Super-Orthogonal Space-Time Block Code Using a ... - CiteSeerX

Super-Orthogonal Space-Time Block Code Using a Unitary Expansion Heechoon Lee∗ , Massimiliano Siti† , Weijun Zhu∗ , and Michael P. Fitz∗ ∗ Department

of Electrical Engineering University of California Los Angeles, Los Angeles, California 90095-1594 Email: {coet, zhuw and fitz}@ee.ucla.edu † STMicroelectronics - Advanced System Technologies Via C. Olivetti, 2 - 20041 Agrate Brianza (Milan) - Italy Email: [email protected]

Abstract— This paper introduces a super-orthogonal spacetime block coding scheme. 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. The performance is compared with orthogonal and quasi-orthogonal block codes in computer simulation.

I. I NTRODUCTION Intensive works of multiple antenna schemes have been performed in wireless environments to increase the data rate toward theoretical limits, after Telatar [1], Foschini and Gans [2] showed that exploiting the multiple transmit/receive antenna increase the outage capacity. In recent years, a number of space-time coding schemes, including space-time block [3], [4] and trellis [5], [6] coding, layered architecture [7], and threaded algebraic space-time [8], have been proposed. Among the proposed space-time coding schemes, orthogonal spacetime code [4] has been of particular attraction due to its simple decoding structure without any loss of performance. Unfortunately, the orthogonal space-time code has the weakness in rate, 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 [9]–[11] to design quasi-orthogonal space-time block codes have been made to overcome the rate limit in orthogonal codes. Jafarkhani’s quasi-orthogonal code [9] doesn’t achieve the full diversity, but quasi-orthogonal codes using constellation rotation [10] and filling the empty threads with a Diophantine number [11] achieve the full diversity. Even though they provide higher data rate, they don’t significantly outperform the conventional orthogonal block code. A desired goal of block codes is to increase the rate while still maintaining the low complexity decoding. A useful idea in the pursuit can be borrowed from noncoherent space-time codes. Hochwald et al. [12], [13] have proposed the unitary & differential unitary space-time block codes, and Hughes [14] also have proposed general framework of differential unitary space-time codes, called space-time group codes. The cardinality of the a space-time block code can be increased by

applying a unitary transformation and this expanded 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 [15], [16], but the literature has not exploited that the super-orthogonal code has the great characteristics as a block code itself. In this paper, a general framework of the block code design and the properties of the super-orthogonal block code are presented. The rest of the paper is organized as follows. Section II reviews the basic backgrounds to help the understanding of this paper, regarding orthogonal block code and performance criterion of space-time code. Section III presents the construction of super-orthogonal block code with the specific design especially for the case of 4 transmit antenna. Decoding algorithm is described in Section IV and simulation results are followed in Section V. Finally, concluding remarks are provided in Section VI. II. BACKGROUNDS A. 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

(1)

where H represents an Lr × Lt channel matrix , D(s) an Lt × Nf transmitted space-time matrix made up of modulated symbol vector s, and N an Lr × 1 additive white Gaussian noise. Here, bold symbol denotes a matrix or vector, and plain symbol denotes a scalar. The receiver is assumed to have the perfect knowledge of CSI. In order to increase the rate 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. · ¸ s −s∗2 (2) D(s1 , s2 ) = 1 s2 s∗1   s1 −s∗2 −s∗3 0 ∗  s2 s∗1 0 s 3  D(s1 , s2 , s3 ) =  (3) s3 0 s∗1 −s∗2  0 −s3 s2 s1 Then, the expanded set of orthogonal space-time matrix is formulated by applying the unitary transformation. The received signal of super-orthogonal block code is given by Y = HD(s)W (s+ ) + N = HD s (s0 ) + N

(4)

where W (s+ ) denotes an Nf × Nf unitary matrix corresponding to additional symbol s+ and D s (s0 = {s, s+ }) ³ ´ + , 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. The detailed design of unitary matrix W 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 code [5], [6]. Assuming the maximum likelihood (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 codeword (D s (β)) over the transmitted one (D s (α)) is asymptotically bounded by [17] Ã ! 2Lr ∆H (α, β) − 1 Lr ∆H (α, β) − 1 P (α, β) ≤ ³ (5) ´Lr SNR∆H (α,β) ∆p (α, β) where ∆H (α, β) is the rank of a signal matrix C s (α, β) defined as C s (α, β) = (D s (α) − D s (β))H (D s (α) − D s (β)) (6) ´ ³ Q ∆H (α,β) λi (α, β) is the product of the and ∆p (α, β) , i=1 non-zero eigenvalues of the signal matrix C s (α, β). (·)H in (6) denotes the complex conjugate transpose of the matrix. ∆H (α, β) and ∆p (α, β) are called the effective Hamming distance and the product measure, respectively. They are key parameters to determine the performance of the spacetime code. Note that the minimum values of ∆H (α, β) and ∆p (α, β) over all possible pairwise errors correspond to the transmit diversity and the coding gain, respectively. When ∆H,min of a space-time code is equal to Lt , 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 [18]–[20] 1 exp {−Lr SNR∆E (α, β)} (7) 4 ³ P ´ ∆H (α,β) where ∆E (α, β) , i=1 λi (α, β) is the sum of nonzero eigenvalues of the signal matrix C s , which is in fact the Euclidean squared distance between the two signal matrices D s (α) and D s (β). The minimum Euclidean distance ∆E,min (α, β) over all possible pairwise errors can be used as an auxiliary design criterion for constructing a good spacetime 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 with maximizing the ∆H,min , ∆p,min and ∆E,min overall. P (α, β) ≤

III. C ODE C ONSTRUCTION In design of the high rate space-time block code, the main work is to expand the cardinality of the orthogonal space-time signal set. The larger the signal set is, the higher rate 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 the super-orthogonal code. The second one is that W do not expand the resulting modulation alphabet in space-time constellation in order not to increase the peakto-average power ratio of the transmitted signal. These two rules restrict that only row-wise/column-wise permutations and phase rotation are allowed in W . The design of 4 transmit super-orthogonal block code is primarily discussed in this paper. The design methodology, however, can be applied to any number of transmit antenna space-time block codes. For the convenience of the design, let us define the following form of unitary matrix, which consist of row-wise/column-wise permutation and phase rotation:   a1 0 0 0  0 a2 0 0   U 0 (a1 , a2 , a3 , a4 ) =  (8)  0 0 a3 0  0 0 0 a4   0 0 0 a1  0 0 a2 0   U 1 (a1 , a2 , a3 , a4 ) =  (9)  0 a3 0 0  a4 0 0 0   0 a1 0 0 a2 0 0 0   U 2 (a1 , a2 , a3 , a4 ) =  (10)  0 0 0 a3  0 0 a4 0   0 0 a1 0  0 0 0 a2   U 3 (a1 , a2 , a3 , a4 ) =  (11) a3 0 0 0  0 a4 0 0

TABLE I U NITARY M ATRIX S ET FOR S QUARED C ONSTELLATION

TABLE III U NITARY M ATRIX S ET FOR C IRCLED C ONSTELLATION (8PSK)

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)

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)

TABLE II P ERFORMANCE M EASURES OF S UPER -O RTHOGONAL B LOCK C ODE U SING QPSK IN 4 T RANSMIT A NTENNA S YSTEM Additional Bits 0 1 2 3 4

∆H,min 4 (100%) 4 (100%) 4 (100%) 2 (0.78%) 2 (1.17%)

∆p,min 16 (9.52%) 4 (5.51%) 4 (8.24%) 4 (8.22%) 4 (8.21%)

∆E,min 8 (9.52%) 8 (4.72%) 8 (2.35%) 8 (1.17%) 8 (0.59%)

where a1 , a2 , a3 and a4 have the form of ejθ , and θ depends on the constellation shape. If the squared modulation like QPSK,16QAM and 64QAM is used, θ can be mπ 2 , and if the circled modulation like 8PSK is used, θ can be mπ 4 to keep the alphabets after the rotation same, where m is an 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 , a unitary matrix set for the squared constellation is formulated in Table I. W (i) with i = 0 to 2n − 1 would be used to transmit n 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 block codes transmitting 1 or 2 additional bits achieve the full diversity, but the codes transmitting 3 or 4 additional bits don’t. On the other hand, as the number of additional bits increases, the Euclidean distance gets larger and larger, which results in relatively small percentage of pairwise errors with the minimum Euclidean distance ∆E,min . As a result of (7), a good performance of the super-orthogonal block code can be guaranteed in a large number of receive antenna environments. Similarly, the unitary matrix set for 8PSK is obtained in Table III. Same as the squared constellation case, the superorthogonal block codes transmitting 1 or 2 additional bits achieve the full diversity, but the codes transmitting 3 or 4

additional bits don’t. It can be easily seen that Jafarkhani’s parameterized class of block code [16] is only the part of our super-orthogonal block code 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 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. Therefore, our design provides more degrees of freedom, i.e., higher rate while providing better performance. IV. D ECODER The standard ML decoding of the super-orthogonal block code 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(2n No ) where No is the decoding complexity of the orthogonal block code. Assuming perfect CSI is available, the simplified ML decoding algorithm is derived as sˆ0 = arg min kY − HD s (s0 )k2F 0 s

= arg min kY − HD(s)W (s+ )k2F + s,s

= arg min kY W (s+ )H − HD(s)k2F +

(12)

s,s

= arg min kY 0 (s+ ) − HD(s)k2F + s,s

where k · k2F is the squared Frobenius norm and Y 0 (s+ ) , Y W (s+ )H . From (12), the decoding algorithm turns into the orthogonal block code’s one [4] for each hypothesis on s+ . Therefore, the decoder performs the exhaustive search on s+ , where each ML metric/decision can be computed with linear

−1

10

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 sˆ0 = arg max T (s|s+ ) +

(13)

H 0 0 0H 0H s˜1 = hH 1 y 1 + y 2 h2 + y 3 h3 + h4 y 4

s˜2 = s˜3 =

− −

H 0 y 0H 2 h1 + h4 y 3 0 0H hH 4 y 2 − y 3 h1

− +

y 0H 4 h3 y 0H 4 h2

(14)

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QPSK: 1.5 bits/s/Hz QPSK: 1.75 bits/s/Hz QPSK: 2.0 bits/s/Hz QPSK: 2.25 bits/s/Hz QPSK: 2.5 bits/s/Hz 8PSK: 2.25 bits/s/Hz 8PSK: 2.5 bits/s/Hz 8PSK: 2.75 bits/s/Hz 8PSK: 3.0 bits/s/Hz 8PSK: 3.25 bits/s/Hz

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Fig. 1. Bit error rate of super-orthogonal code (QPSK and 8PSK) in 4 × 4 system. 0

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where Y 0 (s+ ) = [y 01 y 02 y 03 y 04 ] and H = [h1 h2 h3 h4 ], i.e., y 0i and hi are column vectors comprising Y 0 and H, respectively. Based on the decision metrics (˜ s1 , s˜2 , and s˜3 ) in (14), temporary decisions (ˆ s , s ˆ , and s ˆ ) 1 2 3 and ML metric ¡ ¢ T (ˆ s|s+ ) associated with s+ can be obtained. The temporary ML metric in (13) is also computed as T (ˆ s|s+ ) = T (ˆ s1 |s+ ) + T (ˆ s2 |s+ ) + T (ˆ s3 |s+ )

Bit Error Rate

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+ compared to the orthogonal block code. To demonstrate the simplified ML decoding algorithm, let us consider the decoding of 4 transmit antenna system transmitting 2 additional bits per block (i.e., s+ = 0, 1, 2, and 3). With each s+ , the decision metrics of s1 , s2 and s3 are obtained by

−3

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(15)

where T (ˆ si |s+ ) = Re {˜ s∗i sˆi } − 12 kHk2F |ˆ si |2 for i = 1, 2, and 3. After temporary decisions and ML metrics over all s+ = 0, 1, 2, and 3 are computed, a final decision is made in which the ML metric in (15) is maximized. V. P ERFORMANCE In this section, the performance of the proposed superorthogonal block code is provided via computer simulation. The use of a 4×4 MIMO system is considered with perfect CSI in the receiver. The performance is evaluated in terms of the bit error rate (BER) as a function of Eb /N0 per receive antenna. The BER curves for QPSK, 8PSK, 16QAM and 64QAM are presented in Fig. 1 and Fig. 2 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. The number of receive antennas impacts the performance characteristics. When a small number of receive antennas are used, the performance cannot be improved while increasing the

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Bit Error Rate

s,s

0 hH 2 y1 0 hH 3 y1

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16QAM: 3.0 bits/s/Hz 16QAM: 3.25 bits/s/Hz 16QAM: 3.5 bits/s/Hz 16QAM: 3.75 bits/s/Hz 16QAM: 4.0 bits/s/Hz 64QAM: 4.5 bits/s/Hz 64QAM: 4.75 bits/s/Hz 64QAM: 5.0 bits/s/Hz 64QAM: 5.25 bits/s/Hz 64QAM: 5.5 bits/s/Hz −2

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Eb/N0 (dB)

Fig. 2. Bit error rate of super-orthogonal code (16QAM and 64QAM) in 4 × 4 system.

rate, since the Euclidean distance ∆E (α, β) dominates only in a large number 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 (e.g., QPSK and 8PSK of 2.5 bits/s/Hz in Fig. 1) are compared, the codes using smaller constellation always outperform bigger constellation. Therefore, it is better to use more expanded space-time matrices than to increase the constellation. The performance of the super-orthogonal block code in 2 × 2 system is also shown in Fig. 3. Here, the 2 transmit antenna code in [15] is used. Unlike the 4 transmit antenna code, the performance is getting degraded, when the rate is increased. Since 2 transmit orthogonal space-time matrix is full (i.e., there is no zero), it doesn’t have any advantage for the permutation in unitary transformation, resulting 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

0

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complexity increase in the receiver. The code outperforms the existing quasi-orthogonal block code [11] of same rate and complexity by about 1.5 dB. It has been also shown that the super-orthogonal code has the superiority in more than 2 transmit antenna system.

QPSK: 2.0 bits/s/Hz QPSK: 2.5 bits/s/Hz QPSK: 3.0 bits/s/Hz −1

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

Fig. 3. 0

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

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Fig. 4.

Performance comparison in 4 × 4 system.

cannot be achieved with increased rate. This reveals that the super-orthogonal block code only has the advantage, when more than 2 transmit antennas are employed. Lastly, the performance of the super-orthogonal block code is compared with the quasi-orthogonal block code [11] in Fig. 4. The BER of the orthogonal block code is also shown for the reference. It can be seen that the super-orthogonal block code outperforms the quasi-orthogonal block code by about 1.5 dB in terms of Eb /N0 at the BER of 10−5 . Note that the superorthogonal and quasi-orthogonal codes have approximately same decoding complexity. VI. 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

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