~H)-~H~, (6) where t denotes the complex conjugate transpose. Then the signals are detected individually 2i =argmini(yi-si1i2, i=o,l,...,nt-l. (7) 2; The last operation can be easily implemented with a slicer. The computation of V requires a matrix inversion, for which there exist several reduced-complexity approaches. For slow fading channels, the channel matrix can be regarded as time invariant within a small data block. Therefore, the computation of V only occurs at the start of each block and the ZF solution requires mainly linear processing on each received signal vector within the block. It can be shown that the diversity order provided by the ZF solution is N, - Nt + 1 [6]. The diversity order can be significantly improved by the ML solution. The performance gap between ZF and ML is significant in tight situations (Nt x N,). The gap shrinks as N, - Nt increases. B. Ordered Successive Interference Cancellation The details of the OSIC algorithm can be found in [1][2][3]. We note that in OSIC, the SNR of the first layer selected after nulling has a diversity order of N, - Nt + 1. Assuming error-free nulling, the second layer has diversity order of N, - Nt + 2, and so on. Therefore the diversity provided by OSIC is better than ZF, especially for a large number of antennas, but is worse than ML. We show the performance of ZF, ML, and OSIC in Fig It is clear that the performance gap between ZF and ML is significant. However, for a large number of antennas, especially with N, >Nt, the original BLAST detection algorithm performs close to ML CHANNEL-BASED ADAPTIVE GROUP DETECTION ML detection over all the Nt transmit antennas requires MNt searches, where M is the constellation size. The associated computation complexity is beyond the limit of most systems today, with even moderate M and Nt. However, it might be possible to perform ML search within a group of N,, Ng << Nt, antennas, while nulling/suppressing the interference from the antennas outside the group. This way, the searching complexity is proportional to MNg and can be maintained low by using a small group size Ng. The basic idea of group detection was previously suggested for CDMA multi-user detection [5]. Here, we extend GD to signal detection for systems with multiple transmit and receive antennas. We note that by performing ML search within a group of N,>1 transmit antennas, a significant diversity gain can be achieved over the ZF or MMSE solution used alone, even though Ng is small, such as 2-3. We propose an adaptive grouping detection (AGD) method based on channel characteristics. A. Algorithm The AGD algorithm contains three steps: grouping, subspace projection, and ML search within each group. First, we consider the grouping algorithm. A very simple way is to use fixed grouping, regardless of the channel. For example, for a constant group size of two, we can group Antenna 0 and 1 together, Antenna 2 and 3 together, and so on. But a more sensible way is to group adaptively based on the channel characteristics observed at the receiver. The group size can also vary from group to group, based on the channel and the complexity limitation. Furthermore, the groups do not have to be mutually exclusive - they can overlap. Note that mutually exclusive grouping is more computation efficient than overlapped grouping because the detection results for all the members within a group are obtained simultaneously with the ML search within the group. It is easy to show that overlapped grouping optimized for individual antennas always outperforms mutually exclusive grouping. 1073

3 In the following, we give a simple example of channelbased adaptive grouping method. More sophisticated methods are discussed in [7]. We allow overlapped grouping and optimize grouping for each individual antenna. For example, when detecting signals for Antenna 1, we can group Antenna 1 and 2 together. But then we can group Antenna 2, 3, and 4 together for the detection of signals from Antenna 2. We call the group optimized for the ith antenna Gi. Consider the correlation matrix R = H~H. (8) On the ith row of R, the ith element rii is the matchedfilter output for the ith antenna, while the other elements rij, j # i indicate the interference: both interferer strength and the correlation with the desired signal. We order ~~r~j~~2/~~rjj~~2,j # i and include the corresponding antennas to Gi in a descending order. The group keeps growing until the size hits the complexity limitation or the signal-to-interference is larger than a predesigned threshold. The second step of group detection is interference suppression or subspace projection. To simplify the discussion, we first assume the groups are mutually exclusive and the transmit antennas have been reordered so that the first K1 antennas are in G1, the following KZ antennas are in 42? and so on. Then the main objective of subspace projection is to find a projection matrix P such that where PH = D, (9) D = diag{d1, Da,.. -} (10) is a block diagonal matrix with squared subblocks D1 (KlxKl), DI (KzxKz), and so on. Here we note that AGD is a generalization of ML and ZF. When the group includes all the antennas, AGD becomes ML, and we have P=Ht and D=HtH. On the other hand, if each group size is one, AGD reduces to ZF, and we have P=V and D=I, the identity matrix. Similar discussion can also be made for MMSE. In general, there are many choices of D and as many choices of P. But the goal of projection design should be to jointly null/suppress the interference outside the group, retain the signal engergy within the group, and minimize noise amplification The squared subblocks Di and correspondingly the projection submatrices Pi can be designed individually for each group. In the following, we focus our discussion on the projection design for one group. Without loss of generality, we consider signal detection for the the first group and assume it includes the first K1 antennas (For notation simplification, we drop the subindex "1" from K1 and P1 when it is clear from the context in the following.) We first consider the linear projector used in [5]. The subspace projector is a KxN, matrix P = (QK)-~(R-~)[~:K,:IH~,(11) where QK is the K x K principal submatrix of R-' and (R-l)[l,K,:~ denotes the first K rows of R-'. This method essentially first decorrelates all the matched filtering output using the ZF approach and then restores the correlation of signals within the group. The K xn,. projector P operates on the N, x 1 received signal vector r, and the K-dimensional vector r~ after projection is rk = Pr (12) = (QK)-~(R-')[~:K,:IH~~ (13) = ( ~ ~ 1(~-l)[~:~,:@ - l (HX + n) (14) = (QK>-'XK + Z, (15) where XK is the first K signals in x and z=pn (16) is the K-dimensional complex additive Gaussian noise vector. z has zero mean and covariance matrix R. = E {zz~} = NoPPt = No(QK)-'. (17) It is clear that in this case, the squared subblock Di=(QK)-'. The step after subspace projection is the ML search within the group: where y=rk-(qk)-'xk.?k = arg min yt R,' y (18) xk - arg min yt QKY, (19) xk From this we can extract the detection decision for Antenna 1 51 =XK (1). If the groups Gj, j # 1, individually designed for other antennas coincide with 41, we have also simultaneously obtained the detection decisions for those antennas. The AGD method can be combined with OSIC. In the OSIC algorithm, for a specific layer, the interference from the lower layers (the layers which have already been processed) are cancelled using the decision feedback from those layers, and the interference from the upper layer (the layers which have not been processed) are nulled/suppressed using the ZF or MMSE methods. We replace this ZF/MMSE-based nulling/suppression with the AGD algorithm. Specifically, we group several layers to be processed together and perform ML detection within the group. We may use the search results as the detection decisions of all the layers within the group. Alternatively, we may only use part of the search results for part of the layers within the group, and classify the rest as still unprocessed. After that, another round of interference cancellation and AGD detection will continue until all the layers have been processed. 1074

4 B. Simulation Results In Fig. 4, we compare the performance of various detection algorithms. Four transmit and four receive antennas are used with QPSK modulation. The group for each transmit antenna is individually designed, each with a size of two. There is a large performance gap between ZF and ML. The OSIC algorithm effectively reduces the gap without significant complexity increase over ZF. AGD alone performs close to OSIC at low SNR and is better at high SNR. The performance of AGD combined with OSIC is close to that of ML. IV. MULTI-STEP REDUCED-CONSTELLATION DETECTION When the signal constellation is large, even AGD with a small group size might be too complex. In this section, we show a suboptimal method which further reduces the ML search for high-order modulation. The method can be combined with AGD of any group size, including the global ML methods. Consider the quadratic function f(x) = Ilr - Hx1I2. (20) It can be shown that f(x) is a convex-cup function. Given r and H and with continuous x, the minimum of f(x) is achieved at y=vr and f(y)=o, where V is defined in (6). However, when each element xi of x has a finite constellation, the minimization of f(x) does not have a simple linear solution and the ML search over all possible input is optimal. A. Algorithm Since the function f(x) is convex, the solution under the finite-constellation constraint might lie close to y, the minimization input without the constellation constraint. Therefore, we may perform a local search around y to find the solution we need. This forms the basis of our algorithm, called multi-step reduced-constellation (MSRC) detection. Note that the first-step ZF projection can be generalized to many other processing methods including MMSE or even matched filtering. We can also perform a ML search over coarse constellations, e.g., representing a cluster of constellation points with a single (middle) point, for the first step. We call the particular method which starts with the ZF processing ZFML. We first compute y and then perform a ML search around the neighborhood of y, ils shown in Fig. 1. From each element yi of y, we generate a neighbor list. Then the search area is the union of the combination possibilities over the neighbor lists of all antennas in the group. A simple way to generate the neighbor list is to use a lookup table with the quantized yi as input. For example, for 16-QAM and fixed neighbor size of 4 for each antenna, there are nine entries in the lookup table, with each entry containing 4 constellation points. The lookup table may be fixed and shared among all the antennas. Alternatively, we may optimize the tables individually for each antenna, depending on the channel matrix at the beginning of each block. The size of each neighbor list can also be variable. Once the neighbor lists are generated, our scheme performs ML search over all combinations over the points in the lists just like a regular ML search would do. It is interesting to note that the process can continue in an iterative fashion. That is, once the first-step search result is generated, we can form a second neighbor list around it and perform a second-round search, and so on. The neighbor lists for the second round can be the same as that of the first round, but more generally, the lists can be optimized separately. For example, the second-round list size might be smaller or the geometrical shape/direction may be different depending on the channel and other available information. The algorithm can continue until the most likely data combination is stabilized or a prescribed computation limit is met. The hybrid search algorithm can also be used in combination with OSIC or group detection. Note that the ML search goes through all the combinations of the possible constellation points in the neighbor lists of all the antennas in a group. Therefore, the number of searches is proportional to M,. where M, is the size of the neighbor list (if the size is fixed) and Ng is the number of antennas in a group. Compared with the ML search over the entire constellation, the reduction rate in searches is (g)n9, where M is the size of the original constellation. For 16-QAM7 M=16. If we choose M,=4, the reduction rate is 4N9, which is significant. Even with two-round search, the reduction rate is 4 /2, still quite significant. The suboptimality of the ZFML method lies in the fact that the neighbor list, unless containing all the constellation points, does not always contain the minimization solution, despite the fact that f(x) is a convex function. As shown at the left of Figure 1, the neighbor list of yo contains Point A, B, C, D. But the minimization solution might be Point E. The design of constellations and the neighbor lists might be optimized to enhance the performance. B. Simulation Results In Fig. 5, we show the performance of ZFML. We consider two transmit and two receive antennas with 16- QAM modulation. The neighbor list tables are fixed and the same for each transmit antenna. The size of each neighbor list is also fixed at 4. It is interesting that ZFML alone (without OSIC) already performs fairly close to ML. 1075

5 U-. Md, N-4 QPSK Neighbor List 0.:.A..e....E..... x yo i ~:,g+?llist A.. c. E....,B B. x. D F.. :..._... 5 Antenna 0 Antenna 1 Fig. 1. Neighbor lists in the hybrid search algorithm. lo- o SNR (db) Fig. 4. Performance of the groupdetection algorithm, 4x4, QPSK. t.,... <,,.X , P.:., ~ U-, M. Ne. 16QAM. N = 4W lo- 10 U Itl 10. lo-* SNR (de) Fig. 2. Performance comparison of ZF, OSIC and ML, 2x2, QPSK. a R (de) Fig. 5. Performance of the multi-step hybrid-searching algorithm, 2x2, 16-QAM. lo-ao SNR (de) Fig. 3. Performance comparison of ZF, OSIC and ML, 6x8, QPSK. REFERENCES G. J. Foschini, Layered space-time architecture for wireless communication in a fading environment when using multielement antennas, BLTJ, Autumn, G. J. Foschini, G. D. Golden, R. A. Valenzuela, and P. W. Wolniansky, Simplified processing for high spectral efficiency wireless commun ication employing multi-element arrays, IEEE JSAC, vol. 17, No. 11, Nov G. D. Golden, G. J. Foschini, R. A. Valenzuela, and P. W. Wolniansky, Detection algorithm and initial laboratory results using the V-BLAST space-time communication architecture, Electronics Letters, vol. 35, no. 1, pp , Jan S. Verdu, Multiuser detection, Cambridge University Press. M. K. Varanasi, Group detection for synchronous Gaussian code-division multiple-access channels, IEEE lhns. on Inform. Theory., vol. 41, no. 4, July J..H. Winters, J. Salz, and R. D. Gitlin, The impact of antenna diversity on the capacity of wireless communication systems, IEEE Ipnns. on Commun., vol. 42, no , Feb X. Li, H. Huang, A. Lozano, and G. J. Foschini, Channelbased adaptive group detection for antenna-array systems, in preparation. 1076