Fast-Group-Decodable Space-Time Block Code - Semantic Scholar

Fast-Group-Decodable Space-Time Block Code Tian Peng Ren1, Yong Liang Guan2, Chau Yuen3 and Rong Jun Shen4 1

National University of Defense Technology, Changsha 410073, China 2 Nanyang Technological University, Singapore 639798 3 Institute for Information Research, Singapore 119613 4 General Equipment Department of PLA, Beijing 100080, China Email: [email protected], [email protected] and [email protected]

Abstract—To make the implementation of high-rate STBC realistic in practical systems, group-decodable and fast-decodable code structures have been separately introduced into STBC to reduce the sphere decoding complexity. However, no STBC has both the two code structures until now. In this paper, high-rate fast-group-decodable STBC (FGD-STBC, which has both fastdecodable and group-decodable code structures) is proposed for the first time. We first derive the condition for fast-decodable code structure with the lowest sphere decoding complexity, then we prove that such fast-decodable code structure can be integrated with the group-decodable STBC for FGD-STBC construction. Analysis and simulation show that the proposed FGD-STBC has much lower decoding complexity and comparable performance with the existing fast-decodable STBC (FD-STBC).

I. I NTRODUCTION Multi-input multi-output (MIMO) techniques using multiple antennas have been adopted in many wireless communications standards, e.g., the IEEE 802.16e-2005 standard for mobile broadband wireless access systems [1]. Bell Labs layered space-time (BLAST) system [2] was was designed for maximum spatial multiplexing gain, but at the price of no spatial diversity gain. Using algebraic theory, full-rate full-diversity STBC such as Golden code [3] and perfect codes [4] were designed, but at the price of high decoding complexity. Recently, fast-decodable STBC (FD-STBC) with reduced decoding complexity were presented such as rate-2 FD-STBC for 4 transmit antennas in [5]. Meanwhile group-decodable STBC (GD-STBC) with code rate more than 1 were reported in [6]–[8]. In this paper, fast-group-decodable STBC (FGDSTBC), a class of high-rate reduced-complexity STBC that marries the code structures (hence merits) of FD-STBC and GD-STBC, is defined and presented for the first time. We derive the sufficient and necessary condition for fast-decodable code structure. Then we prove that such fast-decodable code structure can be integrated into the group-decodable STB to form an FGD-STBC. The systematic construction of FGDSTBC with arbitrary transmit antennas is also presented. Finally, an FGD-STBC design example for 4 transmit antennas is designed following this construction. The rest of this paper is organized as follows. In Section II, the system model is described and the existing works are reviewed. The definition and construction of FGD-STBC are proposed in Section III. In Section IV, the comparisons on decoding complexity and performance are presented. This paper is concluded in Section V.

In what follows, bold upper case and lower case letters denote matrices (sets) and vectors, respectively; R and C denote the real and the complex number fields, respectively; (·)R and (·)I stand for the real and imaginary parts of a complex element vector and matrix, respectively; [·]H , rank(·) and | · | denote the complex conjugate transpose, the rank of a matrix and the cardinality of a set, respectively. II. S YSTEM M ODEL AND E XISTING W ORKS A. Signal Model We consider a linear space-time block coding system employing Nt transmit antennas and Nr receive antennas. The transmitted signal sequences are partitioned into independent time blocks for transmission over T symbol durations using STBC matrix X of size T × Nt . Following the signal model in [9], X can be denoted as: XT ×Nt =

L X

sl Cl

(1)

l=1

where sl ∈ R is a real transmitted symbol (i.e., half a complex symbol), Cl ∈ CT ×Nt is called dispersion matrix. Thus, the L code rate is 2T considering complex symbol transmission. The received signals r˜tm of the mth receive antenna at time ˜ = [˜r1 , ˜r2 , · · · , ˜rNr ] = [˜ t can be arranged in a matrix R rtm ] of size T × Nr . Thus, the transmit-receive signal relation can be represented as: ˜ = √ρXH ˜ +Z ˜ R (2) ˜ Nt ×Nr = [h ˜1 , h ˜2 , · · · , h ˜Nr ] is the channel mawhere H ˜ nm are independent entries; Z ˜ = trix whose elements h [˜z1 , ˜z2 , · · · , ˜zNr ] = [˜ ztm ] is the additive noise matrix of size T × Nr whose elements z˜tm are independently, identically distributed (i.i.d.) CN (0, 1); ρ is the average signal-to-noise ratio (SNR) at each receive antenna. The received signal can also be written as [9]: √ r = ρHc + z (3) where 

   r=  

˜rR 1 ˜rI1 .. . ˜rR Nr ˜rINr





     ¯  , h =      

˜R h 1 ˜I1 h .. . ˜R h Nr ˜IN h r



   ,   

 s 1  s2 c=  .. . sL





   , z =     

˜zR 1 ˜zI1 .. . ˜zR Nr ˜zINr



   ,  

H = [h1 , h2 , · · ·  C 0 l  0 Cl Cl =  ..  .. . . 0 0

 , hL ] = ··· 0 ··· 0 .. .. . . · · · Cl

with l = 1, 2, · · · , L.

¯ C1 h    

¯ C2 h

¯ CL h

···

, Cl =



CR l CIl



,

−CIl CR l



2×2

Nr ×Nr

Step 4: Under the vector-to-matrix mapping function g −1 , YR  YR   YR  linearly independent matrices Y1I , Y2I , · · · , YJI can be 1 2 J  YR  obtained from YiI = g −1 (¯yi ) with i = 1, · · · , J. Then i the matrices Y1 , Y2 , · · · ,YJ must satisfy quasi-orthogonality constraint (QOC) [6] as CH Yi = −YH i C(i = 1, · · · , J)

B. Fast-Decodable STBC Consider standard sphere decoder (SD) being used to conduct ML decoding based on QR decomposition of the equivalent channel matrix H in (3): H = QR where Q = [e1 |e2 | · · · |eL ] ∈ R2T Nr ×L is unitary and R ∈ RL×L is uppertriangular. The definition of fast-decodable STBC (FD-STBC) as presented in [5] is as follows:

and hence can be applied as the dispersion matrices in the 2nd group. Note that there may be many solutions in the solution space. Typically, solutions leading to full-rank dispersion matrices, i.e., full symbol-wise diversity, are chosen. As such, the construction of 2-group-decodable STBC with code rate J+1 2T is achieved.

Definition 1 (Fast-Decodable STBC [5]). An STBC allows fast ML decoding if the condition hel1 , hl2 i = 0 (l1 = 1, · · · , K − 1, l2 = l1 + 1, · · · , K) (4) is satisfied, where after QR decomposition, the equivalent channel matrix H = [h1 |h2 | · · · |hL ] = QR, Q = [e1 |e2 | · · · |eL ] and K ≤ L. Then K levels can be removed from the real SD tree. C. Group-Decodable STBC In a group-decodable STBC, the information symbols embedded can be divided into several groups such after channel matched filtering, the ML detection can be decoupled into several submetrics, and only the information symbols in the same group need to be decoded jointly. A definition of Γgroup-decodable STBC was presented in [8] as follows: Definition 2 (Group-Decodable STBC [8]). An STBC with dispersion matrices C1 , · · · , CL is said to be Γ-groupdecodable if H ′ (i) CH p Cq = −Cq Cp , ∀p ∈ Θi , ∀q ∈ Θi′ , i 6= i ; R  R   CR    C C (ii) CiI1 , · · · , CiIk , · · · , iILi are linearly independent i1

ik

Ci

Li

with ik ∈ Θi , k = 1, 2, · · · , Li , where i = 1, 2, · · · , Γ, Θi denotes the setP of indexes of symbols in the ith group with Γ |Θi | = Li and i=1 Li = L.

A systematic construction of 2-group-decodable STBC for Nt transmit antennas over T symbol durations was presented in [8] as follows: Step 1: Pick a seed matrix C of size T ×Nt as the dispersion matrix in the 1st group; Step 2: Based on the seed matrix C, construct the following equation: C¯y = 0 (5)  R Y where C = f (C) of size Nt2 × 2T Nt and ¯y = g( YI ) of size 2T Nt × 1 are given in (7) with mapping functions f and g, I R I R I ′ I R I

n = [cR 1n c1n c2n c2n · · · cT n cT n ] and n = [c1n −c1n c2n I R −cR · · · c − c ]. 2n Tn Tn Step 3: Solve equation (5) and obtain the solution space, which can be represented as {¯y1 , ¯y2 , · · · , ¯yJ } where J = 2T Nt − Nt2 when T ≥ Nt or J = T 2 when T < Nt ;

(6)



                  C = f (C) =                  

1 0 .. . 0

2

′2

3

′3 .. .

N

′N 0 0 .. . 0 0 .. . 0 0

1 − ′1

··· 0 .. . ··· 0 0

0 0 .. . 0 0

− ′1 .. . ··· ···

0

2 .. . ···

3

′3 .. .

N

′N .. . ··· ···

1

2

− ′2 .. . 0 0 .. . ··· ···

··· ··· .. . ··· 0 0 0 0 .. . ··· ··· 0 0 .. . ··· ··· .. . 0 0

··· ··· .. . 0 ··· ··· ··· ··· .. . 0 0 ··· ··· .. . 0 0 .. .

N

′N

0 0 .. .

N 0 0 0 0 .. .

1

− ′1 0 0 .. .

2

− ′2 .. .

N−1

− ′N−1

"

# YR ¯ y = g( I ) Y h i R I R I = y11 y11 · · · yTR1 yTI 1 · · · y1N y1N · · · yTRN yTI N

                                    

(7a)

(7b)

III. FAST-G ROUP -D ECODABLE STBC To data, there is no STBC having both the fast-decodable and group-decodable code structures. In this section, we set forth to achieve this new code design. A. Fast-Decodable Code Structure with Lowest Complexity We first propose a new condition based on the dispersion matrices for fast-decodable code structure as follows, which is different from (4) in Definition 1. Theorem 1. Condition (4) in Definition 1 for fast-decodable code structure with K levels removed from the real SD tree is satisfied if and only if H CH l1 Cl2 = −Cl2 Cl1

with l1 = 1, · · · , K − 1 and l2 = l1 + 1, · · · , K.

(8)

Proof: (Sufficient condition) If (8) is satisfied, following the QOC, the real symbols with index l1 and l2 are orthogonal to each other, i.e., hhl1 , hl2 i = 0

(9)

with l1 = 1, · · · , K − 1 and l2 = l1 + 1, · · · , K. Since e1 = h1 ||h1 || , we can have he1 , hl2 i = 0 with l2 = 2, · · · , K. Since h2 1 ,h2 ie1 e2 = ||hh22 −he −he1 ,h2 ie1 || = ||h2 || , we have he2 , hl2 i = 0 with l2 = 3, · · · , K. Following the same way, we can have that

hel1 , hl2 i = 0

(10)

with l1 = 1, · · · , K − 1 and l2 = l1 + 1, · · · , K. (Necessary condition) The necessary condition is easy to be proved, hence is omitted due to space limitation. Obviously, condition (4) or (8) results in a diagonal submatrix of size K × K at the upper left corner of R. Hence the first K real symbols in the STBC matrix are orthogonal to each other, which implies that fast-decodable code structure can be implemented from an orthogonal STBC. Moreover, the orthogonal STBC with the maximum code rate can guarantee the most levels removed from the real SD tree, i.e., the lowest complexity. B. Definition of FGD-STBC For the group-decodable STBC, although only the information symbols in the same group need to be decoded jointly, the decoding complexity is still high if the group size is large. If the fast decoding can be integrated with the ML decoding conducted for one group of the group-decodable STBC, there will be a new kind of code, named fast-group-decodable STBC (FGD-STBC), which is defined as follows: Definition 3 (Fast-Group-Decodable STBC). An STBC with dispersion matrices C1 , · · · , CL is said to be fast-groupdecodable if H ′ (i) CH p Cq = −Cq Cp , ∀p ∈ Θi , ∀q ∈ Θi′ , i 6= i ; R  R  CR    C C (ii) CiI1 , · · · , CiIl , · · · , iILi are linearly independent i1

il

Ci

Li

with il ∈ Θi , l = 1, 2, · · · , Li ; H (iii) CH il1 Cil2 = −Cil2 Cil1 with il1 , il2 ∈ Θi , l1 = 1, 2, · · · , Ki − 1, l2 = l1 + 1, · · · , Ki and Ki ≤ Li , where i = 1, 2, · · · , Γ, Θi denotes the set of PΓindexes of symbols in the ith group with |Θi | = Li and i=1 Li = L. Then Ki levels are removed from the real SD tree for the ith group.

The relationship between group-decodable, fast-decodable and fast-group-decodable code structures are illustrated graphically in Fig. 1 based on the upper-triangular matrix R after the QR decomposition on the equivalent channel matrix H. C. Construction of FGD-STBC In this subsection, we first prove that the fast-decodable code structure with the lowest complexity can be integrated with 2-group-decodable STBC structure, and then present the construction of FGD-STBC with the fast-decodable code structure in the 2nd group.

Fig. 1. Upper-triangular matrices Rs for the 2-group-decodable, fastdecodable and fast-group-decodable code structures.

Assume that C1 , · · · , CK are the dispersion matrices of an orthogonal STBC with the maximum code rate. Without loss of generality, C1 is chosen as C in (6). Following the construction in Section II-C, the dispersion matrices Yi with i = 1, · · · , J in the 2nd group can be obtained to satisfy YR  YR  H 1 J CH 1 Yi = −Yi C1 and the space { YI , · · · , YI } is of 1

J

H dimension J. Since CH i = 2, · · · , K, 1 Ci = −Ci C1 with CR  CR    R YR 1 { C2I , · · · , CK } is a sub-space of { , · · · , YYJI }. The I I Y 2 1 K J  R  R K following theorem shows that { CC2I , · · · , C } is of dimenI CK 2 sion K − 1.

Theorem 2. If the nonzero matrices C2 , C3 , · · · , CK satisfy  CR   CR  the QOC, the matrices C2I , · · · , CK must be linearly I 2 K independent. Proof: We employ proof by contradiction.  R  R K Suppose that CC2I , · · · , C are not linearly independent, CIK 2 CR   R  R K i.e., without loss of generality, C2I = α3 CC3I +· · ·+αK C CIK 2 3 where not all the scalars α3 , · · · αK ∈ C are zero. Then we can obtain that C2 = α3 C3 + · · · + αK CK

(11)

Since C2 , C3 , · · · , CK satisfy the QOC to each other, H CH 2 Cl = −Cl C2 (l = 3, · · · , K). Based on distributive law, we have H CH 2 (α3 C3 + · · · αK CK ) = −(α3 C3 + · · · αK CK ) C2 H ⇒ CH 2 C2 = −C2 C2 ⇒ C2 = 0

(12)

which is contrary to the original premise “nonzero matrices C2 , · · · ”. Thus Theorem 2 is proved.  R  R  R K Thus, the complement of { CC2I , · · · , C } in { YY1I , · · · , CIK 2 YR  CR  CR  ′ YR  Y1R  J 2 K 1 J }, i.e., { , · · · , } = { , · · · , } − YIJ CI CIK YI1 YIJ CR  CR 2 { C2I , · · · , CK is of dimension J − K + 1. Let I } 2

K

{

CR  K+1 CIK+1

,··· ,

 CR  CR  CR  ′ J+1 } be a basis of { C2I , · · · , CK } , Ci (i = I CI 2

J+1

K

2, · · · , J + 1) will satisfy the QOC with C1 (i.e., condition (i) in Definition 3), and C2 to CJ+1 satisfy condition (ii) in Definition 3, then the resultant STBC is group-decodable with code rate J+1 2T where s1 is in the 1st group, and s2 to sJ+1 are in the 2nd group. In the 2nd group, since C2 to CK satisfy the QOC with each other (i.e., condition (iii) in Definition 3), K − 1 levels will be removed from the real SD tree for the 2nd group. Hence, the resultant STBC is fastgroup-decodable. Note that the orthogonal STBC based on the dispersion matrices C1 to CK has achieved the maximum code rate, which means that at most K − 1 levels are removed from the real SD tree for the 2nd group. Hence the resultant FGD-STBC achieves the lowest complexity as defined in [5] for FD-STBC. From the discussions above, we can summarize the systematic construction of FGD-STBC with 2 groups as follows: Suppose that C1 , · · · , CK are the dispersion matrices of an orthogonal STBC with the maximum code rate. Step 1: Choose C1 as the seed matrix C in (6), then obtain the J-dimension solution space {¯y}, denoted as {¯ y1 , ¯y2 , · · · , ¯yJ }. Step 2: Under the vector-to-matrix mapping function g −1 , YR  YR  YR  YR  obtain { Y1I , Y2I , · · · , YJI } where YiI = g −1 (¯yi ) with 1 2 i J i = 1, · · · , J.  R  R ′  R  R K Step 3: Let the basis of { CC2I , · · · , C } = { YY1I , YY2I , CIK 2 1 YR  CR   CR  CR  2 K+1 · · · , YJI } − { C2I , · · · , CK } be denoted as { ,··· , I CIK+1 2 J K  CR  J+1 }, then the resultant STBC with dispersion matrices CIJ+1 C1 to CJ+1 are fast-group-decodable that achieves the lowest complexity as defined in [5] for FD-STBC. D. Example of FGD-STBC In this subsection, we design an FGD-STBC of size 4×4 from the orthogonal STBC with code rate 3/4 in [10]. The dispersion matrices of this orthogonal STBC are: 

 C1 =  

 C3 = 



 C5 = 

1 0 0 0 0 0 −1 0 0 0 0 1

0 1 0 0

 0 0 0 0  , 1 0  0 1 0 1 0 0 0 1 0 0 0 −1 0 0 0 0 −1 0 1 0 −1 0 0 0 0 0





 C2 = 



   , C4 =  



   , C6 = 

j 0 0 0 0 0 j 0 0 0 0 j

0 j 0 0

0 0 −j 0 0 j 0 0 0 0 −j 0 0 0 0 j j 0 0 0

0 0 0 −j 0 −j 0 0 

j 0  . 0  0



 , 

  , (13)

Let C1 be the seed matrix C in (6), then there will be J = 2T N − N 2 = 16 dispersion matrices in the 2nd group, including C2 to C6 and the other 11 dispersion matrices as follows: 0  −1 C7 =  0 0 

1 0 0 0

0 0 0 −1

  0 0   , C8 =  1  0

0 j 0 0

j 0 0 0

0 0 0 −j

 0 0  , −j  0



 C9 = 



 C11 =  

 C13 =  

 C15 =  

 C17 = 

0 0 0 −1 0 0 −1 0 0 −1 0 0 j 0 0 0 j 0 0 0

0 0 −1 0 0 0 0 1 1 0 0 0 0 −j 0 0 0 −j 0 0

0 1 0 0 1 0 0 0 0 0 0 1 0 0 −j 0 0 0 j 0

1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 j 0 0 0 −j

















   , C10 = 

   , C12 = 

   , C14 = 

   , C16 = 

0 0 0 j 0 0 j 0 0 j 0 0 j 0 0 0

0 0 −j 0 0 0 0 j j 0 0 0 0 j 0 0

0 −j 0 0 j 0 0 j 0 0 0 0 0 0 0 0 0 j j 0 0 0 0 0 j 0 0 j



 .

 j 0  , 0  0   , 

 ,



 ,

(14)

Since the dispersion matrices in (13) and (14) are full rank, the resultant FGD-STBC shown in (17) is full symbolwise diversity and of code rate 17/8 where the real symbol s1 is in the 1st group, while the remaining real symbols si · · · s17 are in the 2nd group. In the 2nd group, since only C2 to C6 satisfy the QOC with each other, 5 levels at most can be removed from the real SD tree for the 2nd group. Hence the resultant FGD-STBC achieves the lowest complexity. Note that following the methodology described above, FGDSTBC can be constructed from not only orthogonal STBC, but also quasi-orthogonal STBC [11]. IV. C OMPARISONS

AND

D ISCUSSIONS

In this section, we compare our proposed FGD-STBC with BLAST [2], perfect code [4] and FD-STBC [5] in a 4×4 MIMO system with 4 bits per channel use. Hence, BLAST, perfect code and rate-2 FD-STBC [5] are simulated with BPSK, BPSK and 4-QAM, respectively. To achieve rate 2, we simulate the FGD-STBC proposed in (17) with s17 removed. A. Complexity Comparison Several important parameters of these codes are compared in Table I where the decoding complexity order O can be presented as [5] G−K+1 2

O = K · Ms

(18)

where the group size G means the maximum number of joint-ML-decoded real symbols, K denotes the number of the levels removed from the real SD tree for ML decoding b these G real symbols above, Ms = 2 R is the size of signal constellation applied and b is the system bit rate. Different from the complexity calculation based on the complex SD in [5], (18) is based on the real SD, which is more explicit. We can see that the proposed rate-2 FGD-STBC has the lowest complexity order among the full-symbolwise-diversity codes. For example, compared with the FD-STBC presented in [5],

XF GD−ST BC =  s1 + js2 + js15 + js16 + js17 −s7 + js8 − s13 + js14   −s3 + js4 − s11 + js12 s5 − js6 − s9 + js10

s7 + js8 + s13 + js14 s1 + js2 + js15 − js16 − js17 −s5 − js6 − s9 − js10 −s3 − js4 + s11 + js12

s3 + js4 + s11 + js12 s5 − js6 + s9 − js10 s1 − js2 + js15 − js16 + js17 −s7 − js8 + s13 + js14

the complexity order of the proposed FGD-STBC is reduced from 4 · 213 to 5 · 211 , or by 68.75%, with b = 4 bits per channel use.

 −s5 − js6 + s9 + js10 s3 − js4 − s11 + js12   s7 − js8 − s13 + js14 s1 − js2 + js15 + js16 − js17 (17)

−1

10

TABLE I C OMPARISONS OF H IGH -R ATE STBC IN A 4×4 MIMO S YSTEM WITH b B ITS /C HANNEL U SE .

−2

10

Code Rate Group size Level removal of Complexity (R) (G) real SD (K) order (O)

BER

−3

10

−4

10

BLAST [2]

4

8

1

2b

Perfect Code [4]

4

32

1

24b

4

BLAST[2], O=2

−5

10

16

Perfect code[4], O=2

13

FD-STBC [5]

2

16

4

4·

FD−STBC best known[5], O=4x2

13b 2 4

11

Proposed rate−2 FGD−STBC, O=5x2 −6

FGD-STBC proposed

2

15

5

5·2

11b 4

B. BER Performance Comparison Assuming that the MIMO channel is quasi-static Rayleigh faded in the sense that the channel coefficients do not change during one codeword transmission, and the channel state information (CSI) is perfectly known at the receiver, we plot the BER curves against signal-to-noise ratio (SNR) per bit in Fig. 2 where the constellation rotations for rate-2 FGDSTBC proposed are obtained by computer search to achieve full diversity. From the simulation results, we can see that the proposed FGD-STBC is able to achieve full diversity, just like perfect code; And the proposed FGD-STBC requires an SNR only 0.4 dB higher than the best known rate-2 FD-STBC with 4-QAM [5], which can be regarded as the penalty to be paid for the complexity reduction. V. C ONCLUSION In this paper, fast-group-decodable STBC (FGD-STBC) with reduced sphere decoding complexity is proposed for the first time. Based on a new condition derived for fast decoding, we show that the lowest sphere decoding complexity is obtained when the fast-decodable code structure is designed from an orthogonal STBC with the maximum code rate. We then prove that such fast-decodable code structure can be integrated with the group-decodable STBC structure, which can reduce the decoding complexity continuously. The systematic construction of FGD-STBC is also presented. Simulation results show that the proposed FGD-STBC has comparable performance to the best known FD-STBC, but with a decoding 13b 11b complexity reduction from 4 · 2 4 to 5 · 2 4 with b bits per channel use.

10

−6

−4

−2

0

2 4 SNR per bit/dB

6

8

10

12

Fig. 2. Performance comparisons in a 4×4 MIMO system with 4 bits per channel use.

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