This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2009 proceedings
A Low ML-Decoding Complexity, High Coding Gain, Full-Rate, Full-Diversity STBC for 4 × 2 MIMO System K. Pavan Srinath
B. Sundar Rajan
Dept of ECE, Indian Institute of science Bangalore 560012, India Email:
[email protected]
Dept of ECE, Indian Institute of science Bangalore 560012, India Email:
[email protected]
Abstract— This paper proposes a full-rate, full-diversity spacetime block code (STBC) with low maximum likelihood (ML) decoding complexity and high coding gain for the 4 transmit antenna, 2 receive antenna (4 × 2) multiple-input multiple-output (MIMO) system that employs 4/16-QAM. For such a system, the best code known is the DjABBA code and recently, Biglieri, Hong and Viterbo have proposed another STBC (BHV code) for 4-QAM which has lower ML-decoding complexity than the DjABBA code but does not have full-diversity like the DjABBA code. The code proposed in this paper has the same ML-decoding complexity as the BHV code for any square M -QAM but has fulldiversity for 4- and 16-QAM. Compared with the DjABBA code, the proposed code has lower ML-decoding complexity for square M -QAM constellation, higher coding gain for 4- and 16-QAM, and hence a better codeword error rate (CER) performance. Simulation results confirming this are presented.
I. I NTRODUCTION A ND BACKGROUND MIMO transmission has attracted special interest due to the enhanced capacity it provides over that provided by the singleinput, single-output (SISO) system. The Alamouti code [1] for two transmit antennas, due to its orthogonality property, allows a low complexity ML-decoder. This scheme led to the generalized complex orthogonal designs (CODs) [2]. Such codes allow the transmitted symbols to be decoupled from one another and single-symbol ML-decoding is achieved over quasi-static Rayleigh fading channels. Another aspect of these codes is that they have full-diversity for any number of transmit antennas and for any arbitrary complex constellation. Unfortunately, they do not have a high code rate. The square COD for 4 transmit antenna has only a rate of 3/4 complex symbols per channel use. With a view of increasing the transmission rate for 4 transmit antennas, a quasi-orthogonal design (QOD) was proposed in [3]. This QOD has a rate of 1 complex symbol per channel use and is double-symbol decodable but does not have full transmit diversity. As an improvement, coordinate interleaved orthogonal designs (CIODs) were proposed [4]. The CIOD for 4 transmit antennas is a single-symbol decodable rate-1 code and importantly, has full transmit diversity for certain complex constellations. Full-diversity single-symbol decodable QODs were subsequently proposed in [5] and [6]. But all of the above
codes are full-rate (see Section II for a definition of full-rate codes) only when the number of receive antennas is one, i.e, for the 4 × 1 MIMO system. Full-rate, full-diversity STBCs are of prime importance in applications like WIMAX. Also of equal importance is low ML-decoding complexity (see Section II for a definition of ML-decoding complexity). For 4 × 2 MIMO transmission, the DjABBA code [7] is a full-rate code and has been reported to have the best performance for the 4 × 2 system employing 4/16-QAM [8], [9] (In this paper the DjABBA code refers to the best code in the family of DjABBA codes, optimized for performance and presented in Chapter 9 of [7]). However this code has a high ML-decoding complexity, of the order of M 6 for any square M -QAM [10]. With a view of decreasing the ML-decoding complexity, a new STBC has been proposed for 4-QAM in [8], for the 4 × 2 system. This code, which we call the BHV √ code, has an ML-decoding complexity of the order of M 4 M for square M -QAM [10]. However, the BHV code does not have full-diversity for 4- and 16-QAM and hence, at a high SNR, it loses out to the DjABBA code in CER performance. In this paper, we propose a new full-rate STBC for 4 × 2 MIMO transmission. Our code is based on the CIOD for 4 transmit antennas [4]. The major contributions of this paper can be summarized as follows: • Our proposed code has an ML-decoding complexity of √ the order of M 4 M for square M -QAM, the same as that of the BHV code and significantly lower than that of the DjABBA code, whose corresponding ML-decoding complexity is of the order of M 6 . • Our code has full-diversity and higher coding gain than the DjABBA code for 4- and 16-QAM (this has been confirmed through exhaustive computer search and we haven’t been able to do the same for constellations of larger size). • Our code outperforms the BHV code for 4- and 16-QAM due to its full-diversity property. • Combining the above, it can be seen that for 4- and 16QAM constellations, our code is the best among known codes for the 4 × 2 system.
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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2009 proceedings
The remaining content of the paper is organized as follows: In Section II, the system model and the code design criteria are given. The proposed STBC is presented in Section III. In Section IV, the ML-decoding scheme for the proposed STBC is discussed. In Section V, simulation results are presented to illustrate the comparison of our code with the DjABBA code and the BHV code. Concluding remarks constitute Section VI. Notations: Throughout, bold, lowercase letters are used to denote vectors and bold, uppercase letters are used to denote matrices. Let X be a complex matrix. Then XT , XH and det(X) denote the transpose, the hermitian and the determinant of X, respectively. sI and sQ denote the real and imaginary parts of a complex number s, respectively, and j represents √ −1. The set of all real and complex numbers are denoted by R and C, respectively. .F and . denote the Frobenius norm and the vector norm, respectively. For a matrix X, the vector obtained by stacking the columns one below the other is denoted by vec (X). The Kronecker product is denoted by ⊗ and IT and OT denote the T × T identity matrix and the null matrix, respectively. For a complex random variable X, X ∼ NC (0, N ) denotes that X has a complex normal distribution with mean 0 and variance N . For any real number m, rnd[m] denotes the operation that rounds off m to the nearest integer, meaning which m if m − m > m − m, rnd[m] = m otherwise T Given a complex vector x = [x1 , x2 , · · · , xn ] , the vector x˜ is defined as ˜ x [x1I , x1Q , · · · , xnQ ]T
ˇ operator acting on it is and for a complex number s, the (.) defined by sI −sQ ˇs . sQ sI ˇ operator can be similarly applied to any m × n matrix The (.) by applying it to all its entries.
Considering an nt ×nr MIMO system, if k = nmin T , where nmin = min (nt , nr ), then the STBC is said to be full-rate. Assuming ML-decoding, the ML-decoding metric that is to be minimized over all possible values of codewords S is given by (2) M (S) = Y − HS2F . Definition 2: (ML-Decoding complexity) The ML decoding complexity is a measure of the maximum number of symbols that need to be jointly decoded in minimizing the ML decoding metric. This number can be k in the worst scenario, k being the total number of information symbols in the code. Such a code is said to have a high ML-decoding complexity, of the order of M k , where M is the size of the signal constellation. If the code has an ML-decoding complexity of order less than M k , the code is said to admit simplified ML-decoding. Definition 3: (Generator matrix) For any STBC S that encodes k information symbols, the generator matrix G is defined by the following equation vec (S) = G˜s,
(3)
T
where s [s1 , s2 , · · · , sk ] is the information symbol vector. G is said to be the generator matrix associated with the information vector s. The code design criteria [11] for an nt × nr MIMO system are as follows: 1) Rank Criterion: To achieve maximum diversity, the ˆ must be full rank codeword difference matrix (S − S) for all possible codeword pairs and the diversity gain is given by nt nr . If full rank is not achievable, then, the diversity gain is given by rnr , where r is the minimum rank of the codeword difference matrix over all possible codeword pairs. 2) Determinant Criterion: For a full ranked STBC, the minimum determinant δmin , defined as H ˆ S−S ˆ (4) δmin min det S − S S=ˆ S
II. S YSTEM M ODEL We consider Rayleigh quasi-static flat-fading MIMO channel with full channel state information (CSI) at the receiver but not at the transmitter. For 4 × 2 MIMO transmission, we have Y = HS + N, (1) where S ∈ C4×4 is the codeword matrix, transmitted over 4 channel uses, N ∈ C2×4 is a complex white Gaussian noise matrix with i.i.d entries ∼ NC (0, N0 ) and H ∈ C2×4 is the channel matrix with the entries assumed to be i.i.d circularly symmetric Gaussian random variables ∼ NC (0, 1). Y ∈ C2×4 is the received matrix. Definition 1: (Code rate) If there are k independent complex information symbols in the codeword which are transmitted over T channel uses, then, the code rate is defined to be k/T complex symbols per channel use.
should be maximized. The coding gain is given by 1/n (δmin ) t . For the 4×2 MIMO system, the objective is to design a code that is full-rate, i.e transmits 2 complex symbols per channel use, has full-diversity and allows simplified ML-decoding. III. T HE P ROPOSED STBC In this section, we present our STBC for the 4 × 2 MIMO system [10]. The design is based on the CIOD for 4 transmit antennas, whose structure is as shown in the top of the next page. Here si , i = 1, · · · , 4 are the information symbols and siI and siQ are the in-phase (real) and quadrature-phase (imaginary) components of si , respectively. Notice that in order to make the above STBC full-rank, the signal constellation A from which the symbols si are chosen should be such that the real part (imaginary part, resp.) of any signal point in A is not equal to the real part (imaginary part, resp.) of any other
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2009 proceedings
⎤ s1I + js3Q −s2I + js4Q 0 0 ⎥ ⎢ s2I + js4Q s1I − js3Q 0 0 ⎥. X(s1 , s2 , s3 , s4 ) = ⎢ ⎣ 0 0 s3I + js1Q −s4I + js2Q ⎦ 0 0 s4I + js2Q s3I − js1Q ⎡
(5)
signal point in A [4]. So if QAM constellations are chosen, they have to be rotated. The optimum angle of rotation has been found in [4] to be 12 tan−1 2 radians and this maximizes the diversity and coding gain. We denote this angle by θg . Our STBC is obtained as follows. Our code matrix, denoted by S ∈ C4×4 , encodes eight symbols x1 , · · · , x8 drawn from a QAM constellation, denoted by Aq . We denote the rotated version of Aq by A. Let si ejθg xi , i = 1, 2, · · · 8, so that the symbols si are drawn from the constellation A. The codeword matrix is defined as
with xi ∈ Aq , i = 1, 2, · · · , 8. Using the equivalent model in (8), the ML-decoding metric can be written as
S(x1 , x2 , · · · , x8 ) X(s1 , s2 , s3 , s4 ) + ejθ X(s5 , s6 , s7 , s8 )P (6) with θ ∈ [0, π/2] and P being a permutation matrix designed to make the STBC full-rate and given by ⎡ ⎤ 0 0 1 0 ⎢ 0 0 0 1 ⎥ ⎥ P=⎢ ⎣ 1 0 0 0 ⎦. 0 1 0 0
If Heq ] = QT vec(Y). where y [y1 , · · · , y16 [h1 h2 · · · h16 ], where hi , i = 1, 2, · · · , 16 are column vectors, then Q and R have the general form obtained by Gram − Schmidt process as shown below
The choice of θ involves maximizing the diversity and coding gain. Here, we take θ to be π/4. This value of θ provides the largest coding gain achievable for this family of codes. It is to be noted that the set of the codeword difference matrices of the CIOD is a subset of the set of the codeword difference matrices of the proposed STBC. The minimum determinant and the coding gain of a code depend on the codeword difference matrices of the code. The minimum determinant for the CIOD, whose codeword has the structure shown in (5) is 10.24 [5] for unnormalized QAM constellations. The value of the minimum determinant for our 4 × 2 code, obtained for unnormalized 4- and 16-QAM is 10.24, which was checked by exhaustive search. This shows that the choice of π/4 in (6) indeed maximizes the coding gain. The resulting code matrix is as shown in the top of the next page. IV. L OW C OMPLEXITY ML-D ECODING In this section, we analyze the ML-decoding complexity of our code. It can be shown that (1) can be written as = Heq x˜q + vec(N), vec(Y) where Heq ∈ R16×16 is given by
ˇ G, Heq = I4 ⊗ H
(8)
(9)
with G ∈ R32×16 being the generator matrix for the STBC associated with the information vector xq , where xq [x1 , x2 , · · · , x8 ]T ,
(Y) − Heq x˜q 2 . M (x˜q ) = vec
(10)
On obtaining the QR decomposition of Heq , we get Heq = QR, where Q ∈ R16×16 is an orthonormal matrix and R ∈ R16×16 is an upper triangular matrix. The ML-decoding metric now can be written as − Rx˜q 2 = y − Rx˜q 2 , M(x˜q ) = QT vec(Y)
(11)
Q = [q1 q2 q3 · · · q16 ], where qi , i = 1, 2, · · · , 16 are mutually orthonormal column vectors and ⎡ ⎤ r1 hT2 q1 hT3 q1 . . . hT16 q1 ⎢ 0 r2 hT3 q2 . . . hT16 q2 ⎥ ⎢ ⎥ ⎢ 0 r3 . . . hT16 q3 ⎥ R=⎢ 0 ⎥, ⎢ . ⎥ .. .. .. .. ⎣ .. ⎦ . . . . 0
0
0
...
r16
where r1 = h1 , q1 = rr11 , i−1 ri = hi − j=1 (hTi qj )qj , qi = rrii , i = 2, · · · , 16. Let R be represented as follows R1 R2 , R= O8 R3 where R1 , R2 and R3 are 8 × 8 matrices, with R1 and R3 being upper triangular matrices. It is observed that R1 specifically has the following structure irrespective of the channel realization (refer [10] for details) ⎡ ⎤ ∗ ∗ 0 0 0 0 0 0 ⎢ 0 ∗ 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ 0 0 ∗ ∗ 0 0 0 0 ⎥ ⎢ ⎥ ⎢ 0 0 0 ∗ 0 0 0 0 ⎥ ⎥ R1 = ⎢ ⎢ 0 0 0 0 ∗ ∗ 0 0 ⎥, ⎢ ⎥ ⎢ 0 0 0 0 0 ∗ 0 0 ⎥ ⎢ ⎥ ⎣ 0 0 0 0 0 0 ∗ ∗ ⎦ 0 0 0 0 0 0 0 ∗ where ’∗’ denotes a possible non-zero entry. The structure of R allows our code to achieve simplified ML-decoding as follows - having fixed the symbols x5 , x6 , x7 and x8 , the
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2009 proceedings
⎤ s1I + js3Q −s2I + js4Q ejπ/4 (s5I + js7Q ) ejπ/4 (−s6I + js8Q ) ⎢ s1I − js3Q ejπ/4 (s6I + js8Q ) ejπ/4 (s5I − js7Q ) ⎥ s2I + js4Q ⎥. S=⎢ ⎦ ⎣ ejπ/4 (s7I + js5Q ) ejπ/4 (−s8I + js6Q ) s3I + js1Q −s4I + js2Q jπ/4 jπ/4 e (s8I + js6Q ) e (s7I − js5Q ) s4I + js2Q s3I − js1Q ⎡
symbols x1 , x2 , x3 and x4 can be decoded independently. In the decoding metric given by (11), it can be observed that the real and imaginary parts of symbol x1 are entangled with one another but are independent of the real and imaginary parts of x2 , x3 and x4 when x5 , x6 , x7 and x8 are conditionally given. Similarly, x2 , x3 and x4 are decoupled from one another although their own real and imaginary parts are coupled with one another. So, in general, the ML-decoding complexity of our code is of the order of M 5 . That is due to the fact that jointly decoding the symbols x5 , x6 , x7 and x8 followed by independently decoding x1 , x2 , x3 and x4 in parallel requires a total of (M 4 )(4M ) = 4M 5 metric computations. However, when square M -QAM is employed, which is usually done in practice, the ML-decoding complexity can be further reduced ˆ , · · · , x8Q ˆ ] denote the decoded as follows. Let ˆ x [xˆ1I , x1Q information vector. Assuming the use of a sphere decoder [12] 1) an 8 dimensional real SD is done to decode the symbols x5 , x6 , x7 and x8 . 2) Next, x1Q , x2Q , x3Q and x4Q are decoded in parallel. This is possible because they are not entangled with one another in the ML-decoding metric, when the symbols x5 , x6 , x7 and x8 are fixed. 3) Following this, x1I , x2I x3I and x4I are decoded using hard limiting as follows
u1 , −M , M , (12) xˆ1I = min max 2rnd 2
u2 xˆ2I = min max 2rnd , −M , M , (13) 2
u3 xˆ3I = min max 2rnd , −M , M , (14) 2
u4 , −M , M , (15) xˆ4I = min max 2rnd 2 where, 8
ˆ − (r(1,2i−1) xˆiI + r(1,2i) xˆiQ ) /r(1,1) u1 y1 − r(1,2) x1Q i=5
u2
y3
− r(3,4) x2Q ˆ −
8
(r(3,2i−1) xˆiI + r(3,2i) xˆiQ ) /r(3,3)
i=5 8
u3 y5 − r(5,6) x3Q ˆ − (r(5,2i−1) xˆiI + r(5,2i) xˆiQ ) /r(5,5) i=5
u4 y7 − r(7,8) x4Q ˆ −
8
(r(7,2i−1) xˆiI + r(7,2i) xˆiQ ) /r(7,7)
i=5
where, r(i,j) denotes the (i, j)th entry of the R-matrix. So, √ in all, we need to make a maximum of 4M 4 M metric
(7)
computations only. Hence, for square M -QAM, the √MLdecoding complexity of our code is of the order of M 4 M . V. S IMULATION R ESULTS In all simulation scenarios, we consider quasi-static Rayleigh flat fading channels. Figure 1 shows the CER performance of our 4 × 2 code, the well known DjABBA code and the BHV code, with all the codes using 4-QAM. Figure 2 shows the CER performance for 16-QAM. Both the plots exhibit a similar trend, with our 4×2 code outperforming both the DjABBA code and the BHV code at high SNR, and the DjABBA code in turn outperforming the BHV code. This can be attributed to the superior coding gain of our 4×2 code. The bad performance of the BHV code at a high SNR is due mainly to the fact that the BHV code does not have full-diversity. Table I gives a comparison between the minimum determinants and the ML-decoding complexities of the above three codes. The minimum determinants of all the codes for 4- and 16QAM have been calculated using exhaustive search without normalizing the constellation energy. However, it has been ensured that the average energy per codeword is maintained uniform for all the codes. It can be seen that our code has a coding gain twice that of the DjABBA’s. In the table, the ML-decoding complexity given for each code is the maximum number of metric computations needed in minimizing the MLdecoding metric. VI. C ONCLUDING REMARKS In this paper, we have presented a full-rate STBC for 4 × 2 MIMO transmission which has full-diversity and high coding gain for 4- and 16-QAM and also enables simplified MLdecoding. For 4- and 16-QAM, our code outperforms the DjABBA code, which has been reported to have the best performance for the 4 × 2 system. For any square M -QAM, our code has the same ML-decoding complexity as the BHV code which, to the best of our knowledge, has the lowest MLdecoding complexity among full-rate codes for a 4 × 2 MIMO system. So, to summarize, among the existing full-rate codes for 4 transmit antennas and 2 receive antennas, the proposed code is the best for 4- and 16-QAM. ACKNOWLEDGEMENT This work was partly supported by the DRDO-IISc program on Advanced Research in Mathematical Engineering. We would also like to thank Mr. Shashidhar, working with Beceem Communications, for useful discussions on the ML-decoding complexity issues.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2009 proceedings
TABLE I
C OMPARISON BETWEEN THE MINIMUM DETERMINANTS AND THE ML- DECODING COMPLEXITIES OF 4 × 2 STBC S
Min Det for ML Decoding complexity Code 4-, 16-QAM for square M -QAM DjABBA code [7] 0.64 4M√6 BHV code [8] 0 4M 4 √ M The proposed code [10] 10.24 4M 4 M 0
0
10
10
−1
Proposed STBC
Proposed STBC
BHV Code
BHV Code
10
DjABBA
DjABBA
−1
10
−2
10
−2
10 CER
CER
−3
10
−4
10
−3
10
−5
10
−4
10 −6
10
−5
−7
10
6
8
10
12
14
16
18
20
10
10
CER performance for 4-QAM R EFERENCES
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SNR in db
SNR in db Fig. 1.
12
Fig. 2.
CER performance for 16-QAM
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