This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.
Orthogonal Space-Time Block Codes design Using Jacket Transform for MIMO Transmission System Wei Song
Moon Ho Lee
Guihua Zeng
Institute of Information and Communication Chonbuk National University Jeonju 561-756 R.Korea Email:
[email protected]
Institute of Information and Communication Chonbuk National University Jeonju 561-756 R.Korea Email:
[email protected]
Department of Electronic Engineering Shanghai Jiaotong University Shanghai 200240, China Email:
[email protected]
Abstract—Hadamard transform has played a great part in Jacket transform. Motivated by Jacket transform, we propose a simple approach for space time block codes (STBC) design by using the Hadamrd in this letter,which achieves full rate, full diversity and employs simple decoding. The orthogonal STBC may be designed easily by using the proposed approach. Especially the performance of the designed orthogonal STBC may be improved greatly.
Index Terms: Orthogonal STBC, MIMO system, Jacket matrix,Full rate,Hadamard transform. I. I NTRODUCTION Compared to the single-antenna communication system [1], the multiple-input and multiple-output (MIMO) systems for wireless communication were demonstrated to provide a potential capacity gain In order to approach the capacity of the MIMO system, the space time block codes (STBC) have received a great amount attentions [1], [2], [3]. Over the past few years, many approaches for the STBCs have been suggested. In summarization, there are typically the following approaches. First, STBCs that admit a simple decoding for arbitrary complex constellations have been studied using the orthogonal designs theory in [2], [4]. Second, designs of STBCs using group and representation theory of groups have been reported in [5], [6]. Third, Hassibi and Hochwald [7] introduced codes that are linear in space and time called linear dispersion codes which absorb STBCs from orthogonal designs as a special case. Finally, the STBC designed using unitary matrices have been investigated in [8], [9]. However, in all previous STBC cases the patterns of the codes design are fixed. In this Letter, we suggest a new model for the orthogonal STBC designs which modulates block diagonal STBC using Jacket transform [10], [11]. It has been proven that the Discrete Fourier Transform (DFT) is a Jacket transform which is invented in [10]. Since a Jacket matrix can be decomposed into multiplication of a Hadamard matrix and a sparse matrix, the Fourier matrix is yielded from the DFT with the following form FN which N 2π −1 X(n) = m=0 x(m)W nm (W = e−i N , 0 ≤ n, m ≤ N − 1) can be expressed as FN = (W nm ) = HN SN PN ,
(1)
where HN is the sylvester Hadamard, SN is a sparse matrix, and PN is a permutation matrix. We note here SN is a diagonal block matrix. Using the permutation matrix PN one may design various pattern for the orthogonal STBC. In this case, the STBC is composed by ˜ = QPN . (2) Q Since | det PN | = 1, the revision with multiplication of a permutation matrix does not change the performance of the code. Motivated by Jacket transform, we propose a novel approach for the STBC in this Letter. II. S YSTEM M ODEL Consider a multiple antenna communication system with M transmit and N receive antennas. Let Hi = [h1,i , h2,i , · · · , hM,i ]T be the channel vector over the M channel uses. The employed channel in this Letter is assumed to be a quasi-static Rayleigh flat fading channel. Let the signal constellation be V = {V1 , V2 , · · · , VL }, and Q(V ) be a transmitted code matrix, where V ∈ V. Then the receive signal vector Y, i.e., a M × N matrix, is, (3) Y = Es QH + n, where H is the M × N channel matrix of Rayleigh-fading coefficients, Es is the average energy at each receiver antenna, and the noise n is modelled as independent samples of a zeromean complex Gaussian random variable with variance N0 /2 per dimension. For the MIMO communication system presented in Eq.(3), it has been proven that the pairwise block error probability under the maximum-likelihood (ML) decoder is as following [12], M −N Es 2 (1 + σ ) , (4) Pe ≤ 4N0 m m=1 where σm denotes the mth singular values of the matrix (Q − Q ). At high SNR, this inequality becomes, −M N Es 1 Pe ≤ . (5) 4N0 | det(Q − Q )|N
978-1-4244-2075-9/08/$25.00 ©2008 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.
Defining a diversity product λ,
B. Four transmitting antennas case
λ = min |det [(Q − Q )]|,
(6)
Q=Q
and
1/2 λ = min det (Q − Q )H (Q − Q ) ,
(7)
Q=Q
where Q and Q are code and error code words matrix, respectively. Evidently, to design ‘good’ STBC, one needs that λ reaches the maximum. Thus, if λ > 0, the code is said to have full diversity. III. E NCODING A PPROACHES A. General encoders We propose a novel approach for the STBC for MIMO communication system based on the encoding criteria. Divide randomly the signal constellation V = {V1 , V2 , · · · , VL } into K subset {S1 , S2 , · · · , SK }. For example, one may construct a generator similar to the Alamouti’s generator. After that, we encode the STBC code as follows, Q = CS = CΛ(S1 , S2 , · · · , SK ),
(8)
where the symbol Λ denotes a diagonal matrix with block elements Si , and C is a Hadamard matrix. we could calculate the following equation, QH Q = S H C H CS,
(9)
where QH is the Hermitian of Q. We know
Then we deal with the 4 transmit antennas case in the following. The transmit code matrix may be composed as Qp = C4 SJ , SJ =
=
(11)
= =
1
min | det(Q − Q )H (Q − Q )| 2
Q=Q
| det(C)|2
12
1
min | det(S − S )H (S − S )| 2 K min | det(C)| | det(Si − Si )| . (12)
S=S
S=S
i=1
The above equation shows that the total diversity product depends only on the multiplication of diversity product of various subsets. Since controlling determinants of various subsets is more easy than that for a high order matrix, the proposed approach is useful for the orthogonal and quasi-orthogonal STBC designs. In addition, the performance of the proposed STBC is controllable.
x2 x∗1 0 0
x1 −x∗2 SJ = 0 0
where In is a identity matrix. Simple calculations gives,
Since the Hadamard matrix is orthogonal matrix, one may obverse that there are no interferences between any difference signal constellation subsets Si and Sj . This property is available for the decoding procedure and diversity, subsequently the high capacity. Evidently, if SiH Sj = σij In , the code Q is orthogonal code; Otherwise it is a quasi-orthogonal STBC with only interference in the various signal constellation subsets. The diversity product of Q is obtained as following,
0 Q34
.
(14)
where the k is the number of symbols the encoder takes as its input in each encoding operation, and the p is the number of transmission periods required to transmit the space-time coded symbols through the multiple transmit antennas. If Qij is an orthogonal design, SJ is an orthogonal STBC. If Qij is quasi-orthogonal matrix, SJ is also a quasi-orthogonal STBC. Evidently, the properties of the code matrix SJ depends on Qij . We consider how to design an orthogonal STBC. We begin with a simple combination with Alamouti’s generator [2] as the block-elements of code matrix SJ , i.e., x1 x2 x3 x4 , Q34 = . (16) Q12 = −x∗2 x∗1 −x∗4 x∗3
(10)
H SK ). QH Q = S H C H CS = Λ(S1H S1 , S2H S2 , · · · , SK
Q12 0
H Thus, we have SJH SJ = diag(QH 12 Q12 , Q34 Q34 ). Apparently, the rate is R = k/p = 1, (15)
In this case,.
C H C = nIn ,
λ
where
(13)
Simply calculation
a0 0 H SJ SJ = 0 0
0 a0 0 0
0 0 x3 −x∗4
0 0 . x4 x∗3
(17)
0 0 a1 0
0 0 . 0 a1
(18)
i = xi − x i , where SJH is the Hermitian of SJ . Assuming x i are the transmit signal and receive signal, where xi and x 2 4 xi |2 , a1 = i=3 | xi |2 , respectively. So we get a0 = i=1 | and (19) λJ = min {(a0 .a1 )} . In the proposed approach, the code matrix QP may be written as, Q12 0 QP = C4 0 Q34
x4 + x∗3 x4 − x∗3 . (20) −x4 − x∗3 −x4 + x∗3 When C4 is 4 by 4 Hadamard matrix(C4 = H2 H2 ), we can x1 − x∗2 x1 + x∗2 = x1 − x∗2 x1 + x∗2
x2 + x∗1 x2 − x∗1 x2 + x∗1 x2 − x∗1
x3 − x∗4 x3 + x∗4 −x3 + x∗4 −x3 − x∗4
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.
λp = min {(b0 .b1 )} .
2
1.98
1.96
Entropy
get an orthogonal code word matrix. Simple calculation gives, b0 0 0 0 0 b0 0 0 , QH (21) P QP = 0 0 b1 0 0 0 0 b1 2 4 xi |2 , b1 = 4 i=3 | xi |2 . One may easily where b0 = 4 i=1 | obtain the diversity product,
1.94
1.92
1.9
1.88
(22)
1.86 −2
−1.5
−1
−0.5
0 w
0.5
1
1.5
2
Similarly, we can get a quasi-orthogonal STBC, QJ = C4 .SJ ,
and
x2 x∗3 0 0
x1 −x∗4 SJ = 0 0
c0 b∗ H QJ QJ = 0 0
0 0 x3 −x∗2
b c1 0 0
0 0 c1 −b∗
0 0 , x4 x∗1
Fig. 1. Relation function between entropy and center weight value w for four symbols.
(24)
0
10
diagonal block 4X4 matrix the proposed 4X4 matrix −1
10
0 0 , −b c0
−2
10
(25)
FER
where
(23)
−3
10
−4
2
2
2
2
10
4 x∗1 x 2 −
x1 | +4| x4 | , c1 = 4| x2 | +4| x3 | , b = where c0 = 4| 4 , and b∗ is complex conjugate of b. One may easily 4 x∗3 x obtain the diversity product, λJ = min (c0 c1 − |b|2 ) . (26)
−5
10
0
5
10 SNR(dB)
15
20
Fig. 2. Performance of the proposed design in (20) using four transmit antennas and one receive antenna over flat fading channel(QPSK).
Decoding algorithms can be derived for STBC with complex signal constellations [4]. The decision statistics x i can be represented by x i =
N
sgnt (i). rtj .hj,∈t (i)∗ .
(27)
t∈η(i) j=1
The decision metric is given by | xi − xi |2 + (2
N M
|hj,t |2 − 1)|xi |2 .
(28)
According to Trace criterion [15], We can analysis the performance of the proposed design. 1 2 Es )||F , (32) Pe ≤ exp −M ||D(x, x 2 4N0 where ||D(x, x )||F is a metric on the codebook of space time
t=1 j=1
Using Cw to take the place of C4 in Eq.(23), we can get a Non-orthogonal STBC,
10
QJ = Cw .SJ ,
10
H Cw Cw
−1
(29)
−2
10
FER
where Cw is a Center Weighted Hadamard matrix [14], 1 1 1 1 1 −w w −1 Cw = (30) 1 w −w −1 , 1 −1 −1 1 and we can get
0
−3
10
Orthogonal Non−orthogonal (w=2) Non−orthogonal (w=1.5) Jafarkhani Quasi−orthogonal
−4
10
−5
10
4 0 0 2 + 2w2 = 0 2 − 2w2 0 0
0 2 − 2w2 2 + 2w2 0
0 0 . 0 4
(31)
0
2
4
6
8 SNR(dB)
10
12
14
16
Fig. 3. Simulation of performance of the others transmitted matrices using four transmit antennas and one receive antenna over flat fading channel(QPSK).
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.
code. This is called “Trace criterion” because ||D(x, x )||2F = T r[QJ (x, x )H QJ (x, x )].
(33)
So we can get )H QJ (x, x )] T r[QJ (x, x 2 2 = 8(|x1 | + |x2 | ) + (4w2 + 4)(|x3 |2 + |x4 |2 ). (34) Assuming |x1 |2 = |x2 |2 = |x3 |2 =|x4 |2 , Ai denote the 4 coefficient of the |xi |2 , and ti = Ai /( i=1 Ai ). Now let H(P ) = −t1 log2 t1 −t2 log2 t2 −t3 log2 t3 −t4 log2 t4 , (35) where t1 =
1 3+w2 ,
t2 =
1 3+w2 , t3
=
1+w2 6+2w2 ,
t4 =
1+w2 6+2w2 .
(36)
2 1 + w2 1 1 + w2 log2 − log2 2 2 2 3+w 3+w 3+w 6 + 2w2
1 + w2 1 + w2 2 log (1 + w ) + log2 2 2 3 + w2 3 + w2 (37) Using First derivative test, we can get w = 0, ±1 when dH(P ) = 0, the function H(P ) has extremum point. In dw the Fig.1, when w2 = 1, H(P ) will be maximum and H(P ) ≤ log2 4 = 2. Simulation of the performance of these codes are are plotted in Fig.2 and Fig.3. One may find the code based on the proposed approach is more optimal than the previous approach in [3], [13]. = log2 (3 + w2 ) −
IV. S IMULATION R ESULTS The system that is radiation power limited requires that the transmitting antennas can send signals at the same time. In all simulation, the four transmit antennas and one receive antenna are considered. The result of simulation in Fig.2 shows that the performance of the proposed design has a remarkable improvement because this method can dispel the zero elements of the diagonal block transmitted matrix. The result of simulation in Fig.3 shows that when w = 1(here using 1.5, 2), the matrix is quasi-orthogonal,whose performance is superior to Jafarkhani case. With the increasing of w, its performance will decrease due to interference. When w = 1, its performance reaches the best because the transmit matrix is orthogonal. V. D ISCUSSIONS AND C ONCLUSIONS Based on the four transmit antennas approach, we can construct eight transmitting antennas orthogonal STBC and more transmit antennas orthogonal STBC. e.g., Qp8 = C8 S8 , where
C8 = C4 ⊗ H2 ,
S8 =
SJ1 0
0 SJ2
,
(39)
Similarly, n transmit antennas orthogonal or quasi-orthogonal STBC can be constructed easily (n = 2m , m = 2, 3, 4....). In conclusion, based on Jacket transform , we have investigated a new design of STBC . The present STBC enjoy the advantage that can be efficiently constructed with good performance. Using this approach, we derive several proposal codes to enrich their family. Simulation shows the modulation of Hadamard transform is optimal. ACKNOWLEDGEMENTS
Simply calculation, we can get H(P ) = −
where SJ1 and SJ2 are block matrix, SJ1 = SJ , Q56 0 . SJ2 = 0 Q78
(38)
This research was supported by the MIC(Ministry of Information and Communication), Korea, Under the ITFSIP(IT Foreign Specialist Inviting Program) supervised by the IITA(Institute of Information Technology Assessment), the Korea Research Foundation Grant funded by the Korean Government(MOEHRD)(KRF−2007 − 521 − D00330) and KICOS K20711000013-07A0100-01310. R EFERENCES [1] H. Jafarkhani, Space-time coding: Theory and Practice, Cambridge, Cambridge University Press, 2005. [2] A.Alamouti, A simple transmitter diversity scheme for wireless communications, IEEE J. Sel.. Areas Comm. Vol. 16, pp. 1451-1458, Oct.1998. [3] H. Jafarkhani, A quasi-orthogonal space time block encodes, IEEE Trans. Commun. Vol.49, pp1-4, 2001. [4] V.Tarokh, H.Jafarkhani, and A.R.Calderbank, Space-time block codes from orthogonal designs, IEEE Trans. Inform. Theory, Vol.45, pp.14561467, July 1999. Also see Correction to ’Space-time block codes from orthogonal designs’, IEEE Trans. Inform. Theory, Vol.46, p.314, Jan 2000. [5] B. Hassibi, B. M. Hochwald, A. Shokrollahi, and W. Sweldens, Representation theory for high-rate multiple-antenna code design, IEEE Trans. Inform. Theory, vol. 47, pp. 2335C2367, Sept. 2001. [6] B. Hughes, Optimal space-time constellations from groups, IEEE Trans. Inform. Theory, vol. 49, pp. 401C410, Feb. 2003. [7] B. Hassibi and B. Hochwald, High-rate codes that are linear in space and time, IEEE Trans. Inform. Theory, vol. 48, pp. 1804C1824, July 2002. [8] B. M. Hochwald and T. L. Marzetta, Unitary space-time modulation for multiple antenna communication in Rayleigh flat-fading, IEEE Trans. Inform. Theory, vol. 46, pp. 543C564, Mar. 2000. [9] T. M. Marzetta, B. Hassibi, and B. M. Hochwald, Structured unitary space-time auto-coding constellations, IEEE Trans. Inform. Theory, vol. 48, pp. 942C950, Apr. 2002. [10] M. H. Lee, B. S. Rajan, and J. Y. Park, ”A generalized reverse Jacket transform,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 48, no. 7, pp. 684-691, Jul. 2001. [11] Moon Ho Lee, ”A New Reverse Jacket Transform and Its Fast Algorithm,” IEEE, Trans. On Circuit and System, vol. 47. no. 1, pp.39-47, Jan. 2000. [12] V. Tarokh, N. Seshadri and A. R. Calderbank, Space-time codes for high data rate wireless communication: performance criterion and code construction, IEEE Trans. Inform. Theory, vol.44, No. 2, pp. 744-765, Mar. 1998. [13] J. Hou, M. Lee, and J. Park, Matrices Analysis of quasi-orthogonal Space-Time Block Codes, IEEE Comm. Lett., Vol.7, No.8, pp. 385-387, 2003. [14] M. H. Lee, The Center Weighted Hadamard Transform, IEEE Trans. Circuits Syst., Vol 36,pp.1247 - 1249. 1989. [15] Ionescu, D.M. New results on space-time code design criteria. IEEE Wireless Communications and Networking Conference (WCNC), 1999, 684-7.
