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:''This article deals with coherent space–timespace–time block codes (STBCs). For differential space–timespace–time block codes, see [[differential space-timespace–time code]]s.
'''Space–timeSpace–time block coding''' is a technique used in [[wireless|wireless communications]] to transmit multiple copies of a data stream across a number of [[antenna (radio)|antenna]]s 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 (physics)|reflection]], [[refraction]] and so on as well as be corrupted by [[thermal noise]] in the [[receiver (radio)|receiver]] means that some of the received copies of the data will be '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–timespace–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|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 paper<ref>{{cite journal|author=I. Emre Telatar|title=Capacity of multi-antenna gaussian channels|journal=Technical Memorandum, Bell Laboratories|date=October 1995|pages=|url=http://mars.bell-labs.com/papers/proof/proof.pdf}}</ref> which, in 1998, inspired [[Gerard Foschini]] to demonstrate<ref>{{cite journal|author=Gerard J. Foschini and Michael. J. Gans|title=On limits of wireless communications in a fading environment when using multiple antennas|journal=Wireless Personal Communications|pages=311–335311–335|volume=6|issue=3|date=January 1998|url=http://www1.bell-labs.com/project/blast/wpc-v6n3.pdf|id={{ISSN|0929-6212}}(paper), {{ISSN|1572-834X}}(online) {{doi|10.1023/A:1008889222784}}}}</ref> the substantial [[channel capacity]] gains in using [[multiple-input multiple-output|antenna arrays at both ends of the link]]. An alternative approach to utilising multiple antennas relies on having multiple transmit antennas and only optionally multiple receive antennas. Proposed by [[Vahid Tarokh]], [[Nambi Seshadri]] and [[Robert Calderbank]], these space–time codes<ref name="sttc">{{cite journal|author=Vahid Tarokh, Nambi Seshadri, and A. R. Calderbank|title=Space–timeSpace–time codes for high data rate wireless communication: Performance analysis and code construction|journal=IEEE Transactions on Information Theory|pages=744–765744–765|volume=44|issue=2|date=March 1998|id={{doi|10.1109/18.661517}}}}</ref>(STCs) achieve significant [[bit error rate|error rate]] improvements over single-antenna [[forward error correction|error-correcting codes]]. Their original scheme was based on [[convolutional code|trellis codes]] but the simpler [[block code]]s were utilised by [[Siavash Alamouti]]<ref name="alamouti">{{cite journal|author=S.M. Alamouti|title=A simple transmit diversity technique for wireless communications|journal=IEEE Journal on Selected Areas in Communications|pages=1451–14581451–1458|volume=16|issue=8|date=October 1998|url=http://www.stanford.edu/~leipoo/ee359/alamouti_1.pdf|id={{doi|10.1109/49.730453}}}}</ref>, and later [[Vahid Tarokh]], [[Hamid Jafarkhani]] and [[Robert Calderbank]]<ref name="stbc">{{cite journal|author=Vahid Tarokh, Hamid Jafarkhani, and A. R. Calderbank|title=Space–timeSpace–time block codes from orthogonal designs|journal=[[IEEE Transactions on Information Theory]]|pages=744–765744–765|volume=45|issue=5|date=July 1999|url=http://www.mast.queensu.ca/~math800/W03/papers/TrkhJafarkCldb_IT99.pdf|id={{doi|10.1109/18.771146}}}}</ref> to develop space–time block-codes (STBCs). STC involves the transmission of multiple redundant copies of data to compensate for [[fading]] and [[thermal noise]]. In the case of STBC, the data stream to be transmitted is encoded in [[block code|blocks]], which are distributed among spaced antennas and across time. 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.
An STBC is usually represented by a [[matrix (mathematics)|matrix]]. Each row represents a time slot and each column represents one antenna's transmissions over time.
==Design of STBCs==
The design of STBCs is based on the so-called diversity criterion derived by Tarokh et. al in their earlier paper on [[space–time trellis codescode]]s.<ref name="sttc" /> Orthogonal STBCs can be shown to achieve the maximum diversity allowed by this criterion.
===Diversity criterion===
\end{bmatrix}
</math>
has to be full-[[rank (linear algebra)|rank]] for any pair of distinct codewords <math>\mathbf{c}</math> and <math>\mathbf{e}</math> to give the maximum possible diversity order of <math>n_Tn_R</math>. If instead, <math>\mathbf{B}(\mathbf{c},\mathbf{e})</math> has minimum rank <math>b</math> over the set of pairs of distinct codewords, then the space-timespace–time code offers diversity order <math>bn_R</math>. An examination of the example STBCs shown [[#Encoding|below]] reveals that they all satisfy this criterion for maximum diversity.
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-timespace–time trellis code|space–time trellis codes]]s 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 1998<ref name="alamouti" />, although he did not coin the term "space–timespace–time block code" himself. It was designed for a two-transmit antenna system and has the coding matrix:
:<math>C_2 = \begin{bmatrix}
s_1 & s_2\\
===Higher order STBCs===
Tarokh et. al discovered, using the arcane theory of orthogonal designs, a set of STBCs<ref name="stbc" /><ref name="perform">{{cite journal|author=Vahid Tarokh, Hamid Jafarkhani, and A. Robert Calderbank|title=Space–timeSpace–time block coding for wireless communications: performance results|journal=IEEE Journal on Selected Areas in Communications|pages=451–460451–460|volume=17|issue=3|date=March 1999|url=http://www.mast.queensu.ca/~math800/W03/papers/TrkhJafarkCldb_JSAC99.pdf|id={{doi|10.1109/49.753730}}}}</ref> 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|decoding]] scheme that goes with their codes under perfect [[channel state information]] assumption.
====3 transmit antennas====
</math>.
These codes achieve rate-1/2 and rate-3/4 respectively, as for their 3-antenna counterparts. <math>C_{4,3/4}</math> exhibits the same uneven power problems as <math>C_{3,3/4}</math>. An improved version of <math>C_{4,3/4}</math> is<ref>{{cite journal|author=G. Ganesan and P. Stoica|title=Space–timeSpace–time block codes: A maximum SNR approach|journal=IEEE Transactions on Information Theory|pages=1650–16561650–1656|volume=47|issue=4|date=May 2001|id={{doi|10.1109/18.923754}}}}</ref>
:<math>
C_{4,3/4}=
==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 three antennas, the maximum possible rate is 3/4<ref name="bounds">{{cite journal|author=Haiquan Wang and Xiang-Gen Xia|title=Upper bounds of rates of complex orthogonal space–timespace–time block codes|journal=IEEE Transactions on Information Theory|pages=2788–27962788–2796|volume=49|issue=10|date=October 2003|id={{doi|10.1109/TIT.2003.817830}}}}</ref>. Codes have been designed 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, <math>T</math> 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!<ref>{{cite journal|author=Weifeng Su, Xiang-Gen Xia, and K. J. Ray Liu|title=A systematic design of high-rate complex orthogonal space-time block codes|journal=IEEE Communications Letters|pages=380–382|volume=8|issue=6|date=June 2004|id={{doi|10.1109/LCOMM.2004.827429}}}}</ref>
It has been [[conjecture]]d<ref name="bounds" />, but not proven, that the highest rate any <math>n_T</math>-antenna code can achieve is
==See also==
*[[Space time coding|Space–timeSpace–time code]]
*[[Space-time trellis code|Space–timeSpace–time trellis code]]
*[[:Category:Wireless communications|Wireless communications]]
*[[Differential space-timespace–time code]]
==References==
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