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{{Short description|WiFi option}}
{{About-distinguish|coherent space–time block codes|Differential space–time code}}
'''Space–time block coding''' is a technique used in [[wireless|wireless communications]] to transmit multiple copies of a data stream across a number of [[
==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|
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|
▲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|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=311–335|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}}</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|author=Vahid Tarokh, Nambi Seshadri, and A. R. Calderbank|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}}</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|author=Vahid Tarokh, Hamid Jafarkhani, and A. R. Calderbank|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|format=PDF|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 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==
===Alamouti's code===
[[File:Alamouti 06.png|thumb|400px|Bit-error ratio performance of the simulated Alamouti transmission over the partially time-invariant MISO and MIMO channels (K = 0.6). Note that the case of Alamouti 2x1 completely matched with the theoretical 2nd order diversity, however, Alamouti 2x2 has the better BER performance due to additional array gain.<ref>[https://www.mathworks.com/help/comm/examples/introduction-to-mimo-systems.html Introduction to MIMO Systems (MathWorks)]</ref>]]
[[Siavash Alamouti]] invented the simplest of all the STBCs in 1998,<ref name="alamouti" /> although he did not coin the term "space–time block code" himself. It was designed for a two-transmit antenna system and has the coding matrix:
:<math>C_2 = \begin{bmatrix}
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This is a very special STBC. It is the '''only''' orthogonal STBC that achieves rate-1.<ref name="sttc" /> 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 number|complex]] modulation symbols. Since almost all [[constellation diagram]]s 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|Rate limits]]' for more detail.
The significance of Alamouti's proposal in 1998 is that it was the first demonstration of a method of encoding which enables full diversity with ''linear'' processing at the receiver. Earlier proposals for [[transmit diversity]] required processing schemes which scaled ''exponentially'' with the number of transmit antennas. Furthermore, it was the first [[open-loop controller|open-loop]] [[transmit diversity]] technique which had this capability. Subsequent generalizations of Alamouti's concept have led to a tremendous impact on the wireless communications industry.
===Higher order STBCs===
Tarokh et al. discovered a set of STBCs<ref name="stbc" /><ref name="perform">{{cite journal|
====3 transmit antennas====
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</math>
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 <math>C_{3,3/4}</math> is that it has uneven power among the symbols it transmits. This means that the signal does not have a [[constant
====4 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|
:<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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:<math>r_t^j = \sum_{i=1}^{n_T}\alpha_{ij}s_t^i + n_t^j,</math>
where <math>\alpha_{ij}</math> is the path gain from transmit antenna <math>i</math> to receive antenna <math>j</math>, <math>s_t^i</math> is the signal transmitted by transmit antenna <math>i</math> and <math>n_t^j</math> is a sample of [[additive white
The maximum-likelihood detection rule<ref name="perform" /> is to form the decision variables
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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|
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
==Quasi-orthogonal STBCs==
These codes exhibit partial orthogonality and provide only part of the diversity gain mentioned [[#Diversity criterion|above]]. An example reported by [[Hamid Jafarkhani]] is:<ref>{{cite journal|author=Hamid Jafarkhani|title=A quasi-orthogonal space–time block code|journal=IEEE Transactions on Communications|pages=1–4|volume=49|issue=1|date=January 2001|doi=10.1109/26.898239|citeseerx=10.1.1.136.1830}}</ref>
:<math>C_{4,1} =
\begin{bmatrix}
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==See also==
*[[Multiple-input and multiple-output]] (MIMO)
*[[
*[[Space–time code]]
*[[Space–time trellis code]]
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==References==
{{
{{DEFAULTSORT:Space-time block code}}
[[Category:Wireless]]
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