Capacity Approaching Super-Orthogonal Space-Time Block Code

Capacity Approaching Super-Orthogonal Space-Time Block Code Heechoon Lee, Student Member, IEEE, and Michael P. Fitz, Senior Member, IEEE

Abstract This paper introduces a super-orthogonal space-time block code (STBC). The code construction is based on the expansion of orthogonal block code via a unitary matrix transformation. By expanding the orthogonal block code, both the code rate can be increased and the performance improved in terms of Eb /N0 , with a moderate increase of the receiver complexity. Exploiting the partial orthogonality of super-orthogonal STBC, a simplified maximum likelihood decoder will be derived. The constrained capacity and performance are compared with orthogonal and quasi-orthogonal block codes in computer simulation.

Index Terms MIMO systems, space-time block code, unitary expansion, orthogonality, constrained capacity.

Heechoon Lee and Michael P. Fitz are with the Department of Electrical Engineering, University of California, Los Angeles, CA 90095-1594 USA (email: [email protected]; [email protected]). February 23, 2006

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Capacity Approaching Super-Orthogonal Space-Time Block Code I. I NTRODUCTION Multiple antenna radio systems have been one of the principal keys in wireless communication systems, since Telatar [1], Foschini and Gans [2] showed that exploiting the multiple transmit/receive antenna increases the outage capacity. In recent years, a variety of space-time coding schemes have been proposed, including layered architecture [3], [4], orthogonal space-time block codes (STBC) [5], [6], quasi-orthogonal STBCs [7], [8], space-time trellis codes (STTC) [9], [10], linear complex-field code [11], linear dispersion code [12] and threaded algebraic code [13]. Some of space-time codes are designed to harvest diversity mostly [5]–[10], others to harvest rate mostly [3], [4], and the others [11]–[13] to take advantages of diversity and rate together. Diversity-oriented codes utilize the maximum spatial diversity usually with a simple maximum likelihood (ML) decoder, but they have drawbacks in rate limitation. On the other hand, rate-oriented codes achieve the high rate, but they suffer from less spatial diversity. Several algebraic codes succeed to attain both diversity and rate at the sacrifice of increased peak power due to the use of non-standard constellations, but they require a variable complexity decoder such as a sphere decoder [14], [15] for the ML estimates. This paper looks at the design of diversity-oriented STBC with higher rate while maintaining a low and fixed complexity ML decoder. Among all the aforementioned space-time coding schemes, orthogonal STBCs [5], [6] have been of particular attraction due to its simple decoding structure without any loss of performance. Unfortunately, the orthogonal STBCs have the weakness in rate, especially as the number of transmit antenna increases. Tarok et al. showed that the full rate orthogonal code for arbitrary complex constellation does not exist, when more than two transmit antennas are used. Some efforts [7], [8], [16] to design quasi-orthogonal STBCs have been made to overcome the rate limit in orthogonal codes. Jafarkhani’s quasi-orthogonal code [7] doesn’t achieve the full diversity, but quasi-orthogonal codes using constellation rotation [16] and filling the empty threads with a Diophantine number [8] achieve the full diversity. Even though they provide higher data rate, February 23, 2006

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they don’t significantly outperform the conventional orthogonal STBC. A useful idea in the pursuit can be borrowed from noncoherent space-time codes. Hochwald et al. [17], [18] have proposed the unitary & differential unitary STBCs, and Hughes [19] also have proposed general framework of differential unitary STBCs, called space-time group codes. The cardinality of a STBC can be increased by applying a unitary transformation and this expanded set of code is denoted a super-orthogonal block code. The impact of this augmented code is that not only can higher rate codes be designed, but also in many situations a better performance is obtained in terms of Eb /N0 , while only adding a moderate amount of increase of receiver complexity. The idea of super-orthogonal code was already published in the context of super-orthogonal trellis code [20]–[22], but the literature has not exploited that the super-orthogonal code has desirable characteristics as a block code itself. In this paper, a general framework of the block code design and the properties of the super-orthogonal STBC are presented. The rest of the paper is organized as follows. Section II reviews the basic backgrounds to help the understanding of this paper and introduce the basic idea of super-orthogonal STBC. Section III presents the construction of super-orthogonal STBC with the specific design especially for the case of 4 transmit antennas. ML decoding algorithm is described in Section IV. Capacity with finite constellations and simulation results are followed in Section V and VI, respectively. Finally, concluding remarks are provided in Section VII. II. F UNDAMENTALS To begin, notation used throughout the paper is defined. Bold symbol denotes a matrix, upperarrowed symbol denotes a vector, and plain symbol denotes a scalar. Superscript H, T , and ∗ represent a complex conjugate transpose, transpose, and complex conjugate, respectively. A. Unitary Expansion of Orthogonal Block Code The multiple-input multiple-output (MIMO) system with Lt transmit and Lr receive antennas is considered. The space-time block has the time duration of Nf . The channel is assumed to be frequency flat Rayleigh fading and channel coefficients are constant during the block, i.e. quasi-static fading is assumed. The received signal is modeled as Y = HD(~s) + N DRAFT

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where H represents an Lr × Lt channel matrix; D(~s) is an Lt × Nf transmitted space-time matrix made up of modulated symbol vector ~s; N is an Lr × Nf additive white Gaussian noise with variance N0 /2 per dimension. The receiver is assumed to have the perfect knowledge of channel state information (CSI). In order to increase the cardinality of orthogonal block code without sacrificing the benefits of simple decoding, a conventional orthogonal block code is used as a base matrix. The orthogonal space-time matrices for Lt = 2 and 4 are shown in (2) and (3), respectively.   ∗ s1 −s2  D(s1 , s2 ) =  s2 s∗1   s1 −s∗2 −s∗3 0  r  ∗ ∗  s s 0 s 4  2 1 3  D(s1 , s2 , s3 ) =   ∗  3 s3 0 s1 −s∗2    0 −s3 s2 s1

(2)

(3)

Then, the expanded set of orthogonal space-time matrix, i.e., super-orthogonal space-time matrix, is formulated by applying unitary transformations. The received signal of super-orthogonal block code is given by Y = HD(~s)W (s+ ) + N (4) = HD s (~ss ) + N where W (s+ ) denotes an Nf × Nf unitary matrix corresponding to an additional symbol s+ and ³ ´ D s (~ss = {~s, s+ }) , D(~s)W (s+ ) is a new super-orthogonal space-time matrix. One more bit per block can be transmitted compared to orthogonal block code with two W matrices, two more bits with four W matrices, and so on. Denote Ns the number of extra bits transmitted. The design criteria for unitary matrix W are given in Section II-B and the detailed design is presented in Section III.

B. Code Construction Criteria The pairwise error probability (PWEP) is commonly used for the performance criterion of the space-time codes [9], [10]. Assuming the ML decoder with perfect CSI in quasi-static fading condition, the PWEP that the optimum decoder makes an erroneous decision in favor of a given February 23, 2006

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~ over the transmitted one (D s (~ codeword (D s (β)) α)) is asymptotically bounded by [23]   ~ −1 2Lr ∆H (~ α, β)   ~ Lr ∆H (~ α, β) − 1 ~ ≤³ P (~ α, β) ´Lr ~ ∆H (~ α,β) ~ SNR ∆p (~ α, β)

(5)

~ is the rank of a where SNR is a signal-to-noise ratio (SNR) per receive antenna and ∆H (~ α, β) ~ defined as signal matrix C s (~ α, β) ~ = (D s (~ ~ H (D s (~ ~ C s (~ α, β) α) − D s (β)) α) − D s (β)) (6) ³ ´ ~ ~ , Q∆H (~α,β) ~ is the product of the non-zero eigenvalues of the signal and ∆p (~ α, β) λ (~ α , β) i i=1 ~ ~ and ∆p (~ ~ are called the effective Hamming distance and the matrix C s (~ α, β). ∆H (~ α, β) α, β) product measure, respectively. They are key parameters to determine the performance of the ~ and ∆p (~ ~ over all possible space-time code. Note that the minimum values of ∆H (~ α, β) α, β) pairwise errors correspond to the transmit diversity and the coding gain, respectively. When ∆H,min of a space-time code is equal to Lt , it is said that the code achieves the full transmit diversity. On the other hand, the PWEP can be characterized especially for a moderate or large number of receive antennas. By utilizing the central limit theorem for Gaussian random variables, the PWEP can also be represented as [24]–[26] n o 1 ~ ~ P (~ α, β) ≤ exp −Lr SNR∆E (~ α, β) (7) 4 ³ P ´ ~ ~ , ∆H (~α,β) ~ is the sum of non-zero eigenvalues of the signal matrix where ∆E (~ α, β) λi (~ α, β) i=1 C s , which is in fact the Euclidean squared distance between the two signal matrices D s (~ α) ~ The minimum Euclidean distance ∆E,min (~ ~ over all possible pairwise errors and D s (β). α, β) can be used as an auxiliary design criterion for constructing a good space-time code. In code construction in Section III, all of abovementioned performance measures ∆H,min , ∆p,min , and ∆E,min are taken into consideration. The main goal is to design the set of unitary matrices W which maximizes the ∆H,min , ∆p,min and ∆E,min overall. III. C ODE C ONSTRUCTION In design of the high rate super-orthogonal STBC, the main work is to expand the cardinality of the orthogonal space-time signal set. The larger the signal set is, the higher the rate that DRAFT

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can be achieved. However, there are some rules to take into account in the design of matrix W . The first one is to maximize the minimum effective Hamming distance ∆H,min , minimum product measure ∆p,min and minimum Euclidean distance ∆E,min to guarantee that the increased rate doesn’t harm the performance of super-orthogonal STBC. The second one is that W does not expand the resulting modulation alphabet in order not to increase the peak-to-average power ratio of the transmitted signal and to be backward compatible to standard modulations. These two rules restrict that only row-wise/column-wise permutations and phase rotation that maintain the constellation are allowed in W . The designs of 2 and 4 transmit antenna super-orthogonal STBC are discussed in the following. The design methodology, however, can be applied to any number of transmit antennas.

A. Design for Lt = 2 Since the orthogonal space-time matrix for 2 transmit antennas is a full matrix, only phase rotations are considered for the expansion. Then, the unitary matrix can be the form of   1 0 . W = 0 ejθ Constellation preserving θ can be

mπ 2

(8)

with m = 0, 1, 2, 3 for general QAM and PSK constella-

tions. Using these set of unitary matrices, additional 2 more bits per block can be transmitted. Recall the rate-diversity tradeoff [10], [27] ¶ µ d + 1 log2 |Ωs |, R ≤ Lt − Lr

(9)

from the Singleton bound where R is the achievable transmission rate in bits per channel use (bits/s/Hz); d is the diversity gain; Ωs is the output constellation from each transmit antenna. The use of a symmetrical space-time constellation is assumed. The maximum rate in the case of full diversity (i.e., d = Lt Lr ) is upper-bounded by R ≤ log2 |Ωs |.

(10)

Since Alamouti space-time matrix already attains the highest rate with full diversity, the superorthogonal STBC in 2 transmit antennas cannot achieve the full diversity. Improvements, however, can be obtained when the super-orthogonal STBC is combined with other coding [20]–[22]. February 23, 2006

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B. Design for Lt = 4 Unlike Lt = 2, both permutation and phase rotation are considered in the design of W , since the base orthogonal space-time matrix is not symmetrical to permutation. For the convenience of the design, the following forms of unitary matrices are defined, which consist of the phase rotations and all the independent1 column-wise permutations:  a1 0 0  0 a 0 2  U 0 (a1 , a2 , a3 , a4 ) =   0 0 a3  0 0 0  0 0 0  0 0 a 2  U 1 (a1 , a2 , a3 , a4 ) =   0 a3 0  a4 0 0  0 a1 0  a 0 0  2 U 2 (a1 , a2 , a3 , a4 ) =  0 0 0  0 0 a4  0 0 a1  0 0 0  U 3 (a1 , a2 , a3 , a4 ) =  a 3 0 0  0 a4 0

0



 0   0  a4  a1  0   0  0  0  0   a3   0  0  a2    0  0

(11)

(12)

(13)

(14)

where a1 , a2 , a3 and a4 have the form of ejθ , and phase rotation factor θ depends on the constellation shape. If the squared modulation such as QPSK, 16QAM and 64QAM is used, θ can be

mπ , 2

and if MPSK is used, θ can be

2mπ M

to keep the constellation after the rotation the

same, where m is an arbitrary integer. W can be any of these matrices and specific designs are followed. After some numerical searches to optimize ∆H,min , ∆p,min and ∆E,min altogether, a unitary matrix set for the squared constellation is formulated in Table I. W (i) with i = 0 to 2Ns − 1 1

Unitary matrix set achieving the full rank.

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TABLE I U NITARY M ATRIX S ET FOR S QUARED C ONSTELLATION IN 4 T RANSMIT A NTENNAS

W (0)

= U 0 (1, 1, 1, j)

W (1)

= U 1 (1, 1, j, 1)

W (2)

= U 2 (1, j, 1, 1)

W (3)

= U 3 (j, 1, 1, 1)

W (4)

= U 0 (1, 1, 1, j) · U 4

W (5)

= U 1 (1, 1, j, 1) · U 4

W (6)

= U 2 (1, j, 1, 1) · U 4

W (7)

= U 3 (j, 1, 1, 1) · U 4

W (8)

= U 0 (1, 1, 1, j) · U 8

W (9)

= U 1 (1, 1, j, 1) · U 8

W (10)

= U 2 (1, j, 1, 1) · U 8

W (11)

= U 3 (j, 1, 1, 1) · U 8

W (12)

= U 0 (1, 1, 1, j) · U 12

W (13)

= U 1 (1, 1, j, 1) · U 12

W (14)

= U 2 (1, j, 1, 1) · U 12

W (15)

= U 3 (j, 1, 1, 1) · U 12

where

U 4 = Diag(1, 1, −1, −1) U 8 = Diag(1, −1, −1, 1) U 12 = Diag(1, −1, 1, −1)

would be used to transmit Ns more additional bits per block. Therefore, up to 4 more bits per block can be transmitted using unitary matrices in Table I. The resulting ∆H,min , ∆p,min and ∆E,min are illustrated in Table II for the QPSK. Here, the percentage inside parenthesis indicates the percentage of the pairwise errors with the minimum ∆H,min , ∆p,min and ∆E,min over all possible pairwise errors. Note that the super-orthogonal STBCs transmitting 1 or 2 additional bits achieve full diversity, but the codes transmitting 3 or 4 additional bits don’t achieve it. On the other hand, as the number of additional bits increases, the Euclidean distance gets relatively larger, which results in small percentage of pairwise errors with the minimum Euclidean distance ∆E,min . As a result of (7), better performance of the super-orthogonal STBC is expected with a large number of receive antennas despite the non-full diversity characteristics. The effect of this large Euclidean distance will be confirmed via block error rate (BLER) characterization in Section VI. The highest rate with full diversity in 4 transmit antenna super-orthogonal STBC can be investigated through the rate-diversity tradeoff as well. Due to an asymmetrical structure of the February 23, 2006

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TABLE II P ERFORMANCE M EASURES OF S UPER -O RTHOGONAL B LOCK C ODE U SING QPSK IN 4 T RANSMIT A NTENNAS

Additional Bits

∆H,min

∆p,min

∆E,min

0

4 (100%)

16 (9.52%)

8 (9.52%)

1

4 (100%)

4 (5.51%)

8 (4.72%)

2

4 (100%)

4 (8.24%)

8 (2.35%)

3

2 (0.78%)

4 (8.22%)

8 (1.17%)

4

2 (1.17%)

4 (8.21%)

8 (0.59%)

base space-time matrix (3), the rate-diversity inequality is different from (9) in symmetrical space-time constellation case. Provided that the unitary matrices consist of only permutation and phase rotation, the maximum cardinality of super-orthogonal STBC with diversity d is (4|Ωs |3 )(Lt −d/Lr +1) where s1 , s2 , and s3 assume the use of constellation Ωs . A factor of 4 indicates the location of zero from permutation, i.e., the codeword from each transmit antenna can have one of the following: (ωs , ωs , ωs , 0), (ωs , ωs , , 0, ωs ), (ωs , 0, ωs , ωs ), (0, ωs , ωs , ωs ) where ωs ∈ Ωs . As a result, the rate-diversity inequality for 4 transmit antennas is given by µ ¶µ ¶ d 3 log2 |Ωs | + 2 R ≤ Lt − +1 . (15) Lr Nf Therefore, the maximum rate with full diversity for 4 transmit antennas is constrained by R≤

3 log2 |Ωs | + 2 . 4

(16)

Equation (16) confirms that the presented super-orthogonal STBC with 4 transmit antennas achieves the highest rate with full diversity according to the rate-diversity tradeoff. Similarly, the unitary matrix set for 8PSK is obtained in Table III. Same as the squared constellation case, the super-orthogonal STBCs transmitting 1 or 2 additional bits achieve full diversity, but the codes transmitting 3 or 4 additional bits do not achieve full diversity. It can be easily seen that Jafarkhani’s parameterized class of block codes [22] are only a subset of our super-orthogonal STBC, which is based on the unitary expansion. Jafarkhani’s code uses only U 0 type of unitary transformation. In 2 transmit antenna system, there is no difference DRAFT

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TABLE III U NITARY M ATRIX S ET FOR C IRCLED C ONSTELLATION IN 4 T RANSMIT A NTENNAS (8PSK)

W (0)

= U 0 (1, 1, 1, 1)

W (1)

= U 1 (1, 1, ej3π/4 , ej3π/4 )

W (2)

= U 2 (1, ej3π/4 , 1, −ejπ/4 )

W (3)

= U 3 (1, ej3π/4 , ej3π/4 , 1)

W (4)

= U 0 (1, 1, 1, 1) · U 4

W (5)

= U 1 (1, 1, ej3π/4 , ej3π/4 ) · U 4

W (6)

= U 2 (1, ej3π/4 , 1, −ejπ/4 ) · U 4

W (7)

= U 3 (1, ej3π/4 , ej3π/4 , 1) · U 4

W (8)

= U 0 (1, 1, 1, 1) · U 8

W (9)

= U 1 (1, 1, ej3π/4 , ej3π/4 ) · U 8

W (10)

= U 2 (1, ej3π/4 , 1, −ejπ/4 ) · U 8

W (11)

= U 3 (1, ej3π/4 , ej3π/4 , 1) · U 8

W (12)

= U 0 (1, 1, 1, 1) · U 12

W (13)

= U 1 (1, 1, ej3π/4 , ej3π/4 ) · U 12

W (14)

= U 2 (1, ej3π/4 , 1, −ejπ/4 ) · U 12

W (15)

= U 3 (1, ej3π/4 , ej3π/4 , 1) · U 12

where

U 4 = Diag(1, 1, −1, −1) U 8 = Diag(1, −1, −1, 1) U 12 = Diag(1, −1, 1, −1)

between two designs, since the 2 transmit orthogonal space-time matrix in (2) is permutationinvariant when the use of symmetric complex constellation is assumed. However, the 4 transmit orthogonal space-time matrix in (3) is not symmetric about permutation any more due to zeros in the space-time constellation. Therefore, our design provides more degrees of freedom, i.e., higher rate while providing better performance.

IV. M AXIMUM L IKELIHOOD D ECODER The standard ML decoding of the super-orthogonal STBC can be accomplished by exhaustive search over all possible codewords. Then, the complexity of standard ML decoding process is exponential with the data rate. However, the ML decoding of super-orthogonal code can be greatly simplified exploiting the orthogonality given a fixed unitary matrix W . By exploiting this structure, the decoding complexity becomes O(2Ns No ) where No is the decoding complexity of the orthogonal STBC. Assuming perfect CSI is available, the simplified ML decoding algorithm February 23, 2006

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is derived as ~sˆs = arg min kY − HD s (~ss )k2F ~ss

= arg min kY − HD(~s)W (s+ )k2F ~s,s+

H

= arg min kY W (s+ ) − ~s,s+

(17)

HD(~s)k2F

= arg min kZ(s+ ) − HD(~s)k2F ~s,s+

where k · k2F is the squared Frobenius norm and Z(s+ ) , Y W (s+ )H . From (17), the decoding algorithm turns into the orthogonal block code’s one [6] for each hypothesis on s+ . Therefore, the decoder performs the exhaustive search on s+ , where each ML metric/decision can be computed with linear complexity. At the end of the search, the final ML decision can be made over all temporary decisions. Finally, the simplified ML decoding algorithm can be rewritten as ~sˆs = arg max T (~s|s+ ) ~s,s+

(18)

where T (~s|s+ ) represents the ML metric of the orthogonal block code given s+ . Thus, overall complexity of the decoder is approximately proportional to the cardinality of additional symbol s+ (2N s ) multiplied by the decoding complexity of the orthogonal STBC. To demonstrate the simplified ML decoding algorithm, let us consider the decoding of 4 transmit antenna system transmitting Ns additional bits per block (i.e., s+ = 0, 1, · · · , 2Ns − 1). With each s+ , the decision metrics of s1 , s2 and s3 are obtained by s˜1 = ~hH z1 + ~z2H~h2 + ~z3H~h3 + ~hH z4 1 ~ 4 ~ s˜2 = ~hH z1 − ~z2H~h1 + ~hH z3 − ~z4H~h3 2 ~ 4 ~

(19)

s˜3 = ~hH z1 − ~hH z2 − ~z3H~h1 + ~z4H~h2 3 ~ 4 ~ where Z(s+ ) = [~z1 ~z2 ~z3 ~z4 ] and H = [~h1 ~h2 ~h3 ~h4 ], i.e., ~zi and ~hi are column vectors comprising Z and H, respectively. Based on the decision metrics (˜ s1 , s˜2 , and s˜3 ) in (19), temporary decisions ¡ ¢ (ˆ s1 , sˆ2 , and sˆ3 ) and ML metric T (~sˆ|s+ ) associated with s+ can be obtained. The temporary ML metric (18) is also computed as T (~sˆ|s+ ) = T (ˆ s1 |s+ ) + T (ˆ s2 |s+ ) + T (ˆ s3 |s+ ) (20) q where T (ˆ si |s+ ) = Re {˜ s∗i sˆi } − 21 43 kHk2F |ˆ si |2 for i = 1, 2, and 3. After temporary decisions and ML metrics over all s+ (= 0, 1, · · · , 2Ns − 1) are computed, a final decision is made in which the ML metric (20) is maximized. DRAFT

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V. C ONSTRAINED C APACITY Information theoretic capacity gives good insight into the performance of systems using constrained modulations. MIMO quasi-static fading channel is the most commonly used model, when the performance of STBC is evaluated. In this situation, average and outage capacity are usually used as capacity measure. It is known that i.i.d. Gaussian-distributed inputs are optimal [1], [2] to maximize the capacity, but Gaussian inputs are not feasible for the practical implementation. Instead, a constrained modulation such as finite constellation (e.g., QPSK, 16QAM, and 64QAM etc.) and/or space-time code is employed. In the case of some STBCs, capacity was computed still assuming Gaussian inputs. The capacity of Alamouti STBC was shown to achieve the open-loop capacity of a 2 × 1 system despite its decoupled nature [28]. The capacity of quasi-orthogonal STBC was computed in [29], [30] and it was shown that the quasi-orthogonal STBC attains a significant fraction of the open-loop capacity of a 4 × 1 system [29]. However, the capacity assuming the Gaussian inputs doesn’t reflect the performance very well in realistic situation. In this paper, the capacity will be computed in the case of constrained constellation rather than Gaussian inputs. Consider X and Y to be transmit and receive vectors or matrices, respectively. Assuming the channel model Y = HX + N ,

(21)

and equiprobable transmit signals, the average capacity is given by [See Appendix] "

·

h

P

C = log |ΩX | − EH EX EN log

0 i 0 X ∈ΩX fY |X (HX + N |X ) fN (N ) 0

¸# (22)

where ΩX is the cardinality of transmit signal X; EH , EN , and EX denote the expectation over H, N , and X, respectively; f is the likelihood function. Therefore, the average capacity (22) can be computed via Monte Carlo simulation with many realizations over channel H, noise N , and transmit signal X. This computation can be applied to any multiple antenna communication systems, but it would be numerically expensive. Thanks to the partial orthogonality of super-orthogonal STBC, the capacity computation can February 23, 2006

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be simplified. The likelihood computation in (22) is decomposed as Xµ

¶Nf Lr · 1 1 √ fY |X 0 (Y |X ) = exp − kY 2No 2πN 0 0 ~ss X ∈ΩX · X X µ 1 ¶Nf Lr 1 √ = exp − 2No 2πN0 s+ ~s | X

0

¸ −

HD s (~ss )k2F

¸ ° °2 H °Y W (s+ ) − HD(~s)° . F {z } ,L(s+ ,~s)

(23) For a given s+ , L(s+ , ~s) term in (23) can be further decomposed into individual symbols in ~s. For instance, for 4 transmit antenna super-orthogonal STBC, L(s+ , ~s) is replaced by L(s+ , ~s) = L1 (s+ , s1 ) · L2 (s+ , s2 ) · L3 (s+ , s3 ).

(24)

Therefore, O(2Ns |Ωs |3 ) likelihood computations can be reduced to O(2Ns · 3|Ωs |). The average capacity of super-orthogonal STBCs in 2 × 2 and 4 × 4 systems are computed in Fig. 1 as a function of SNR per receive antenna. The capacity of spatial multiplexing is also shown for the comparison. In Fig. 1, ‘QPSK+1’ denotes the super-orthogonal STBC employing QPSK with 1 additional bit. It is seen that the capacity of orthogonal STBC can be systematically increased toward that of spatial multiplexing by using the super-orthogonal STBC. In some cases, the capacity of super-orthogonal STBC outperforms that of spatial multiplexing. This means that a certain capacity can be easily achieved with less complexity. When the capacities of super-orthogonal codes using different constellations with same rate are compared (e.g., ‘QPSK+4’ and ‘8PSK+1’ of 2.5 bits/s/Hz in Fig. 1(b)), the codes using smaller constellation always outperform bigger constellation. Therefore, as a rule it is better to use more expanded space-time matrices than to increase the constellation. The capacities of super-orthogonal and quasi-orthogonal [8] STBCs using 16QAM (4.0 bits/s/Hz) are compared in Fig. 2 with different number of receive antennas. Here, the capacity of quasiorthogonal STBC is computed using the partial orthogonality in similar way as the one for super-orthogonal STBC. It is easily seen that the super-orthogonal STBC outperforms the quasiorthogonal STBC. When increasing the number of receive antennas, the capacity gap between two STBCs is getting bigger thanks to the design of super-orthogonal STBC optimized with the Euclidean distance. DRAFT

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8

Alamouti: QPSK Super: QPSK+1 Super: QPSK+2 Alamouti: 8PSK Super: 8PSK+1 Super: 8PSK+2 Alamouti: 16QAM Super: 16QAM+1 Super: 16QAM+2 Alamouti: 64QAM Super: 64QAM+1 Super: 64QAM+2 Spat Mult: QPSK Spat Mult: 8PSK Spat Mult: 16QAM

Average Capacity (bits/s/Hz)

7

6

5

4

3

2

1

0 −5

0

5

10

15

20

25

SNR (dB) (a) 2 × 2 system 5

4.5

Average Capacity (bits/s/Hz)

4

3.5

3

2.5

2

Orthogonal: QPSK Super: QPSK+1 Super: QPSK+2 Super: QPSK+3 Super: QPSK+4 Orthogonal: 8PSK Super: 8PSK+1 Super: 8PSK+2 Super: 8PSK+3 Super: 8PSK+4 Orthogonal: 16QAM Super: 16QAM+1 Super: 16QAM+2 Super: 16QAM+3 Super: 16QAM+4 Orthogonal: 64QAM Super: 64QAM+1 Super: 64QAM+2 Super: 64QAM+3 Super: 64QAM+4 Spat Mult: BPSK Spat Mult: QPSK

1.5

1

0.5

0 −10

−5

0

5

10

15

SNR (dB) (b) 4 × 4 system Fig. 1.

Constrained capacity of super-orthogonal STBCs.

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4

Average Capacity (bits/s/Hz)

3.5

3

2.5

2

Super: Lr=1 Super: Lr=2 Super: Lr=3 Super: L =4 r Quasi: L =1 r Quasi: Lr=2 Quasi: L =3 r Quasi: Lr=4

1.5

1

0.5

0 −10

−5

0

5

10

15

20

SNR (dB)

Fig. 2. Capacity comparison between super-orthogonal and quasi-orthogonal STBCs using 16QAM (4.0 bits/s/Hz) with different number of receive antennas.

VI. P ERFORMANCE The performance of the proposed super-orthogonal STBC is provided via computer simulation. The use of a 4 × 4 MIMO system is considered with perfect CSI in the receiver. In this paper, the performance is evaluated and analyzed in terms of block error rate (BLER) as a function of Eb /N0 per receive antenna, but analogous results are observed in the bit error rate [31]. The BLER curves for QPSK, 8PSK, 16QAM and 64QAM are presented in Fig. 3 with respect to the different rate. In each constellation, the lowest rate code corresponds to the conventional orthogonal block code. It can be seen that the super-orthogonal block codes outperform the orthogonal codes in every case. It is remarkable that better performance can be achieved, while increasing the rate. Note that the highest and second highest rate code in each constellation are not full diversity code, as mentioned in Section III. Even if they are not full diversity code, their performance still show the full rank characteristics. It mainly results from the significantly increased Euclidean distance and partially from relatively low percentage of pairwise errors with minimum effective Hamming distance ∆H,min (< Lt ), as can be seen in Table II. DRAFT

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15

0

10

−1

10

−2

Block Error Rate

10

−3

10

−4

10

Orthogonal: QPSK (1.5 bits/s/Hz) Super: QPSK+1 (1.75 bits/s/Hz) Super: QPSK+2 (2.0 bits/s/Hz) Super: QPSK+3 (2.25 bits/s/Hz) Super: QPSK+4 (2.5 bits/s/Hz) Orthogonal: 8PSK (2.25 bits/s/Hz) Super: 8PSK+1 (2.5 bits/s/Hz) Super: 8PSK+2 (2.75 bits/s/Hz) Super: 8PSK+3 (3.0 bits/s/Hz) Super: 8PSK+4 (3.25 bits/s/Hz)

−5

10

−6

10

−7

10

−4

−2

0

2

4

6

8

10

Eb/N0 (dB) (a) QPSK and 8PSK 0

10

−1

10

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Block Error Rate

10

−3

10

−4

10

−5

10

−6

10

−4

Orthogonal: 16QAM (3.0 bits/s/Hz) Super: 16QAM+1 (3.25 bits/s/Hz) Super: 16QAM+2 (3.5 bits/s/Hz) Super: 16QAM+3 (3.75 bits/s/Hz) Super: 16QAM+4 (4.0 bits/s/Hz) Orthogonal: 64QAM (4.5 bits/s/Hz) Super: 64QAM+1 (4.75 bits/s/Hz) Super: 64QAM+2 (5.0 bits/s/Hz) Super: 64QAM+3 (5.25 bits/s/Hz) Super: 64QAM+4 (5.5 bits/s/Hz) −2

0

2

4

6

8

10

12

14

E /N (dB) b

0

(b) 16QAM and 64QAM Fig. 3.

Block error rate of super-orthogonal STBC in 4 × 4 system.

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10

Alamouti: QPSK (2.0 bits/s/Hz) Super: QPSK+1 (2.5 bits/s/Hz) Super: QPSK+2 (3.0 bits/s/Hz) −1

10

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Block Error Rate

10

−3

10

−4

10

−5

10

−6

10

−5

0

5

10

15

20

25

Eb/N0 (dB)

Fig. 4.

Block error rate of super-orthogonal code (QPSK) in 2 × 2 system.

The number of receive antennas impacts the performance characteristics. When a small number of receive antennas are used, it has been observed that the performance cannot be improved while ~ dominates only in a large number increasing the rate, since the Euclidean distance ∆E (~ α, β) of receive antennas. Nonetheless, the rate can still be increased in a small number of receive antenna case. When the performance of super-orthogonal codes using different constellations with same rate are compared, again the codes using smaller constellation always outperform bigger constellation, which confirms the result in capacity. The performance of the super-orthogonal STBC in 2×2 system is also shown in Fig. 4. Unlike the 4 transmit antenna code, the performance is getting degraded, when the rate is increased. Since the 2 transmit antenna code doesn’t have advantage for the permutation in unitary transformation, it results in the worse distribution of the minimum Hamming distance, product measure and Euclidean distance. In this case, even if the number of receive antenna is increased, better performance cannot be achieved with increased rate. This reveals that the super-orthogonal STBC has more benefits, when more than 2 transmit antennas are employed. DRAFT

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17 0

10

−1

10

−2

Block Error Rate

10

−3

10

−4

10

−5

10

−6

10

−7

10

−4

Orthogonal: QPSK (1.5 bits/s/Hz) Super: QPSK+2 (2.0 bits/s/Hz) Quasi: QPSK+2 (2.0 bits/s/Hz) Orthogonal: 16QAM (3.0 bits/s/Hz) Super: 16QAM+4 (4.0 bits/s/Hz) Quasi: 16QAM+4 (4.0 bits/s/Hz) −2

0

2

4

6

8

10

Eb/N0 (dB)

Fig. 5.

Performance comparison in 4 × 4 system.

Lastly, the performance of the super-orthogonal STBC is compared with the quasi-orthogonal STBC [8] in Fig. 5. The BLER of the orthogonal STBC is also shown for the reference. It can be seen that the super-orthogonal STBC outperforms the quasi-orthogonal STBC by about 1.3 dB in terms of Eb /N0 at the BLER of 10−4 . Note that the super-orthogonal and quasi-orthogonal codes have approximately same decoding complexity. VII. C ONCLUSIONS In this paper, a general framework for the design of the super-orthogonal block code was presented. The code construction is based on the unitary transformation of the orthogonal block code. The code is designed according to the general PWEP criterion. The simplified ML decoder was derived exploiting the partial-orthogonality. The presented super-orthogonal block codes achieve both higher rate and performance improvement together at the expense of slight complexity increase in the receiver. It has been shown that the capacity of the code approaches to that of spatial multiplexing for small number of transmit antennas. The code outperforms the existing quasi-orthogonal block code [8] of same rate and complexity by about 1.3 dB in February 23, 2006

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Eb /N0 . It has been also shown that the super-orthogonal code has more superiority in more than 2 transmit antenna systems. A PPENDIX C ONSTRAINED C APACITY OF M ULTIPLE A NTENNA S YSTEMS The average capacity for multiple transmit antenna system is represented by the mutual information: C = EH [I(X; Y |H)] = EH [H(X) − H(X|Y , H)]

(25)

= H(X) − EH [H(X|Y , H)] . where H(·) denotes the entropy function. Assuming the equiprobable transmit signals, and given H = h, the conditional entropy of X given Y can be manipulated as · i¸ h 1 H(X|Y , H = h) = EX EY |X log p(X|Y ) ¸ ·Z fY (y) = EX fY |X (y|X) log dy fY |X (y|X)pX (X) Y P ¸ ·Z 0 0 0 0 X 0 ∈ΩX fY |X (y|X )pX (X ) = EX fY |X (y|X) log dy fY |X (y|X)pX (X) Y P ·Z ¸ 0 0 X 0 ∈ΩX fY |X (hX + n|X ) = EX fN (n) log dn fN (n) N P · 0 i¸ h 0 X 0 ∈ΩX fY |X (hX + N |X ) = EX EN log , fN (N )

(26)

where Bayes’ rule, pX 0 (X 0 ) = pX (X), and fY |X (y|X) = fY |X (hX + n|X) = fN (n) are used for the derivation. From (25) and (26), the average capacity with constrained constellation is derived as

"

·

h

C = log |ΩX | − EH EX EN log

P X 0 ∈ΩX

¸# fY |X 0 (HX + N |X 0 ) i . fN (N )

(27)

Similar derivation exists in [32]. R EFERENCES [1] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” AT&T Bell Labs Internal Tech. Memo., June 1995. [2] G. J. Foschini and W. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Communication, vol. 6, pp. 314–335, Mar. 1998. DRAFT

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[23] M. P. Fitz, J. Grimm, and S. Siwamogsatham, “A new view of performance analysis techniques in correlated Rayleigh fading,” in Proc. IEEE Wireless Communications and Networking Conf., vol. 1, Sep. 1999, pp. 139–144. [24] Z. Chen, J. Yuan, and B. Vucetic, “An improved space-time trellis coded modulation scheme on slow Rayleigh fading cannels,” in Proc. IEEE Int. Conf. Commun., vol. 4, June 2001, pp. 1110–1116. [25] E. Biglieri, G. Taricco, and A. Tulino, “Performance of space-time codes for a large number of antennas,” IEEE Trans. Info. Theory, vol. 48, no. 7, pp. 1794–1803, July 2002. [26] D. Aktas, H. E. Gamal, and M. P. Fitz, “On the design and maximum-likelihood decoding of spacetime trellis codes,” IEEE Trans. Commun., vol. 51, no. 6, pp. 854–859, June 2003. [27] H. E. Gamal and A. R. Hammons, “Algebraic space-time codes for block fading channels,” in Proc. IEEE Int. Symp. Inform. Theory, Washington, D. C. USA, June 2001, p. 152. [28] C. Papadias, “On the spectral efficiency of space-time spreading schemes for multiple antenna cdma systems,” in Proc. Asilomar Conf. Signals, Systems, and Computers, Pacific Grove, CA USA, Oct. 1999, pp. 639–643. [29] C. B. Papadias and G. J. Foschini, “Capacity-approaching space-time codes for systems employing four transmitter antennas,” IEEE Trans. Info. Theory, vol. 49, no. 3, pp. 726–733, Mar. 2003. [30] A. Sezgin and T. J. Oectering, “On the outage probability of quasi-orthogonal space-time codes,” in Proc. IEEE Info. Theory Workshop, San Antonio, TX USA, Oct. 2004, pp. 381–386. [31] H. Lee, M. Siti, W. Zhu, and M. P. Fitz, “Super-orthogonal space-time block code using a unitary expansion,” in Proc. IEEE Vehicular Tech. Conf., vol. 4, Los Angeles, CA USA, Sep. 2004, pp. 2513–2517. [32] E. Baccarelli, “Evaluation of the reliable data rates supported by multiple-antenna coded wireless links for QAM transmissions,” IEEE J. Select. Areas Commun., vol. 19, no. 2, pp. 295–304, Feb. 2001.

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