Quasi-Orthogonal STBC with Minimum Decoding Complexity: Further Results Chau Yuen1, Yong Liang Guan1, Tjeng Thiang Tjhung2 1
Nanyang Technological University, Singapore,
[email protected],
[email protected] 2 Institute for Infocomm Research, Singapore,
[email protected] for eight transmit antennas. Hence with GCLT in [11], QOSTBC can always achieve full diversity by requiring only half the number of symbols required for joint detection than CR [7-10]. However, such decoding complexity reduction by GCLT can only be achieved for square or rectangular QAM constellations.
Abstract - A new class of Quasi-Orthogonal Space-Time Block Code (QO-STBC) namely Minimum-DecodingComplexity QO-STBC (MDC-QOSTBC) has recently been proposed in the literature. In this paper, we analyze some of its essential code parameter and code property. Specifically, we derive its maximum achievable code rate expression for any number of transmit antennas. We also derive the closedform expression of its diversity product and use it to obtain the optimum constellation rotation in order to achieve optimum decoding performance. Other performance benefits of MDC-QOSTBC, such as low decoding complexity, good coding gain and power distribution property, flexibility in supporting any constellation and number of transmit antennas, are also presented and discussed.
To obtain minimum decoding complexity for QO-STBC with any constellation, Co-ordinate Interleaved Orthogonal Design (CIOD) was proposed in [13, 14], and MinimumDecoding-Complexity QO-STBC (MDC-QOSTBC) was proposed [12], to achieve real symbol-pair-wise ML decoding for any complex constellation. In this paper, our intention is to analyze some essential code parameter and code property of the MDC-QOSTBC, as these analyses were not reported in [12] due to space constraint. Firstly, we derive the close-form expression of the maximum achievable code rate of MDC-QOSTBC. Next, we derive analytically the optimum CR angle for MDCQOSTBC with square and rectangular QAM constellation. For PSK constellation, computer search are employed to find the optimum CR angle. Next, we compare the performance of MDC-QOSTBC with CIOD and others QO-STBCs.
I. INTRODUCTION Orthogonal Space-Time Block Code (O-STBC) has been proposed in the literature [1-3] to provide full transmit diversity with simple maximum likelihood (ML) decoding complexity. For the case of PSK constellation, ML decoding of O-STBC requires complex symbol-wise decoding; while for the case of square or rectangular QAM constellation, ML decoding of O-STBC only requires real symbol-wise decoding. Despite these advantages, O-STBC will not be bandwidth efficient if more than two transmit antennas and complex constellation are used, because its maximum code rate will be less than one. As a result, researchers have proposed Quasi-Orthogonal STBC (QO-STBC).
The organization of this paper is as follow. Section II reviews the generic signal model of linear STBC and the construction of MDC-QOSTBC. Section III gives a closed form on the maximum achievable code rate of MDCQOSTBC. Section IV gives the derivation and search of optimum CR angle for MDC-QOSTBC to achieve optimum decoding performance and Section V gives a performance comparison of MDC-QOSTBC with other scheme. Finally Section VI concludes the paper.
QO-STBC achieves a higher code rate than O-STBC at the expense of allowing higher decoding complexity [4-6]. QO-STBC can achieve full diversity if constellation rotation (CR) technique is used [7-10]. With CR, the ML decoding of QO-STBC will be complex symbol-pair-wise decodable (i.e. it requires joint detection of two complex symbols) for full rate QO-STBC for four transmit antennas [7-10], or 3/4-rate QO-STBC for eight transmit antennas [9]; while joint detection of four complex symbols will be required for full rate QO-STBC for eight transmit antennas [10]. In a later publication [11], group-constrained linear transformation (GCLT) has been proposed as a mean to optimize the decoding performance of QO-STBC at a reduced decoding complexity than CR for the case of square and rectangular QAM constellation. Specifically, it is showed in [11] that QO-STBC with GCLT will be real symbol-pair-wise decodable for full rate QO-STBC for four transmit antennas, or 3/4-rate QO-STBC for eight transmit antennas; while complex symbol-pair-wise decodable for full rate QO-STBC
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II. REVIEW OF MDC-QOSTBC A. Signal Model for Linear STBC Suppose that there are Nt transmit antennas, Nr receive antennas, and an interval of T symbols in which the propagation channel condition is almost time-invariant and known to the receiver. The transmitted signal can then be written as a T × Nt matrix G that governs the transmission over the Nt antennas during the T interval. It is assumed that the data sequence has been broken into blocks with K symbols in each block for transmission over T symbol periods of time and that x1, x2, …, xK are the complex symbols chosen from an arbitrary constellation. The code rate of a STBC is defined as R = K/T. By following the model in [15], a STBC G can be expressed as:
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G = ∑ q =1 ( xqR A q + jxqI B q ) K
MDC-QOSTBC is constructed from O-STBC, the proof of Theorem 1 will be divided into the following 2 cases.
(1)
xq = xqR + jxqI , and superscipt R and I represent the real and imaginary of a symbols respectively. Matrices Aq and Bq are called the “dispersion matrices” and are of size T × Nt.
where the transmitted symbols are
Case 1: Three to eight transmit antennas In [2], it has been shown that O-STBC has a code length of 2
The construction of an MDC-QOSTBC [12] can be described by the four mapping rules in (2). An MDCQOSTBC which consists of K sets of dispersion matrices denoted as {A, B} for Nt transmit antennas with code length T can be constructed by an O-STBC which consists of K/2 sets of dispersion matrices denoted as {A, B} for Nt/2 transmit antennas with code length T/2. 0 0 A jB u #1: A u = u #3: A K / 2 + u = 0 j 0 A B u u (2) jA u 0 0 Bu #2: B u = #4: B K / 2 + u = 0 jA u Bu 0
(iii)A uH B v = B vH A u
1≤ u ≠ v ≤ K /2
log 2 N t
n = 2
(5)
= 2 N t 2 for 2 ≤ N t ≤ 4
(6)
Since MDC-QOSTBC constructed by (2) has the same code rate as the half-size O-STBC used to construct it. Hence, by substituting n = 2 N t 2 = 2 N t 4 , where N t = N t / 2 , into (5), we get the following maximum achievable code rate of MDC-QOSTBC for three to eight transmit antennas:
RMDC-QOSTBC =
(3)
1 ≤ u, v ≤ K / 2
=
1 + log 2 ( 2 N t 4 ) for 3 ≤ Nt ≤ 8 2 N t 4 (7) 2 + log 2 ( Nt 4 ) 2 N t 4
Hence the first part of Theorem 1 is proved. Case 2: For more than eight transmit antennas and above When the number of transmit antennas is more than four, it has been shown in [3] that a non-square O-STBC has a higher code rate than square O-STBC, and the maximum code rate of the former is:
III. MAXIMUM ACHIEVABLE CODE RATE OF MDC-QOSTBC From the construction of MDC-QOSTBC in (2), it can be seen that the maximum code rate of an MDC-QOSTBC is related to the maximum code rate of an O-STBC.
RO-STBC =
2+n 2n
n = 2 N t 2
(8)
Using a similar approach as that in Case 1, the second part can be easily proved too. Hence Theorem 1 is proved. ■
Theorem 1: The maximum achievable code rate of an MDCQOSTBC is:
However, it should be noted that, although the MDCQOSTBC constructed by non-square O-STBC in Case 2 has a higher code rate, it has a higher code length as well, which leads to longer decoding latency. For example, for 16 number of transmit antennas, the MDC-QOSTBC constructed from the non-square O-STBC in [3] will have a code rate of 5/8 and a code length of 112, whereas the MDC-QOSTBC constructed by the square O-STBC in [2] will have a slightly lower code rate of 4/8 and a much shorter code length of just 16.
2 + log 2 n N where n = t , 3 ≤ Nt ≤ 8. 2 n 4 = (4) 1 + n where n = N t , N > 8. t 2n 4
where x represents the upper integer floor of x. Proof of Theorem 1: It has been shown in [3] that a nonsquare O-STBC for more than 4 transmit antennas has a higher achievable code rate than square O-STBC. Since
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log2 N t
where n = 2
From (2), it is clear that the MDC-QOSTBC is constructed by O-STBC with half of the transmit antennas. Since the number of transmit antennas for the MDCQOSTBC considered here is three to eight, the size of OSTBC is two to four correspondingly. Under this range of transmit antenna number, the following equality holds:
From (2), MDC-QOSTBC will have a code rate that is the same as the half-size O-STBC used to construct it. Although the construction method expressed in (2) generates MDCQOSTBC with even number of transmit antennas, [12] has shown that by removing any number of the codeword column, the resultant code is still a valid MDC-QOSTBC.
RMDC-QOSTBC
1 + log 2 n n
In (5), N t is the number of antennas for the O-STBC.
where 1 ≤ u ≤ K/2, and the dispersion matrices {A, B} of an O-STBC have the following properties [2]: A uH A u = I Nt / 2 (i) 1≤ u ≤ K /2 B uH Bu = I Nt / 2 A uH A v = − A vH A u B uH B v = −B Hv B u
and the maximum code rate of
RO-STBC =
B. Construction of MDC-QOSTBC
(ii)
log2 N t
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IV. PERFORMANCE OPTIMIZATION OF MDC-QOSTBC
constellations. Hence CR technique can be used to avoid this situation. Assuming that xi is now the rotated symbols, and ∆i is the error in the symbol xi, the determinant expression in (12) can now be represented in (14) in terms of the error of an un-rotated constellation symbol, Λi:
Similar to the QO-STBC in [4-6] and the CIOD or ACIOD in [13, 14], MDC-QOSTBC [12] is not able to achieve full diversity without CR. In this section, we give a study on the optimization of MDC-QOSTBC based on CR.
∆ iR = Λ iR cos θ − Λ iI sin θ
A. Diversity Product of MDC-QOSTBC We first derive the closed form expression of the diversity product of MDC-QOSTBC, which will subsequently be used to search for the optimum CR angle. Due to the structure of MDC-QOSTBC as shown in (2), and the properties of the dispersion matrices of an O-STBC as shown in (3), we can obtain the following expression for MDCQOSTBC’s codeword distance matrix, Ace:
( Λ R )2 cos ( 2θ ) − 2Λ R Λ I sin ( 2θ ) − i i i det = I 2 ( Λ i ) cos ( 2θ )
0 0 +β I Nt / 2 I Nt / 2
I Nt / 2 0
K /2
i =1
i =1
Theorem 2: The optimum CR angle of MDC-QOSTBC for square and rectangular QAM is ½ tan-1(½). And the corresponding diversity product is:
(9)
ζ =
α = ∑ (∆iR ) 2 + (∆iI ) 2 , β = 2 ∑ −∆ iR ∆ iI + ∆ RK / 2 + i ∆ IK / 2 + i , and ∆i Since the decoding performance of a space-time code has been shown to be dependent on the rank and minimum determinant (for the case of a full diversity code) of a codeword distance matrix [16], the determinant of the Ace of MDC-QOSTBC can be derived from (9) to be: Nt / 2
1 2 Nt
min [ det( A ce ) ]
1/ 2T
(10)
In order to achieve full diversity, the diversity product has to be non-zero, and in order to achieve optimum coding gain, its value has to be maximized. Following [7-10], when considering the minimum value of the determinant in (11), we can assume that only one of the symbols (e.g. symbol xi) makes an error and the rest of the symbols are error free, hence the determinant can be further simplified as follows when considering only the worst case: ( ( ∆ ) + ( ∆ ) + 2∆ ∆ ) × det = ( ( ∆ iR ) 2 + ( ∆ iI )2 − 2∆ iR ∆ iI ) R 2 i
I 2 i
2 2 = ( ∆ iR ) − ( ∆ iI )
R i
I i
(15)
Nt
•
2 (ΛR, ΛI) = ± (0, dmin) Î det1 = d min cos(2θ )
•
2 (ΛR, ΛI) = ± (dmin, 0) Î det 2 = d min cos(2θ )
•
2 (ΛR, ΛI) = ± (dmin, dmin)Î det 3 = −2d min sin(2θ )
•
2 (ΛR, ΛI) = ± (dmin, -dmin)Î det 4 = 2d min sin(2θ )
Nt Nt
Nt
Note that det1 = det2 and det3 = det4. In order to maximize the smaller value between det1 and det3, we equate det1 and det3 to get: cos(2θ opt ) = 2sin(2θ opt ) ⇒ θ opt = tan −1 (1/ 2) 2 = 13.280
Nt 2
(16)
Hence the minimum determinant and the corresponding diversity product are:
(12)
Nt
4 det min = 5
It can be easily seen from (12) that, without CR, the determinant and hence the diversity product will be zero due to the possible occurrence of ∆R = ∆I in most standard
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Nt / T d min
The I and Q components of a 4-QAM symbol can be viewed as two independent 2-PAM symbols. Hence ΛR, ΛI ∈{0, ±dmin} where dmin is the minimum Euclidian distance between two constellation points, and ΛR, ΛI cannot be both zero in (14). To maximize the minimum determinant value in (14), the following four cases of (ΛR, ΛI) and their resultant determinant values as per (14) are considered:
(11)
C≠E
N t / 4T
For 4-QAM
A related code parameter that has commonly been used to characterize the decoding performance of a STBC [9] is called the diversity product, ζ, which is defined as:
ζ =
4 2 Nt 5 1
Proof of Theorem 2: To derive the optimum CR angle, we start with the constellation 4-QAM, and then we will generalize to M-ary square and rectangular QAM.
represents the possible error of symbols xi.
det = det( A ce ) = (α + β )(α − β )
(14)
B. Optimum CR Angle for Square and Rectangular QAM
where C and E represent two different codewords, K
Nt
where θ represents the angle of CR.
A ce = (C − E) H (C − E) I Nt / 2 =α 0
(13)
∆ iI = Λ iR sin θ + Λ iI cos θ
Nt / 2
4 ζ = 2 Nt 5 1
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(17)
N t / 4T
d
Nt / T min
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For M-ary Square QAM
0.35 QPSK 8PSK
We now generalize the above results to larger QAM constellations. Consider the M-ary square-QAM constellation, where the I and Q components of a symbol can be viewed as
0.3
Diversity Product
0.25
two independent M -ary PAM symbols. The following four cases of (ΛR, ΛI) and their resultant determinant values as per (14) are considered:
0.15
0.1
Nt
•
2 (ΛR, ΛI) = ± (0, ndmin) Î det 5 = n 2 d min cos(2θ )
• •
2 (ΛR, ΛI) = ± (mdmin, 0) Î det 6 = m 2 d min cos(2θ ) R I (Λ , Λ ) = ± (mdmin, ndmin) Î
Nt
•
2 det 7 = d min ( (m2 − n2 ) cos(2θ ) − 2mn sin(2θ ) ) (ΛR, ΛI) = ± (mdmin, -ndmin) Î 2 det 8 = d min ( (m2 − n2 ) cos(2θ ) + 2mn sin(2θ ) )
Nt
0.05
Nt
0
0
10
20
30
40
50
60
70
80
90
Angle of Constellation Rotation
Figure 1 Optimization of PSK rotation angle for MDCQOSTBC D. Full Diversity for Odd Number of Transmit Antennas? As mentioned in [12], by removing any column of an MDC-QOSTBC constructed by (2), the resultant code is still an MDC-QOSTBC. By using the following two theorems, we shall further prove that, a full-diversity MDC-QOSTBC can still maintain full diversity after any number of columns is removed from its codewords.
where dmin represents the minimum Euclidian distance between the PAM constellation points, m and n are integers such that 1 ≤ m, n ≤ M − 1 , and M is the dimensionality of the QAM constellation. To maximize the smaller value of det5 to det8 for all valid values of m and n, consider first the smallest value of m = n = 1. For this case, det5 to det8 are identical to det1 to det4, hence the optimum θ value for det5 to det8 is the same as that for (16), i.e. θopt = ½ tan-1(½), and the corresponding det5 to det8 values are identically detmin in (17).
Theorem 3 [Theorem 4.3.15 of 17]: Let A be a Hermitian matrix with size n, and r be an integer with 1 ≤ r ≤ n, and let Ar denote any r-by-r principal submatrix of A (obtained by deleting n-r rows and the corresponding columns from A). For each integer k such that 1 ≤ k ≤ r, we have
Next, consider m, n > 1. In this case, it can be shown that with θ = θopt = ½ tan-1(½),
λk ( A) ≤ λk ( A r ) ≤ λk + n − r ( A )
(18)
Nt
where λk(A) represents the kth eigenvalue of matrix A.
Nt
Theorem 4: By removing any column in the codewords of an MDC-QOSTBC achieving full diversity, the resultant MDCQOSTBC can still achieve full diversity.
det 5 = n 2 Nt det min
; det 7 = (m 2 − mn − n 2 ) det min
det 6 = m 2 Nt det min
; det 8 = (m 2 + mn − n 2 ) det min
which are all greater than or equal to detmin for interger m, n > 1. Hence the minimum determinant value for MDCQOSTBC with M-ary square-QAM constellation occurs when m = n = 1, and it is optimized when θ = θopt.
Proof of Theorem 4: The removal of any column from a STBC can be represented as follows: G = GT where G of size T × N t is the new STBC after column removal, and T is a truncated version of an identity matrix after column removal at the same column locations as G (hence T is an rectangular matrix of size N t × N t ). For example, to remove the last column of G (in order to reduce the number of transmit antennas from Nt = 4 to N t = 3), the following T should be used:
For M-ary Rectangular QAM This is similar to the case of square-QAM, besides than the m and n may have different maximum value. However in the above derivation, the maximum value of m and n has been shown to be irrelevant to the worst-case value of the determinant value, since m = n = 1 is the one that dominant the minimum determinant value. So θopt in (16) applied to rectangular-QAM as well. Hence Theorem 2 is proved. ■ C. Optimum CR Angle for PSK
1 0 0 0 T = 0 1 0 0 0 0 1 0
Unlike QAM, the analytical derivation of the optimum CR angle for PSK is unfortunately not as tractable, hence we rely on computer search to find the optimum PSK CR angle for MDC-QOSTBC. As shown in Figure 1, the optimum CR angle is 31.70 for QPSK, and 4.90 for 8PSK. Likewise can be done for the other PSK constellations.
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0.2
T
In such cases, the new STBC will have the following codeword distance matrix:
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A ce = TH (C − E)H (C − E)T = TH A ce T = N t -by-N t submatrix of A ce Since A ce and Ace are both Hermitian matrixes, Theorem 3 applies. As a result, if Ace does not have any zero eigenvalue (since G is assumed to achieve full diversity), A ce will not have any zero eigenvalue too, and G can also achieve full diversity. Therefore, Theorem 4 is proved. ■
corresponding to a reduced diversity product as shown earlier in Table 1. However it should be noted that QO-STBC with CR has a higher decoding complexity, as shown in Table 1 too. 0
10
QO-STBC CIOD MDC-QOSTBC
-1
10
-2
10 FER
V. PERFORMANCE COMPARISON Table 1 shows the comparisons of diversity product and ML decoding complexity (in terms of number of real symbols required for joint detection (JD)) between O-STBC, QOSTBC with CR, QO-STBC with GCLT, MDC-QOSTBC and CIOD for 4-QAM and 8-PSK constellation. It can be seen that, MDC-QOSTBC has a lower decoding complexity than QO-STBC with CR. This decoding complexity reduction is accompanied by a slight reduction in the diversity product, which, interestingly, is also observed for CIOD and QOSTBC with GCLT. Hence it appears that this is a fundamental price to pay to achieve a lower decoding complexity. Although QO-STBC with GCLT can achieve the same decoding complexity as MDC-QOSTBC in this case, MDC-QOSTBC supports any complex constellation, while GCLT is only applicable for square or rectangular QAM constellations [11]. For the case of PSK constellations, MDC-QOSTBC has the same ML decoding complexity as OSTBC, but MDC-QOSTBC also achieves a higher spectral efficiency because it has full code rate while O-STBC has a code rate of ¾.
-4
10
-5
10
STBC
O-STBC
Bps/ Diver. Hz Prod.
1
QO-STBC CR [9]
2
0.3536
4
QO-STBC GCLT [11]
2
0.3344
2
MDCQOSTBC
2
0.3344
2
CIOD [13]
2
0.3344
2
2.25 0.2209 3
0.1674
0.1453
3 Tx Antennas
2
2
5 Tx Antennas
Diversity Product
Po
Diversity Product
Po
CIOD [13]
0.2504
50%
0.2489
50%
ACIOD [14]
0.3564
33%
0.2946
20%
MDCQOSTBC
0.3641
0
0.2996
0
4
Not reported
Table 2 and Figure 3 show that our proposed MDCQOSTBC can achieve a much higher diversity product (hence much lower FER) than CIOD, and slightly higher diversity product (hence slightly lower FER) than ACIOD. As shown in Table 2, our proposed MDC-QOSTBC also has much better Po than the CIOD and ACIOD codes. In fact, MDCQOSTBC achieves the ideal value of Po as it does not require
We show in Figure 2 the frame error rate (FER) performance comparison between QO-STBC, CIOD and MDC-QOSTBC with 4QAM constellation for four transmit antennas. These results show that both CIOD and our proposed MDC-QOSTBC suffer a slight 0.4 dB loss in FER performance compared to the QO-STBC with CR,
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20
Table 2 Comparison of STBCs for three and five tx antennas
N.A. for PSK 3
15
Corresponding comparisons between MDC-QOSTBC, CIOD and ACIOD with 4QAM constellation for the cases of three (CIOD and MDC-QOSTBC for three transmit antennas are obtained by removing the last column from their counterparts for four transmit antennas) and five (CIOD and MDC-QOSTBC for five transmit antennas are obtained by removing the first and last two columns from their counterparts for eight transmit antennas (based on the guideline given in [13]) transmit antennas are presented in Table 2 and Figure 3. Po in Table 2 represents the probability that an antenna transmits the “zero” symbol, hence it should be kept as low as possible so as to achieve a low peak-toaverage power ratio [14]. Codes with lower Po are referred to as better power distribution properties.
Real Bps/ Diver. Real syms Hz Prod. syms for JD for JD
0.4082
10
Figure 2 Simulation performance of STBCs with spectral efficiency of 2 bps/Hz for four transmit antennas
8-PSK
1.5
5
SNR
Table 1 Comparison of STBCs for four tx antennas 4-QAM
-3
10
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any transmit antennas to transmit zero at all, while ACIOD for three transmit antennas requires 1/3 of the transmit antennas to be turned off at any one time. Hence our proposed MDC-QOSTBC is more versatile in supporting both odd and even number of antennas, whereas CIOD only performs well for even number of antennas and ACIOD only supports odd number of antennas.
antennas, and it also has much better power distribution properties than CIOD or ACIOD. VII. REFERENCES [1] Tarokh, V.; Jafarkhani, H.; Calderbank, A.R., “Spacetime block codes from orthogonal designs”, IEEE Trans. on Information Theory, vol:45, July 1999, pp:1456-1467. [2] Ganesan, G.; Stoica, P., “Space-time block codes: a maximum SNR approach”, IEEE Trans. on Information Theory, vol:47, May 2001, pp:1650-1656. [3] K. Lu; S. Fu; X. Xia, “Close Forms Designs of Complex Orthogonal Space-Time Block Codes of Rates (k+1)/(2k) for 2k-1 or 2k Transmit Antennas”, ISIT 2004, pp: 307. [4] Jafarkhani, H., “A quasi-orthogonal space-time block code”, IEEE Trans. on Communications, vol:49, Jan. 2001, pp:1-4. [5] Tirkkonen, O.; Boariu, A.; Hottinen, A., “Minimal nonorthogonality rate 1 space-time block code for 3+ Tx antennas”, ISSSTA 2000, pp:429-432. [6] Papadias, C.B.; Foschini, G.J., “Capacity-approaching space-time codes for systems employing four transmitter antennas”, IEEE Transactions on Information Theory, vol:49, Mar 2003, pp: 726- 733. [7] Tirkkonen, O., “Optimizing Space-Time Block Codes by Constellation Rotations”, FWCW 2001, pp:59-60. [8] Sharma, N.; Papadias, C.B., “Improved quasi-orthogonal codes through constellation rotation”, IEEE Transactions on Communications, vol:51, March 2003, pp: 332- 335. [9] Su, W.; Xia, X., “Quasi-Orthogonal Space-Time Block Codes with Full Diversity”, Globecom 2002, pp: 1098 – 1102. [10] C. Yuen; Y. L. Guan; T. T. Tjhung, “Full-Rate FullDiversity STBC with Constellation Rotation”, VTC 2003Spring, pp: 296 –300. [11] C. Yuen; Y. L. Guan; T. T. Tjhung, “Improved QuasiOrthogonal STBC with Group-Constrained Linear Transformation”, Globecom 2004. [12] C. Yuen; Y. L. Guan; T. T. Tjhung, “Construction of Quasi-Orthogonal STBC with Minimum Decoding Complexity”. ISIT 2004, pp:309. Also accepted for publication in IEEE Trans. on Wireless Communications. [13] Zafar Ali Khan; B.Sundar Rajan, “Space-Time Block Codes from Co-ordinate Interleaved Orthogonal Designs”, ISIT 2002, pp: 275. [14] Md. Zafar Ali Khan; B. Sundar Rajan; Moon Ho Lee, “Rectangular Co-ordinate Interleaved Orthogonal Designs”, Globecomm 2003, pp. 2004-2009. [15] B. Hassibi; B. M. Hochwald, “High-Rate Codes that are Linear in Space and Time”, IEEE Trans. on Information Theory, vol:48, Jul 2002, pp: 1804 –1824. [16] Tarokh, V.; Seshadri, N.; Calderbank, A.R., “Space-time codes for high data rate wireless communication: performance criterion and code construction”, IEEE Trans. on Information Theory, vol:44, Mar. 1998, pp:744-765. [17] Roger A. Horn; Charles R. Johnson, “Matrix Analysis”, Cambridge University Press, 1985.
0
10
-1
FER
10
-2
10
-3
10
CIOD ACIOD MDC-QOSTBC CIOD ACIOD MDC-QOSTBC
-4
10
-5
10
5
3 Tx Ant, Rate 1 3 Tx Ant, Rate 1 3 Tx Ant, Rate 1 5 Tx Ant, Rate 3/4 5 Tx Ant, Rate 3/4 5 Tx Ant, Rate 3/4 10
15
20
SNR
Figure 3 Simulation performance of STBCs with 4-QAM csonstellation for three and five transmit antennas VI. CONCLUSIONS In this paper, the maximum achievable code rate of MDC-QOSTBC is derived analytically. Due to the structure of MDC-QOSTBC, we could also derive a closed-form determinant expression for its codeword distance matrix, and use it to obtain the optimum constellation angle for the MDCQOSTBC to achieve full diversity and maximum coding gain. For square and rectangular QAM constellations, the optimum CR angle is derived analytically to be 13.280. On the other hand, the optimum constellation angle for PSK constellation is found by computer search to be 31.70 for QPSK and 4.90 for 8PSK. In addition, we also show that by removing any column in the codeword of a full-diversity MDC-QOSTBC designed originally for even number of transmit antennas, the resultant code is still an MDC-QOSTBC which can now support odd number of transmit antennas with full diversity. We further show that MDC-QOSTBC can achieve the same low decoding complexity as O-STBC when PSK constellations are used, but at a higher code rate than OSTBC. Compared with QO-STBC with constellation rotation (CR), MDC-QOSTBC can always achieve a lower ML decoding complexity with less than 0.4 dB loss in coding gain. Compared with QO-STBC with group-constrained linear transformation (GCLT), MDC-QOSTBC has the same decoding performance and decoding complexity, but it can support any complex constellation while QO-STBC with GCLT can only support square and rectangular QAM constellation. Finally, MDC-QOSTBC has much better decoding performance than CIOD for odd number of transmit
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