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{{Short description|WiFi option}}
{{About-distinguish|coherent
'''Space–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 the transmitted signal must traverse a potentially difficult environment with [[scattering]], [[Reflection (physics)|reflection]], [[refraction]] and so on and may then be further corrupted by [[thermal noise]] in the [[Receiver (radio)|receiver]] means that some of the received copies of the data may be closer to the original signal than others. This redundancy results in a higher chance of being able to use one or more of the received copies to correctly decode the received signal. In fact, [[Space–time code|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==
Most work on wireless communications until the early 1990s had focused on having an antenna array at only one end of the wireless link — usually at the receiver.<ref>E. Larsson and P. Stoica,''Space-Time Block Coding For Wireless Communications''. Cambridge University Press, UK, 2003 (Chinese Edition, 2006).</ref> Seminal papers by [[Gerard J. Foschini]] and Michael J. Gans,<ref>{{cite journal|author1=Gerard J. Foschini |author2=Michael. J. Gans |
An alternative approach to utilizing 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|author1=Vahid Tarokh |author2=Nambi Seshadri |author3=A. R. Calderbank |
▲Most work on wireless communications until the early 1990s had focused on having an antenna array at only one end of the wireless link — usually at the receiver.<ref>E. Larsson and P. Stoica,''Space-Time Block Coding For Wireless Communications''. Cambridge University Press, UK, 2003 (Chinese Edition, 2006).</ref> Seminal papers by Gerard J. Foschini and Michael J. Gans,<ref>{{cite journal|author1=Gerard J. Foschini |author2=Michael. J. Gans |lastauthoramp=yes |title=On limits of wireless communications in a fading environment when using multiple antennas|journal=Wireless Personal Communications|pages=311–335|volume=6|issue=3|date=January 1998|doi=10.1023/A:1008889222784}}</ref> Foschini<ref>{{cite journal|author=Gerard J. Foschini|title=Layered space-time architecture for wireless communications in a fading environment when using multi-element antennas|journal=Bell Labs Technical Journal |pages=41–59|volume=1|date=Autumn 1996|doi=10.1002/bltj.2015|issue=2}}</ref> and Emre Telatar<ref>{{cite journal|author=I. Emre Telatar|title=Capacity of multi-antenna gaussian channels|journal=European Transactions on Telecommunications|date=November 1999|pages=585–595|volume=10|doi=10.1002/ett.4460100604|issue=6|url=http://infoscience.epfl.ch/record/125918}}</ref> enlarged the scope of wireless communication possibilities by showing that for the highly scattering environment substantial capacity gains are enabled when antenna arrays are used at both ends of a link.
▲An alternative approach to utilizing 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|author1=Vahid Tarokh |author2=Nambi Seshadri |author3=A. R. Calderbank |last-author-amp=yes |title=Space–time codes for high data rate wireless communication: Performance analysis and code construction|journal=IEEE Transactions on Information Theory|pages=744–765|volume=44|issue=2|date=March 1998|doi=10.1109/18.661517|citeseerx=10.1.1.112.4293 }}</ref> (STCs) achieve significant [[bit error rate|error rate]] improvements over single-antenna systems. 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=1451–1458|volume=16|issue=8|date=October 1998|doi=10.1109/49.730453}}</ref> and later [[Vahid Tarokh]], [[Hamid Jafarkhani]] and [[Robert Calderbank]]<ref name="stbc">{{cite journal|author1=Vahid Tarokh|author2=Hamid Jafarkhani|author3=A. R. Calderbank|last-author-amp=yes|title=Space–time block codes from orthogonal designs|journal=[[IEEE Transactions on Information Theory]]|pages=744–765|volume=45|issue=5|date=July 1999|url=http://www.mast.queensu.ca/~math800/W03/papers/TrkhJafarkCldb_IT99.pdf|doi=10.1109/18.771146|url-status=dead|archive-url=https://web.archive.org/web/20091229160912/http://www.mast.queensu.ca/~math800/W03/papers/TrkhJafarkCldb_IT99.pdf|archive-date=2009-12-29|df=|citeseerx=10.1.1.138.4537}}</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 hope that some of them may arrive at the receiver in a better state than others. In the case of STBC in particular, 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.
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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–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–time trellis code]]s which provide both diversity and coding gain since they spread a conventional trellis code over space and time.
==Encoding==
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===Higher order STBCs===
Tarokh et al. discovered a set of STBCs<ref name="stbc" /><ref name="perform">{{cite journal|author1=Vahid Tarokh|author2=Hamid Jafarkhani|author3=A. Robert Calderbank|
====3 transmit antennas====
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</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|author1=G. Ganesan |author2=P. Stoica |
:<math>
C_{4,3/4}=
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==Decoding==
One particularly attractive feature of orthogonal 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 <math>t</math>, the signal <math>r_t^j</math> received at antenna <math>j</math> is:
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==Rate limits==
Apart from there being no full-rate, complex, orthogonal STBC for more than 2 antennas, it has been further shown that, for more than two antennas, the maximum possible rate is 3/4.<ref name="bounds">{{cite journal|author1=Haiquan Wang |author2=Xiang-Gen Xia |
It has been proven<ref name="COD">{{cite journal|author=Xue-Bin Liang|title=Orthogonal Designs With Maximum Rates|journal=IEEE Transactions on Information Theory|pages=2468–2503|volume=49|issue=10|date=October 2003|doi=10.1109/TIT.2003.817426}}</ref> that the highest rate any <math>n_T</math>-antenna code can achieve is
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: <math>r_\max = \frac{n_0 + 1}{2n_0},</math>
where <math>n_T = 2n_0</math> or <math>n_T = 2n_0 - 1</math>, if no linear processing is allowed in the code matrix (the above maximal rate proved in<ref name="COD"/> only applies to the original definition of orthogonal designs, i.e., any entry in the matrix is <math>0, c_i, -c_i, c_i^*,</math>, or <math>-c_i^*</math>, which forces that any variable <math>c_i</math> can not be repeated in any column of the matrix). This rate limit is conjectured to hold for any complex orthogonal
==Quasi-orthogonal STBCs==
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==See also==
*[[Multiple-input and multiple-output]] (MIMO)
*[[
*[[Space–time code]]
*[[Space–time trellis code]]
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{{DEFAULTSORT:Space-time block code}}
[[Category:Wireless]]
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