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[8] W. Seo, H. Song, J. Lee, and D. Hong, “A new asymptotic analysis of throughput enhancement from selection diversity using a high SNR approach in multiuser systems,” IEEE Trans. Wireless Commun., vol. 8, no. 1, pp. 55–59, Jan. 2009. [9] H. Wang, C. Woo, and D. Hong, “A new subcarrier oriented handover scheme in downlink OFDMA cellular systems,” in Proc. IEEE VTC—Fall, Sep. 2005, vol. 3, pp. 1618–1622. [10] J. M. Peha, “Approaches to spectrum sharing,” IEEE Commun. Mag., vol. 43, no. 2, pp. 10–12, Feb. 2005. [11] R. Zhang and Y. C. Liang, “Exploiting hidden power-feedback loops for cognitive radio,” in Proc. 3rd IEEE DySPAN, Oct. 2008, pp. 1–5. [12] S. Buljore, H. Harada, S. Filin, P. Houze, K. Tsagkaris, O. Holland, K. Nolte, T. Farnham, and V. Ivanov, “Architecture and enablers for optimized radio resource usage in heterogeneous wireless access networks: The IEEE 1900.4 working group,” IEEE Commun. Mag., vol. 47, no. 1, pp. 122–129, Jan. 2009. [13] A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Process. New York: McGraw-Hill, 2002. [14] A. Goldsmith, Wireless Communications. New York: Cambridge Univ. Press, 2005.
Reduced-Complexity ML Signal Detection for Spatially Multiplexed Signal Transmission Over MIMO Systems With Two Transmit Antennas Hyun-Myung Woo, Jaekwon Kim, Member, IEEE, Joo-Hyun Yi, Member, IEEE, and Yong-Soo Cho, Member, IEEE
Abstract—In this paper, we address spatially multiplexed (SM) multiple-input–multiple-output (MIMO) systems that are expected to be used in next-generation mobile communication systems that require highspeed data transmission. The data-transmission rate of an SM MIMO system increases in proportion to the number of spatial streams it transmits. However, it is hard to implement large numbers of spatial streams due to the spatial limits of mobile units and the high cost of installing many antennas. In this paper, we focus on the case of two spatial streams and propose a novel signal-detection technique. We first define the bidirectional detectability of a 2-D complex vector, and using the fact that only a small number of vectors, including the maximum-likelihood (ML) solution vector, satisfy the bidirectional detectability, we reduce the search space, thereby enabling efficient ML signal detection at the receiver. Index Terms—Maximum-likelihood (ML) signal detection, multiple input–multiple output (MIMO).
I. I NTRODUCTION Multiple-input–multiple-output (MIMO) techniques enable highspeed data transmission without increasing the required transmission power or spectral bandwidth. To meet the requirements of high datatransmission rate in next-generation mobile communication systems, a Manuscript received July 9, 2009; revised September 20, 2009. First published October 13, 2009; current version published February 19, 2010. This work was supported in part by Samsung Electronics Company, Ltd., and in part by the Ubiquitous Computing and Network Project, Knowledge and Economy Frontier R&D Program, Ministry of Knowledge Economy, Korea, under Subproject UCN 09C1-C2-11T. The review of this paper was coordinated by Prof. E. Bonek. H.-M. Woo and J. Kim are with the Computer and Telecommunications Engineering Division, Yonsei University, Wonju 220-710, Korea (e-mail:
[email protected];
[email protected]). J.-H. Yi is with the DMC Research Center, Samsung Electronics, Suwon 443-742, Korea (e-mail:
[email protected]). Y.-S. Cho is with the School of Electrical and Electronics Engineering, Chung-Ang University, Seoul 156-756, Korea (e-mail:
[email protected]). Digital Object Identifier 10.1109/TVT.2009.2034459
spatially multiplexed (SM) MIMO technique has actively been investigated [1], [2]. This SM MIMO technique is successfully combined with an orthogonal frequency-division multiplexing (OFDM) system that is known to be robust to multipath fading channels. However, the advantages of SM MIMO come at the cost of installing multiple antennas, which is particularly costly for mobile stations. Therefore, most recent communication systems adopting an SM MIMO technique incorporate a two-spatial-stream mode, which requires installing two antennas [3], [4]. In this paper, we address signal detection for MIMO systems with two spatial streams. The conventional maximum-likelihood (ML) detection method achieves optimal performance; however, its complexity is high. Denoting the constellation point set and its size as Ω and |Ω|, respectively, ML detection requires |Ω|2 squared Euclidean distance calculations. It has been shown that only |Ω| squared Euclidean distance calculations are sufficient to find the ML symbol vector for the two-spatial-stream case [5]. In this paper, we propose a novel detection method that uses bidirectionally detectable ML (BDML) signal detection. First, we define the bidirectional detectability of a 2-D complex vector. Exploiting the fact that a small set of transmitted symbol vectors, including the ML solution vector, satisfy the bidirectional detectability, we show that the required number of squared Euclidean distance calculations can be even smaller than |Ω|. We also compare the proposed BDML with sphere decoding (SD) [6], [7] and show the superiority of BDML from the complexity perspective. In this paper, a boldfaced lowercase letter denotes a vector, and a boldfaced uppercase letter denotes a matrix. The entries are denoted as italic lowercase letters. CN (0, σ 2 ) denotes the circularly symmetric complex Gaussian noise with zero mean and variance σ 2 . The operation [·]T denotes the transpose, [·]H denotes the Hermitian transpose, · denotes the l2 norm, | · | denotes the absolute value, and Re{·} and Im{·} represent the real and imaginary parts of a complex number.
II. M ULTIPLE -I NPUT –M ULTIPLE -O UTPUT S YSTEMS W ITH T WO S PATIAL S TREAMS We consider an SM MIMO system with two transmit antennas and nR receiving antennas. The relation between the transmitted signal x = [x1 x2 ]T and the received signal y = [ y1 y2 · · · ynR ]T is expressed as follows: y = Hx + z
(1)
where xi , i = 1, 2 is the transmitted signal from the ith transmit antenna, yj , j = 1, 2, . . . , nR is the received signal at the jth receiving antenna, H is the nR × 2 matrix where entry hj,i , j = 1, 2, . . . , nR , and i = 1, 2 is the channel gain between the ith transmission antenna and the jth receiving antenna. In this paper, the channel gains are assumed to be independent identically distributed (i.i.d.) Rayleigh fading. The noise z = [ z1 z2 · · · znR ]T is circularly symmetric complex Gaussian, i.e., the entries are i.i.d., and zj ∼ CN (0, σz2 ), j = 1, 2, . . . , nR . The transmitted symbols are assumed to be drawn from |Ω|-ary quadrature amplitude modulation (QAM). In this paper, we consider only square QAMs, whose constellation points contain integer real and imaginary parts. There exist other types of QAM constellations, i.e., type-I and type-II constellations [8], both of which contain constellation points on concentric circles. The square QAM constellations are referred to as type-III constellations in [8].
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III. M AXIMUM L IKELIHOOD D ETECTION OF T WO S PATIAL S TREAMS In this section, we first summarize conventional ML detection methods: ML and modified ML (MML) detection methods. We then propose a novel detection method, i.e., BDML detection. A. Conventional ML Detection In this paper, we assume ideal channel estimation at the receiver. When the noise has a Gaussian distribution, the ML signal detection is described as follows: xML = arg max P (y|x) x∈Ω2
= arg min y − Hx2 . x∈Ω2
(2) (3)
From (3), we can see that the ML signal-detection method requires |Ω|2 squared Euclidean distance calculations. The signal vector corresponding to the smallest metric value is considered the transmitted signal vector. As is well known, this method shows the optimal performance; however, its complexity is high when the constellation is large. B. MML Detection
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operations, exploiting the fact that the real and imaginary parts of x1 are integers. Thus, a − bx1 requires 8 × nR multiplications, excluding the complexity of the shift operations. The norm calculation h2 2 requires 2 × nR real multiplications. When square QAMs are assumed, the decision boundaries are described as inequalities involving integer values. For example, when 16-QAM is assumed, the decision boundaries are {−2, 0, 2} for both real and imaginary parts. Using h2 2 , we can change the decision boundaries into {−2h2 2 , 0, 2h2 2 }. The scaling of the decision boundaries can be implemented as shift operations. In the case of 16-QAM, only one-time shift operations are necessary. Now, using the new decision boundaries, we perform the slicing of a − bx1 . Consequently, the construction of SMML,1 requires 10 × nR real multiplications in total. However, if type-I or type-II constellations are assumed, the real and imaginary parts of x1 are not integers. Thus, the multiplication of b and x1 cannot be implemented as shift operations; instead, four real multiplications are required for each x1 . Assuming that all x1 in Ω have noninteger real and imaginary parts, the construction of SMML,1 requires 10 × nR + 4 × |Ω| real multiplications. ML vector detection xML = arg
If we use the fact that the globally optimal vector is one of the locally optimal vectors, the high complexity of ML detection can be reduced [5]. We let Q(ˆ x) denote the slicing function that is defined as follows: ˆ|. x ¯ = Q(ˆ x) = arg min |x − x x∈Ω
(4)
Then, the ML signal vector can be found as follows: SMML,1 or SMML,2 generation
¯2|x1 ] x1 ∈ Ω SMML,1 = [ x1 x ¯1|x2 x2 ]T x2 ∈ Ω SMML,2 = [ x
T
min
x∈SMML,1
y − Hx2 .
(7)
Comparing (3) and (7), we can see that |Ω|2 squared Euclidean distance calculations are required for ML detection. However, only |Ω| squared Euclidean distance calculations are required for MML detection. We note that SMML,2 can be used instead of SMML,1 . C. Proposed BDML Detection
(5) (6)
x ¯1|x2 = where x ¯2|x1 = Q((1/h2 2 )(a − bx1 )), Q((1/h2 2 )(c − dx2 )), hi , i = 1, 2 is the ith column H H H vector of H, a = hH 2 y, b = h2 h1 , c = h1 y, and d = h1 h2 . We note that SMML,i , i = 1, 2 is composed of signal point vectors, and the ML vector is one of the |Ω| vectors in SMML,1 or SMML,2 . In obtaining SMML,1 , all the constellation points are tried as x1 . Obviously, one of the constellation points is the ML solution symbol x1,ML . In [5], it was shown that, when one of the two ML solution symbols is known, the other one can be found by the simple slicing of (4). Therefore, the ML solution vector is included in SMML,1 . Although SMML,2 is not the same as SMML,1 , the ML solution vector is also included in SMML,2 . Therefore, we can regard SMML,i with |Ω| vectors as a smaller search space for the ML solution vector than the conventional search space Ω2 with |Ω|2 vectors. Since xML ∈ SMML,i , i = 1, 2, we can find the ML solution vector xML in step 2. Let us deal with the complexity of MML step 1. Since the complexity of obtaining SMML,1 is the same as that of obtaining SMML,2 , we deal with only SMML,1 . The complexity of constructing SMML,1 depends on the complexity of the slicing H operation Q((1/h2 2 )(hH 2 y − h2 h1 x1 )). The entries of y and H are complex numbers with noninteger real and imaginary parts. Thus, the calculation of a requires 4 × nR real multiplications, assuming that one complex multiplication is equivalent to four real multiplications, and the calculation of b requires 4 × nR real multiplications. When square QAMs are assumed, the multiplication of b and x1 can be implemented as shift
We first define the bidirectional detectability of a 2-D symbol vector; then, using the fact that a small set of symbol vectors, including the ML vector, satisfy the bidirectional detectability, we further reduce the search space of MML. Definition (Bidirectional Detectability of a Vector x ∈ Ω2 ): When y and H are given, a 2-D complex vector x = [ x1 x2 ]T ∈ Ω2 is defined to be bidirectionally detectable if and only if the following condition is satisfied:
x1 x ¯1|x2 = . x2 x ¯2|x1
(8)
We note that the bidirectional detectability of a vector depends on both the channel matrix H and the received signal vector y. Theorem: The ML solution vector is bidirectionally detectable and results in the following equation:
xML
x1,ML x ¯1|x2,ML = = . x2,ML x ¯2|x1,ML
(9)
For a detailed proof, see [5, eq. (12)]. The aforementioned property implies that, when one of the two ML symbols is known, the other one can be found by the simple slicing of (4). We also note that the ML solution vector depends on both the channel matrix H and the received signal vector y. Using the definition of the bidirectional detectability of a 2-D vector and the fact that the ML solution vector is bidirectionally detectable, we propose the BDML detection method, which proceeds as follows: SBDML,1 generation SBDML,1 =
[ x1
x ¯2|x1 ]T x1 ∈ Ω .
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(10)
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Fig. 1. Squares described by (17) and (18).
The new set SBDML,1 is the same as SMML,1 ; thus, BDML step 1 is the same as MML step 1. SBDML,2 generation SBDML,2 =
[x ¯1|x2
x2 ]T x2 ∈ Ω .
(11)
The constellation point set Ω = {¯ x2|x1 |x1 ∈ Ω} is obtained in BDML step 1. We note that |SBDML,1 | = |Ω|; however, |SBDML,2 | = |Ω | ≤ |Ω|. SBDML generation The set of bidirectionally detectable symbol vectors can be found as follows: SBDML = SBDML,1 ∩ SBDML,2 .
(12)
ML vector detection xML = arg
min
x∈SBDML
y − Hx . 2
Now, we investigate the relationship between the channel condition and the number of bidirectionally detectable vectors. From (8), we can see that the bidirectional detectability of a vector depends on both the channel matrix H and the received signal vector y. According to (8), ˆ2 ]T is bidirectionally detectable, the following if a vector x = [ x ˆ1 x two conditions need to be simultaneously satisfied: hH h2 hH z 1 (dmin + jdmin ) < Δx1 + 1 2 Δx2 + 1 2 2 h1 h1 1 (dmin + jdmin ) 2
