structing improved high-rate space-time codes. The proposed code construction is based on a concatenation of an orthogonal space- time block code and an ...
Improved High-Rate Space-Time Codes via Orthogonality and Set Partitioning Siwaruk Siwamogsatham and Michael P. Fitz Department of Electrical Engineering The Ohio State University Columbus, OH 43210, USA
Abstract— In this paper, we propose a new technique for constructing improved high-rate space-time codes. The proposed code construction is based on a concatenation of an orthogonal spacetime block code and an outer M-TCM encoder. However, unlike the existing STB-MTCM schemes which are rate-lossy, the proposed construction yields higher-rate space-time codes by expanding the cardinality of the orthogonal space-time block code before concatenating with an outer M-TCM encoder. An advantage of the proposed construction is that the standard techniques for designing a good TCM codes, such as the classic set partitioning concept, can be adopted to realize the STB-MTCM designs with large coding gains. We present several design examples of improved fullrate space-time codes for a system with 2 transmit antennas. Simulation results show that the new space-time codes considerably outperform the existing ST-TCM designs. For example, the new 4-state 2-bits/symbol QPSK space-time code performs even better than the original 32-state design, while performance of the new 32-state QPSK code is only 1.5 dB away from the outage probability limit. Moreover, decoding complexity of the proposed M-TCM construction is made reasonably low by exploiting signal orthogonality.
I. I NTRODUCTION Over the past several years, there has been a great deal of research to improve performance of wireless communications in fading environments by exploiting transmitter and/or receiver diversity. The pioneering work by Telatar, Foschini and Gans [1], [2] showed that multiple antennas in a wireless communication system can greatly improve performance. For transmit antennas and receive antennas, it was shown that with spatial independence there were essentially levels of diversity available and there were independent parallel channels that could be established. These information theoretic studies spawned two lines of work; one where the number of independent channels is large [3] and one where the number of independent channels is small [4]. This work is concerned with the latter research paradigm where a small number of transmit and receive antennas are utilized. The goal in this paradigm is to achieve as large a throughput as possible while retaining highly reliable demodulation. For wireless communication systems with a fair number of parallel channels, space-time (ST) coding is the commonly accepted approach and has been shown to be a very effective di-
This work was supported by National Science Foundation through CCR0073505
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versity strategy. The standard design criteria for constructing a good space-time code were proposed in [4], [5]. These design criteria have been commonly used in the literature to construct many classes of space-time codes. The focus of this research is on the elegant class of the space-time trellis coded modulation (ST-TCM). Since the introduction of the pioneering ST-TCM designs by Tarokh, Seshadri, and Calderbank (TSC) [4], there has been a great deal of research aimed at improving the performance of the original TSC ST-TCM designs. Because the original TSC ST-TCM codes were handcrafted, they were not the optimum designs. In recent years, much research has been completed which proposed code constructions or performed systematic searches for more optimal convolutional encoders [6], [7], [8], [9], [10], [11], [12], [13], yielding space-time codes with improved worst-case error performance. However, only marginal or moderate gains of average performance over the original TSC designs were mostly obtained from these research efforts. The study by Aktas and Fitz [14] on distance spectrum of these existing space-time codes suggested that tweaking the convolutional encoders to improve the worst-case error performance was not always a productive means for constructing improved space-time codes. In this work, we propose a new design approach that produces high-rate space-time codes that largely outperform the existing ST-TCM designs. The previously proposed high-rate ST-TCM never really used the revolutionary idea of Ungerboeck [15] that was constellation expansion and set partitioning. In this paper we intend to tie in this idea of Ungerboeck to the design of space-time codes. A common design approach that bridges the designs of space-time codes and the conventional single-antenna TCM schemes is a concatenation of an orthogonal space-time block code [16], [17] and an outer TCM or M-TCM encoder (STBMTCM). An advantage of the STB-MTCM construction is that many aspects for constructing a good space-time codes can be conveniently borrowed from the rich and well-understood literature of the classic TCM designs. In particular, the essential concept of signal partitioning [18] can be adopted for designing a good STB-MTCM codes. In fact, the existing TCM designs for the AWGN channels can be optimally used as an outer code [19]. Nonetheless, the main weakness of the traditional STB-MTCM schemes (e.g. [19], [20], [21], [22], [23]) is that the overall transmission rate is reduced because the inner block
code is at best a full-rate code and the outer TCM encoder must have redundancy. In order to enable a high data throughput via a concatenated STB-MTCM-based scheme, the inner space-time block code must be expanded before being concatenated with an outer MTCM encoder. The auxiliary codewords can be formulated by applying some unitary transformations to the original signal set. The expanded space-time block code may not necessarily be a full-rank code but the outer M-TCM encoder can be designed such that the resulting space-time code achieves full diversity. In this work, several new full-rate space-time codes for a system with 2 transmit antennas are derived based on the proposed technique. Simulation results show that the new space-time codes considerably outperform the existing ST-TCM designs. For example, the new 4-state 2-bits/sybol QPSK space-time code achieves even better performance than the original TSC 32-state design, while the new 32-state QPSK code performs about 1.5 dB away from the outage probability limit. Moreover, decoding complexity of the proposed expanded-STBCTCM construction is reasonably low because codeword orthogonality is exploited to simplify computation for the optimal demodulator. A. System Model We consider a space-time wireless communication system antennas at the transmitter and antennas at the rewith ceiver. The transmitter employs a concatenated coding scheme where an M-TCM encoder with multiplicity of is used as an outer code and an space-time block code/constellation is used as an inner code. First, the information to be transmitted are converted into a stream of -bit words. At the coding interval, each -bit word is encoded by an outer M-TCM encoder of rate into bits, which are then transformed by an inner block code of rate to output an space-time codeword
µ +1µ
µ +1µ
½´ ½´
¾´ ¾´
.. .
.. . +
..
.
´ µ ´ +1µ .. . ´ + -1µ
where
defines the bit energy per receive antenna,
is an
transmitted codeword matrix formed as
(2) is an channel vector whose
element denotes the complex channel distortion between the transmit antenna and the receiver, and is the addi-
tive white Gaussian noise (AWGN) with a covariance matrix With the quasi-static fading assumption, the channel distortion coefficients are independent zero-mean Guassian random variables with a unit variance, and the signal-to-noise ratio . The (SNR) per receive antenna is computed as receiver is assumed to have perfect knowledge of channel state information (CSI).
.
B. Performance Criterion Basically, the performance criterion for the space-time system under consideration resembles the standard one [5], [4], derived from an analysis of the pairwise error probability (PWEP) which is the probability that the optimum decoder makes an erroneous decision in favor of a given codeword ) over the trans(
mitted one ( ). For the maximum likelihood (ML) decoder with perfect CSI in quasi-static fading, the PWEP is asymptotically bounded by [24]
¡ ´ µ ¡ ´ µ
¾
½ ½
(3)
defined (4)
where is the rank of a signal matrix as
and is the product of the non + -1µ ¾ ´ -1µ zero eigenvalues of the signal matrix . Note that with corresponding to the symbol linearly modulated and may be referred to as the effective and transmitted via the antenna at time . The overall trans½´
mission rate of this space-time code is defined as bits/symbol. The modulation symbols are chosen from a constellation with a unit average energy. The channel between a transmit and a receive antenna is modeled as a frequency non-selective flat Rayleigh fading process. The channel is assumed to be constant during a data frame but fades independently from frame to frame and from antenna to antenna, i.e., the quasi-static fading assumption. At the receiver, assuming ideal timing information, an vector of matched filter output is formed for maximum likelihood se denoting the observation quence decoding, with frame length. This observation vector can be expressed as
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(1)
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Hamming distance and the product measure, respectively. The standard design criteria for constructing a good space-time code are the minimum values of the effective Hamming distance and product measure over all possible pairwise error events (i.e., and ) for a given code, which shall be maximized. Additionally, the sum of eigenvalues of the signal ma trix is a good auxiliary perfor mance measure [12], which shall also be maximized, especially for a system with a large number of antennas. Since we consider a concatenated STB-MTCM scheme, it is useful to define performance measures associated with the (inner) space-time codewords at each coding interval. That is, we define the signal matrix at the coding interval as
(5)
such that
(6)
designing a good space-time code. We denote the four subsets by
The rank of , the product, and the sum of non-zero eigenvalues of this signal matrix are defined by Æ , , respectively. Using the standard maÆ , and Æ trix algebra, it follows that
Æ Æ
That is, Æ , Æ , and mized at every coding interval .
(7) (8)
Æ
(9)
Æ
shall be maxi-
II. N EW S PACE -T IME C ODE C ONSTRUCTION The proposed code construction is based on a concatenated STB-MTCM scheme in which an orthogonal spacetime block code is an inner code. Traditionally, this concatenated scheme yields rate-lossy space-time codes. However, a full-rate space-time code is possible here because the inner orthogonal space-time block code is expanded before being concatenated with an outer encoder. A simple design rule is then imposed on the outer encoder to guarantee that the resulting space-time code achieves full-diversity. A. Expanded Orthogonal Space-Time Block Code The goal here is to expand the cardinality of the orthogonal space-time signal set, in order that sufficient signal points are available to allow a high-rate STB-MTCM design. Note that we do not require that the Hamming distance achieves the maximum possible value for every codeword pair in the entire signal set, but it is desirable that this holds for codewords in each given signal subset. To do this, first the standard space-time block code is used as a building subset. A distinct block-code subset can then be generated from the original signal subset by applying certain unitary transformations, i.e., if is an orthogonal block code for an input , another orthogonal block code can be formed by
(11) (12) (13) (14)
Note that only the first two subsets would be valid for the case of BPSK modulation. B. Set Partitioning An advantage of an M-TCM-based construction is that the essential concept of set partitioning can be realized. In the pro as the posed design, we utilize the product measure Æ criterion to partition codewords within a given full-rank block code subset. Given that at the coding interval a pair of codewords are labeled with signals from a given full-rank block code, e.g., , we have and
Æ
(15)
which is conveniently related to the squared Euclidean distance of the input labels and . Therefore, the classic set partitioning technique [18], which employs the squared Euclidean distance as the partition criterion, can be adopted to partition the signals from the same full-rank block code subset. For the codewords from different full-rank block code subsets, as the partition criterion. we utilize the sum measure Æ After numerically examining the sum measure between different signal subsets, the higher-level partitions can be done as shown in Figure 1. A (s 0 , s1), B (s 0 , s1 ), J (s 0, s1), K (s 0 , s1)
A (s 0 , s1), B (s 0 , s1)
J (s 0, s1), K (s 0 , s1)
(10)
where and are some and diagonal unitary matrices, respectively. A union of these signal subsets forms an expanded orthogonal block code. In this work, we limit our attention to the 2-transmitter case. We start with the Alamouti signals as the building subset. We found that the following four signal subsets are sufficient for
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A (s 0 , s1)
B (s 0 , s1)
J (s 0, s1)
K (s 0 , s1)
Fig. 1. Higher-level partitions
C. Design Rules Since an expanded space-time code is not necessarily a fullrank code, an appropriate design rule must be imposed on the
A 0 A1
outer M-TCM encoder to guarantee that the resulting STBMTCM construction achieves full diversity. Because , , , or is a full-rate full-diversity space-time code, we can arrive at a simple design rule for constructing a full-diversity STB-MTCM code; i.e., “transition branches leaving from (or merging to) each state are uniquely labeled with codewords from the same full-rank block code.” Additionally, to achieve large coding gains, we can also employ design rules similar to the standard ones in [15]; e.g., “transition branches leaving from and/or merging to each state are uniquely labeled with codewords from the same upper-level signal partition.”
B 1 B0
A 0 A1
B 1 B0
A 1 A0
III. D ESIGN E XAMPLES In this paper, we present several design examples of the new full-rate QPSK and 8-PSK space-time codes for a 2-transmitter system. In this case, the M-TCM with multiplicity of 2 is used as an outer encoder, and thus 16 and 64 incoming and outgoing transitions are needed to achieve the desired code rate of 2 bits/symbol and 3 bits/symbols, respectively. For the 2-state, 4-state, and 8-state QPSK codes, we design the outer M-TCM encoder to have 8 parallel transitions from a given state to another. We use the standard method to divide each of the four full-rank block code subsets into 2 partitions with a cardinality of 8, e.g,
¼ ½
¼ ½
¼¾ ½¼ ½¾
¼¿
½½
½¿
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B 1 B0 J0
A 1 A0 B 0 B1
Fig. 2. Improved 2-state, 4-state, 8-state, 2-bits/symbol QPSK codes
J0
K 0 K1
J1
K 1 K0
J1
¼½
A 0 A1
and similarly for , , , and . The proposed simple design rule is then used to construct the STB-MTCM codes that achieve full-diversity. The new 2-state, 4-state, and 8-state 2bits/symbol QPSK space-time codes are depicted in Figure 2. Note that for the 2-state and 4-state encoders, it is sufficient to use only 2 block code subsets, but it is usually better to use all four subsets for a larger encoder. The proposed 2-state, 4-state, and 8-state QPSK space-time codes achieve full diversity and have a minimum product measure , which is better than that achieved by the original TSC ST-TCM with 32 states. The large coding advantage results from the exploitation of signal partitioning and symmetries. Simulation results verify that the new 4-state and 8-state space-time codes outperform the original 32-state TSC space-time code. For an encoder with larger memory, we may allow a smaller number of parallel transitions to improve coding gains. For the 32-state QPSK code, we employ an M-TCM encoder with 2 parallel branches. In this case, each full-rank block code subset is divided into 8 smaller partitions, e.g., ¼¼
B 0 B1
similarly, for , , , , , for i = 0, 1, 2, and 3. The new 32-state 2-bits/symbol QPSK space-time code is depicted in Figure 3. Simulation results show performance of this code is about 1.5 dB away from the outage probability limit. The same design concepts can be used for 8-PSK modulation. For brevity of this presentation, we only include an example of an improved 8-state 3-bits/symbol 8-PSK space-time code. In this case, we partition each subset of block code into 4 sub-partitions each with a cardinality of 16, e.g., (defining and ) ¼ ½ ¾ ¿
½ ½ ¿ ¿ ½ ¿ ¿ ½ ½ ¿ ¿ ½ ½ ¿ ½ ¿
and similarly for , , for i = 0, 1, 2, and 3. The new 8-state 8-PSK code is shown in Figure 4. IV. S IMPLIFIED ML D ECODING AND C OMPLEXITY C ONSIDERATION
In general, it is typical more difficult to decode an M-TCM scheme than a single-dimension TCM counterpart because a
267 264
A00 A10 A 02 A 12 A 01 A11 A 03A 13
tion is needed. The key idea is to combine the standard simplified ML decoding algorithm and the concept of signal partitions. To enable reduced-complexity decoding, we shall partition a signal set assigned to parallel transitions into smaller signal partitions such that the standard simplified ML decoding algorithm can be directly applied in each smaller signal partition. For example, consider computation of the ML branch met. First, ric for a partition of parallel branches labeled with we divide codewords assigned to these parallel branches into 2 subsets: and , in which the standard reduced-complexity decoding algorithm can be conveniently utilized to obtain the most-likely candidate within each subset. Then, the ML decision between these candidates is taken and the associated likelihood is the branch metric of the current path. More explicitly, after some manipulation by using orthogonality and ignoring constant terms, the likelihood function for a at the decoding interval transition labeled with given the observation vector at the receiver for this interval is and the channel vector at the re ceiver is (assuming perfect CSI at the decoder) is given as
B10 B00 B 12 B 02 B 11 B 01 B 13 B 03 J 10 J 00 J 12 J02 J 11 J 01 J 13 J 03 K00 K10 A02 A12 B12 B02 J 12 J 02 K02 K12 A01 A11 B11 B01 J 11 J 01 K01 K11 A03 A13 B13 B03 J 13 J 03 K03 K13 A10 A00 B00 B10 J 00 J 10 K10 K00 A12 A02 B02 B12 J 02 J 12 K12 K02 A11 A01 B01 B11 J 01 J 11 K11 K01 A13 A03 B03 B13 J 03 J 13 K13 K03
K02 K12 K01 K11 K 03 K13 A 01 A 11 A 03 A13 A 00A 10 B 11 B 01 B 13 B 03 B 10 B 00 J 11 J01 J 13 J 03 J 10 J 00 K01 K11 K03 K13 K 00 K10 A 03 A 13 A 00 A10 A 02A 12 B 13 B 03 B 10 B 00 B 12 B 02 J 13 J03 J 10 J 00 J 12 J 02 K03 K13 K00 K10 K 02 K12 A 00 A 10 A 02 A12 A 01A 11 B 10 B 00 B 12 B 02 B 11 B 01 J 10 J00 J 12 J 02 J 11 J 01 K00 K10 K02 K12 K 01 K11 A 12 A 02 A 11 A01 A 13A 03 B 02 B 12 B 01 B 11 B 03 B 13 J 02 J12 J 01 J 11 J 03 J 13 K12 K02 K11 K01 K 13 K03 A 11 A 01 A 13 A03 A 10A 00 B 01 B 11 B 03 B 13 B 00 B 10 J 01 J11 J 03 J 13 J 00 J 10 K11 K01 K13 K03 K 10 K00 A 13 A 03 A 10 A00 A 12A 02 B 03 B 13 B 00 B 10 B 02 B 12 J 03 J13 J 00 J 10 J 02 J 12 K13 K03 K10 K00 K 12 K02 A 10 A 00 A 12 A02 A 11A 01 B 00 B 10 B 02 B 12 B 01 B 11 J 00 J10 J 02 J 12 J 01 J 11 K10 K00 K12 K02 K 11 K01
(16)
where
Fig. 3. Improved 32-state 2-bits/symbol QPSK code
A p0 Ap2 Ap1 Ap3 J p0 Jp2 Jp1 Jp3
Then, by defining the following estimates:
Bp2 Bp1 Bp3 Bp0
K p2 Kp1 Kp3 Kp0
A p1 Ap3 Ap0 Ap2 J p1 Jp3 Jp0 Jp2 Bp2 Bp0 Bp3 Bp1 K p2 Kp0 Kp3 Kp1 Fig. 4. Improved 8-state 3-bits/symbol 8-PSK code
potentially much larger number of transition branches are required for an M-TCM encoder to achieve the same date rate. This can be a main disadvantage of an M-TCM scheme. In the proposed design, however, we can exploit signal orthogonality to ease data decoding since the orthogonal space-time block code is used as the inner code. The simplified ML decoding algorithm is described in the sequel. Note that the standard reduced-complexity decoder for an orthogonal space-time block code was provided in [16], [17]. Nonetheless, it may not be directly applicable to the proposed concatenated STB-MTCM scheme. An appropriate modifica-
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and exploiting linearity, the most-likely candidates are given by and . The ML decision among these two candidates is taken and the corresponding likelihood function is the branch metric for this partition. By analogy, similar computations are proceeded to calculate metrics for all distinct branches. Once all branch metrics are computed, the Viterbi algorithm is applied as usual to search for the path with the largest accumulated metric. Table I lists decoding complexity estimates for the proposed STB-MTCM codes with and without the simplified ML decoder, as well as the standard ST-TCM codes with a generic Viterbi ML decoder, with a single receiver. Note that the values listed in parentheses denote decoding complexity if a generic ML decoder is employed. Decoding complexity is evaluated in terms of the total number of basic arithmetic and mathematics operations utilized per symbol, assuming the decoders are efficiently implemented such that redundant computations are smartly stored for multiple usages. Note that these basic operations include numerical comparison, real-value addition, subtraction, and multiplication.
3-bits/symbol
156 184 240 352
664 784 1024
52 (268) 104 (512) 352 (728)
198 (2022) 212 (2036) 704 (2240)
2 bit/sec/Hz, QPSK, 2 Tx, 1 Rx, 128 symbols/frame
0
10
4−state TSC 8−state TSC 4−state YB 8−state YB 2−state new 4−state new 8−state new 8−state IMYL outage probability FER
ST-TCM 4-state 8-state 16-state 32-state STB-MTCM 4-state 8-state 32-state
2-bits/symbol
−1
10
TABLE I D ECODING C OMPLEXITY
It should now be apparent that the computational complexity of the ML decoder is greatly reduced via the proposed decoding algorithm, particularly for high-rate cases. Without the simplified ML decoder, a high-rate STB-MTCM construction is rather less practical. This underscores significance of the use of an orthogonal space-time block code as an inner encoder. The ability to achieve excellent performance with relatively low decoding complexity at high transmission rate makes the proposed design technique particularly attractive. It should be pointed out also that a similar STB-MTCM scheme was independently proposed in [25], but simplified decoding was not considered.
−2
10
10
10.5
11
11.5
12
12.5 Eb/N0 (dB)
13
13.5
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14.5
15
Fig. 5. Performance of the 2-bits/symbol QPSK space-time codes 2 bit/sec/Hz, QPSK, 2 Tx, 1 Rx, 128 symbols/frame
16−state TSC 32−state TSC 16−state YB 32−state YB 4−state new 8−state new 32−state new outage probability
−1
10
FER
V. S IMULATION R ESULTS This section illustrates performance comparisons between the proposed vs. existing space-time codes. The simulated performance is evaluated in forms of the frame error rate (FER) as a function of signal-to-noise ratio, where a frame consists of 128 channel symbols including trellis-terminating bits. A single receiver is assumed in the simulations. Figures 5 and 6 depict performance of the proposed 2bits/symbol QPSK space-time codes vs. the existing ST-TCM designs. Note that the Yan and Blum (YB) codes [9] are the existing optimized QPSK ST-TCM codes for the 2-transmitter case. It can be seen that the best-known 4-state design provides only a fractional performance gain over the original 4-state TSC design, while the proposed 4-state QPSK code achieves a large 2-dB performance advantage. More interestingly, the proposed 4-state QPSK code even outperforms the original 32-state TSC design and the performance is only about 2.5 dB away from the outage probability. This level of performance is comparable to that of the original 64-state ST-TCM and an elegant space-time turbo-TCM scheme such as [26]. The performance gain is more pronounced with the new 32-state code whose performance is only 1.5 dB away from the outage probability limit. On the other hand, the 32-state YB code performs roughly the same as the new 8-state code. Also, the new 2-state code performs better than the 8-state YB code. We also include a comparison of performance of the proposed codes and the IMYL code [25] which was obtained independently from a similar design concept. It is worthwhile to
14
−2
10
10
10.5
11
11.5
12
12.5 Eb/N0 (dB)
13
13.5
14
14.5
15
Fig. 6. Performance of the 2-bits/symbol QPSK space-time codes
distinguish [25] from the proposed work. In [25], only 2 block code subsets: ¼ ½ and ¼ ½ were proposed in their designs, while we used 4 signal subsets in this work. Using more signal points can typically give better designs. We can see that the proposed 8-state QPSK code performs better than the 8-state IMYL code. In [25], only the 8-state and 16-state codes were derived, while we also unveil the new 4-state design which outperforms the original 32-state TSC design, as well as the new 32-state design whose performance is only 1.5dB away from the fundamental limit. We also present the new 2-state code. Note that we did not include performance of the 16-state IMYL code since it performed virtually the same as the 8-state one. We did not show performance of the new 16state code due to the similar reason. We also derived the sev-
3 bits/sec/Hz, 8−psk, 2 Tx, 1 Rx, 128 symbols/frame
0
10
FER
8−state TSC 16−state TSC 32−state TSC 4−state new 8−state new 32−state new 64−state new outage probability −1
10
−2
10
10
10.5
11
11.5
12
12.5 Eb/N0 (dB)
13
13.5
14
14.5
15
Fig. 7. Performance of the 3-bits/symbol 8-PSK space-time codes
eral new 3 bits/symbol 8-PSK space-time codes. In addition, we show performance of several new improved 3-bits/symbol 8-PSK space-time codes in Figure 7. It is noted that the new 8-state code achieves similar performance as that of the original 32-state TSC design, and further performance improvement can be achieved by a larger encoder. VI. C ONCLUSIONS In this paper, we proposed a novel technique for designing high-rate space-time codes that achieved significant performance gains over the existing ST-TCM designs. The proposed 4-state 2-bits/sybol QPSK space-time code considerably outperformed the original 4-state ST-TCM design and performed even better than the original 32-state design. The performance of the new 32-state QPSK code was only 1.5 dB away from the outage probability limit. The major performance improvement was due to an exploitation of the set partitioning concept, which is essential to the design of a good trellis code, but was not possible in the original ST-TCM designs. Additionally, signal orthogonality was exploited to keep decoding complexity of the proposed STB-MTCM construction relatively low. The ability to achieve excellent performance with relatively low decoding complexity at high transmission rate made the proposed design technique particularly attractive. R EFERENCES [1] I. Telatar, “Capacity of multi-antenna Gaussian channels.” AT&T Bell Labs Technical Report, 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, March 1998. [3] G. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas.” Bell Labs Technical Journal, 1996. [4] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: performance and code construction,” IEEE Trans. Inform. Theory, March 1998.
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