Space–time block code

This article deals with coherent space–time block codes (STBCs). For differential space–time block codes, see differential space-time codes.

Space–time block coding is a technique used in wireless communications to transmit multiple copies of a data stream across a number of antennas and to exploit the various received versions of the data to improve the reliability of data-transfer. The fact that transmitted data must traverse a potentially difficult environment with scattering, reflection, refraction and so on as well as be corrupted by thermal noise in the receiver means that some of the received copies of the data will 'better' than others. This redundancy results in a higher chance of being able to use one or more of the received copies of the data to correctly decode the received signal. In fact, space–time coding combines all the copies of the received signal in an optimal way to extract as much information from each of them as possible.

Introduction

Until 1995, most work on wireless communications focused on having an antenna array at only one end of the wireless link — usually at the receiver. In 1995, Emre Telatar published a seminal paper¹ which, in 1998, inspired Gerard Foschini to demonstrate² the substantial channel capacity improvements available by correctly using antenna arrays at both ends of the link. Shortly afterwards, Siavash Alamouti and Vahid Tarokh et. al demonstrated^3,4 how to use these multiple-input multiple-output systems to achieve significant error rate improvement — the scheme they invented is called space–time coding. In fact, Tarokh et. al invented two types of space–time code: space–time trellis codes (STTCs) and space–time block codes (STBCs); the latter are the topic of this article.

STC relies on the transmission of multiple redundant copies of data in the hope that some of them may arrive at the receiver in a state that means they can be usefully relied upon for correct decoding. In the case of STBC, the data-stream to be transmitted is encoded in 'blocks', as in (block coding) which are distributed across space (meaning antennas) and time — hence the name. Note that while it is necessary to have multiple transmit antennas, it is not necessary to have multiple receive antennas although to do so improves performance. This process of receiving diverse copies of the data is known as diversity reception and is what was largely studied until Foschini's 1998 paper.

The usual, and simplest representation of STBCs is in matrix form. In this form, each row represent a time-slot and each column represents one antenna's transmissions over time e.g.

${\mbox{time-slots}}{\begin{matrix}{\mbox{transmit antennas}}\\\left\downarrow {\overrightarrow {\begin{bmatrix}s_{11}&s_{12}&\cdots &s_{1n_{T}}\\s_{21}&s_{22}&\cdots &s_{2n_{T}}\\\vdots &\vdots &&\vdots \\s_{T1}&s_{T2}&\cdots &s_{Tn_{T}}\end{bmatrix}}}\right.\end{matrix}}$.

Here, $s_{ij}$ is the modulated symbol to be transmitted from antenna $j$ in time-slot $i$. There are to be $T$ time-slots and $n_{T}$ transmit antennas as well as $n_{R}$ receive antennas. This block is usually considered to be of 'length' $T$

The code rate of an STBC measures how many symbols per time-slot it transmits on average over the course of one block⁴. If a block encodes $k$ symbols, the code-rate is

$r={\frac {k}{T}}$.

Unfortunately, only one known standard STBC can achieve full-rate (rate-1) — Alamouti's code.

Orthogonality

STBCs as originally introduced, and as usually studied, are orthogonal. This means that the STBC is designed such that the vectors representing any pair of columns taken from the coding matrix is orthogonal. The result of this is simple, linear, optimal decoding at the receiver. Its most serious disadvantage is that all but one of the codes that satisfy this criterion must sacrifice some proportion of their data rate (see Alamouti's code).

There are also 'quasi-orthogonal STBCs' that allow some inter-symbol interference but can achieve a higher data rate, and even a better error-rate performance, in harsh conditions.

Design of STBCs

The design of STBCs is based on the so-called diversity criterion derived by Tarokh et. al in an earlier paper⁵. Orthogonal STBCs can be shown to achieve the maximum diversity allowed by this criterion.

Diversity criterion

Call a codeword

$\mathbf {c} =c_{1}^{1}c_{1}^{2}...c_{1}^{n_{T}}c_{2}^{1}c_{2}^{2}...c_{2}^{n_{T}}...c_{T}^{1}c_{T}^{2}...c_{T}^{n_{T}}$

and call an erroneously decoded received codeword

$\mathbf {e} =e_{1}^{1}e_{1}^{2}...e_{1}^{n_{T}}e_{2}^{1}e_{2}^{2}...e_{2}^{n_{T}}...e_{T}^{1}e_{T}^{2}...e_{T}^{n_{T}}$.

Then the matrix

$\mathbf {B} (\mathbf {c} ,\mathbf {e} )={\begin{bmatrix}e_{1}^{1}-c_{1}^{1}&e_{2}^{1}-c_{2}^{1}&\cdots &e_{T}^{1}-c_{T}^{1}\\e_{1}^{2}-c_{1}^{2}&e_{2}^{2}-c_{2}^{2}&\cdots &e_{T}^{2}-c_{T}^{2}\\\vdots &\vdots &\ddots &\vdots \\e_{1}^{n_{T}}-c_{1}^{n_{T}}&e_{2}^{n_{T}}-c_{2}^{n_{T}}&\cdots &e_{T}^{n_{T}}-c_{T}^{n_{T}}\\\end{bmatrix}}$

has to be full-rank for any pair of distinct codewords $\mathbf {c}$ and $\mathbf {e}$ to give the maximum possible diversity order of $n_{T}n_{R}$. If instead, $\mathbf {B} (\mathbf {c} ,\mathbf {e} )$ has minimum rank $b$ over the set of pairs of distinct codewords, then the space-time code offers diversity order $bn_{R}$. An examination of the example STBCs shown below reveals that they all satisfy this criterion for maximum diversity.

As an aside, note that STBCs offer only diversity gain (compared to single-antenna schemes) and not coding gain. There is no coding scheme included here - the redundancy purely provides diversity in space and time. This is contrast with space–time trellis codes which provide both diversity and coding gain since they spread a conventional trellis code over space and time.

Encoding

Alamouti's code

Alamouti invented the simplest of all the STBCs in 1998³, although he did not coin the term "space–time code" himself. It was designed for a two-transmit antenna system and has the coding matrix:

$C_{2}={\begin{bmatrix}s_{1}&s_{2}\\-s_{2}^{*}&s_{1}^{*}\end{bmatrix}}$.

It is readily apparent that this is a rate-1 code. It takes two time-slots to transmit two symbols. Using the optimal decoding scheme discussed below, the bit-error rate (BER) of this STBC is equivalent to $2n_{R}$-branch maximal ratio combining (MRC). This is a result of the perfect orthogonality between the symbols after receive processing — there are two copies of each symbol transmitted and $n_{R}$ copies received.

This is a very special STBC. It is the only orthogonal STBC that achieves rate-1⁴. That is to say that it is the only STBC that can achieve its full diversity gain without needing to sacrifice its data rate. Strictly, this is only true for complex modulation symbols. Since almost all constellation diagrams rely on complex numbers however, this property usually gives Alamouti's code a significant advantage over the higher-order STBCs even though they achieve a better error-rate performance. See 'Rate limits' for more detail.

Higher order STBCs

Tarokh et. al discovered^4,6, by computer search, a set of STBCs that are particularly straightforward, and coined the scheme's name. They also proved that no code for more than 2 transmit antennas could achieve full-rate. Their codes have since been improved upon (both by the original authors and by many others). Nevertheless, they serve as clear examples of why the rate cannot reach 1, and what other problems must be solved to produce 'good' STBCs.

They also demonstrated the simple, linear decoding scheme that goes with their codes.

3 transmit antennas

Two straightforward codes for 3 transmit antennas are:

$C_{3,1/2}={\begin{bmatrix}s_{1}&s_{2}&s_{3}\\-s_{2}&s_{1}&s_{4}\\-s_{3}&s_{4}&s_{1}\\-s_{4}&-s_{3}&s_{2}\\s_{1}^{*}&s_{2}^{*}&s_{3}^{*}\\-s_{2}^{*}&s_{1}^{*}&s_{4}^{*}\\-s_{3}^{*}&s_{4}^{*}&s_{1}^{*}\\-s_{4}^{*}&-s_{3}^{*}&s_{2}^{*}\end{bmatrix}}\quad {\mbox{and}}\quad C_{3,3/4}={\begin{bmatrix}s_{1}&s_{2}&{\frac {s_{3}}{\sqrt {2}}}\\-s_{2}^{*}&s_{1}^{*}&{\frac {s_{3}}{\sqrt {2}}}\\{\frac {s_{3}^{*}}{\sqrt {2}}}&{\frac {s_{3}^{*}}{\sqrt {2}}}&{\frac {\left(-s_{1}-s_{1}^{*}+s_{2}-s_{2}*\right)}{2}}\\{\frac {s_{3}^{*}}{\sqrt {2}}}&-{\frac {s_{3}^{*}}{\sqrt {2}}}&{\frac {\left(s_{2}+s_{2}^{*}+s_{1}-s_{1}^{*}\right)}{2}}\end{bmatrix}}$.

These codes achieve rate-1/2 and rate-3/4 respectively. These two matrices give examples of why codes for more than two antennas must sacrifice rate — it is the only way to achieve orthogonality. One particular problem with $C_{3,3/4}$ is that it has uneven power among the symbols it transmits. This means that the signal does not have a constant envelope and that the power each antenna must transmit has to vary, both of which are undesirable. Modified versions of this code that overcome this problem have since been designed.

4 transmit antennas

Two straightforward codes for 4 transmit antennas are:

$C_{4,1/2}={\begin{bmatrix}s_{1}&s_{2}&s_{3}&s_{4}\\-s_{2}&s_{1}&s_{4}&s_{3}\\-s_{3}&s_{4}&s_{1}&-s_{2}\\-s_{4}&-s_{3}&s_{2}&s_{1}\\s_{1}^{*}&s_{2}^{*}&s_{3}^{*}&s_{4}^{*}\\-s_{2}^{*}&s_{1}^{*}&s_{4}^{*}&s_{3}^{*}\\-s_{3}^{*}&s_{4}^{*}&s_{1}^{*}&-s_{2}^{*}\\-s_{4}^{*}&-s_{3}^{*}&s_{2}^{*}&s_{1}^{*}\end{bmatrix}}\quad {\mbox{and}}\quad {}C_{4,3/4}={\begin{bmatrix}s_{1}&s_{2}&{\frac {s_{3}}{\sqrt {2}}}&{\frac {s_{3}}{\sqrt {2}}}\\-s_{2}^{*}&s_{1}^{*}&{\frac {s_{3}}{\sqrt {2}}}&-{\frac {s_{3}}{\sqrt {2}}}\\{\frac {s_{3}^{*}}{\sqrt {2}}}&{\frac {s_{3}^{*}}{\sqrt {2}}}&{\frac {\left(-s_{1}-s_{1}^{*}+s_{2}-s_{2}*\right)}{2}}&{\frac {\left(-s_{2}-s_{2}^{*}+s_{1}-s_{1}^{*}\right)}{2}}\\{\frac {s_{3}^{*}}{\sqrt {2}}}&-{\frac {s_{3}^{*}}{\sqrt {2}}}&{\frac {\left(s_{2}+s_{2}^{*}+s_{1}-s_{1}^{*}\right)}{2}}&-{\frac {\left(s_{1}+s_{1}^{*}+s_{2}-s_{2}^{*}\right)}{2}}\end{bmatrix}}$.

These codes achieve rate-1/2 and rate-3/4 respectively, as for their 3-antenna counterparts. $C_{4,3/4}$ exhibits the same uneven power problems as $C_{3,3/4}$. An improved version of $C_{4,3/4}$ is⁷

$C_{4,3/4}={\begin{bmatrix}s_{1}&s_{2}&s_{3}&0\\-s_{2}^{*}&s_{1}^{*}&0&s_{3}\\-s_{3}^{*}&0&s_{1}^{*}&-s_{2}\\0&-s_{3}^{*}&s_{2}^{*}&s_{1}\end{bmatrix}}$,

which has equal power from all antennas in all time-slots.

Decoding

One particularly attractive feature of STBCs is that maximum likelihood decoding can be achieved at the receiver with only linear processing. In order to consider a decoding method, a model of the wireless communications system is needed.

At time $t$, the signal $r_{t}^{j}$ received at antenna $j$ is:

$r_{t}^{j}=\sum _{i=1}^{n_{T}}\alpha _{ij}s_{t}^{i}+n_{t}^{j}$,

where $\alpha _{ij}$ is the path gain from transmit antenna $i$ to receive antenna $j$ and $n_{t}^{j}$ is a sample of additive white Gaussian noise (AWGN).

The maximum-likelihood detection rule⁶ is to form the decision variables

$R_{i}=\sum _{t=1}^{n_{T}}\sum _{j=1}^{n_{R}}r_{t}^{j}\alpha _{\epsilon _{t}(i)j}\delta _{t}(i)$

where $\delta _{k}(i)$ is the sign of $s_{i}$ in the $k$^th row of the coding matrix, $\epsilon _{k}(p)=q$ denotes that $s_{p}$ is (up to a sign difference), the $(k,q)$ element of the coding matrix, for $i=1,2$...$n_{T}$ and then decide on constellation symbol $s_{i}$ that satisfies

$s_{i}=\arg {}\min _{s\in {\mathcal {A}}}\left|R_{i}-s\right|^{2}+\left(-1+\sum _{k,l}^{}\left|\alpha _{kl}\right|^{2}\right)\left|s\right|^{2}$,

with ${\mathcal {A}}$ the constellation alphabet. Despite its appearance, this is a simple, linear decoding scheme that provides maximal diversity.

Rate limits

Apart from there being no full-rate complex STBC for more than 2 antennas, it has been further shown that, for more than 3 antennas, the maximum possible rate is 3/4⁸. Codes have been designed⁹ which achieve a good proportion of this, but they have very long block-length and are unsuitable for practical use. This is because decoding cannot proceed until all transmissions in a block have been received, so a longer block-length, $T$ results in a longer decoding delay. One particular example, for 16 transmit antennas, has rate-9/16 and a block length of 22 880 time-slots!

It has been conjectured⁸, but not proven, that the highest rate any $n_{T}$-antenna code can achieve is

$r_{\mathrm {max} }={\frac {n_{0}+1}{2n_{0}}}$,

where $n_{T}=2n_{0}$ or $n_{t}=2n_{0}-1$.

Quasi-orthogonal STBCs

These codes exhibit partial orthogonality and provide only part of the diversity gain mentioned above. An example reported by Jafarkhani is¹⁰:

$C_{4,1}={\begin{bmatrix}s_{1}&s_{2}&s_{3}&s_{4}\\-s_{2}^{*}&s_{1}^{*}&-s_{4}^{*}&s_{3}^{*}\\-s_{3}^{*}&-s_{4}^{*}&s_{1}^{*}&s_{2}^{*}\\s_{4}&-s_{3}&-s_{2}&s_{1}\end{bmatrix}}$.

The orthogonality criterion only holds for columns (1 and 2), (1 and 3), (2 and 4) and (3 and 4). Crucially, however, the code is full-rate and still only requires linear processing at the receiver, although decoding is slightly more complex that for orthogonal STBCs. Results show that this Q-STBC outperforms (in a bit-error rate sense) the fully-orthogonal 4-antenna STBC over a good range of signal-to-noise ratios (SNRs). At high SNRs, though (above about 22dB in this particular case), the increased diversity offered by orthogonal STBCs yields a better BER. Beyond this point, the relative merits of the schemes have to be considered in terms of useful data throughput.

Q-STBCs have also been developed considerably from the basic example shown here.

See also

• Space–time code
• Space–time trellis code
• Wireless communications

References

1. "I. Emre Telatar", "Capacity of multi-antenna gaussian channels", Technical Memorandum, Bell Laboratories, Oct. 1995.
2. Gerard. J. Foschini and Michael. J. Gans, "On limits of wireless communications in a fading environment when using multiple antennas", Wireless Personal Communications, vol. 6, no. 3, pp. 311–335, Jan. 1998.
3. S.M. Alamouti, "A simple transmit diversity scheme for wireless communications", IEEE J. Select. Areas Commun., vol. 16, no. 8, pp. 1451–1458, Oct. 1998.
4. Vahid Tarokh, Hamid Jafarkhani, and A. R. Calderbank, "Space–time block codes from orthogonal designs", IEEE Trans. Inform. Theory, vol. 45, no. 5, pp 1456–1467, Jul. 1999.
5. Vahid Tarokh, Nambi Seshadri, and A. R. Calderbank, "Space–time codes for high data rate wireless communication: Performance analysis and code construction," IEEE Trans. Inform. Theory, vol. 44, pp. 744–765, Mar. 1998.
6. Vahid Tarokh, Hamid Jafarkhani, and A. Robert Calderbank, "Space–time block coding for wireless communications: performance results", IEEE J. Select. Areas Commun., vol. 17, pp. 451–460, Mar. 1999.
7. G. Ganesan and P. Stoica, "Space–time block codes: A maximum SNR approach", IEEE Trans. Inform. Theory, pp. 1650–1656, Apr. 2001.
8. H. Wang and X.-G. Xia, "Upper bounds of rates of space–time block codes from complex orthogonal designs", IEEE Trans. Inform. Theory, vol. 49, pp. 2788–2796, Oct. 2003.
9. Weifeng Su, Xiang-Gen Xia, and K. J. Ray Liu, "A systematic design of high-rate complex orthogonal space-time block codes", IEEE Commun. Lett., vol. 8, no. 6, pp. 380–382, Jun. 2004.
10. Hamid Jafarkhani, "A quasi-orthogonal space–time block code", IEEE Trans. Commun.", vol. 49, no. 1, pp. 1–4, Jan. 2001.