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==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<sup>[[#References{{ref|1]]</sup>telatar}} which, in 1998, inspired [[Gerard Foschini]] to demonstrate<sup>[[#References{{ref|2]]</sup>limits}} the substantial [[channel capacity]] improvements available by correctly using antenna arrays at ''both'' ends of the link. Shortly afterwards, [[Siavash Alamouti]] <sup>[[#References{{ref|3]]</sup>alamouti}} and [[ Vahid Tarokh]] , [[Hamid Jafarkhani]] and [[RobRobert Calderbank]] <sup>[[#References{{ref|4]]</sup>stbc}} demonstrated how to use these [[multiple-input multiple-output]] systems to achieve significant [[bit error rate|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 code|space–time trellis code]]s (STTCs){{ref|sttc}} 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 code|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 (mathematics)|matrix]] form. In this form, each ''row'' represent a time-slot and each ''column'' represents one antenna's transmissions over time e.g.
Here, <math>s_{ij}</math> is the [[modulation|modulated]] symbol to be transmitted from antenna <math>j</math> in time-slot <math>i</math>. There are to be <math>T</math> time-slots and <math>n_T</math> transmit antennas as well as <math>n_R</math> receive antennas. This block is usually considered to be of 'length' <math>T</math>
The [[code rate]] of an STBC measures how many symbols per time-slot it transmits on average over the course of one block<sup>[[#References{{ref label|stbc|4]]</sup>|a}}. If a block encodes <math>k</math> symbols, the code-rate is
:<math>r = \frac{k}{T} </math>.
==Design of STBCs==
The design of STBCs is based on the so-called ''diversity criterion'' derived by Tarokh et. al in an earlier paper<sup>[[#References{{ref label|sttc|5]]</sup>|a}}. Orthogonal STBCs can be shown to achieve the maximum diversity allowed by this criterion.
===Diversity criterion===
==Encoding==
===Alamouti's code===
Alamouti invented the simplest of all the STBCs in 1998<sup>[[#References{{ref label|alamouti|3]]</sup>|a}}, 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:
:<math>C_2 = \begin{bmatrix}
s_1 & s_2\\
===Higher order STBCs===
Tarokh et. al discovered<sup>[[#References|4]],[[#References|6]]</sup>, by computer search, a set of STBCs{{ref label|stbc|4|b}}{{ref|perform}} 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.
</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<sup>[[#References{{ref|7]]</sup>maxsnr}}
:<math>
C_{4,3/4}=
where <math>\alpha_{ij}</math> is the path gain from transmit antenna <math>i</math> to receive antenna <math>j</math> and <math>n_t^j</math> is a sample of [[additive white Gaussian noise|additive]] [[white noise|white]] [[Gaussian noise]] ([[AWGN]]).
The maximum-likelihood detection rule<sup>[[#References{{ref label|perform|6]]</sup>|a}} is to form the decision variables
:<math>R_i = \sum_{t=1}^{n_T}\sum_{j=1}^{n_R}r_t^j\alpha_{\epsilon_{t}(i)j}\delta_t(i)</math>
where <math>\delta_k(i)</math> is the sign of <math>s_i</math> in the <math>k</math><sup>th</sup> row of the coding matrix, <math>\epsilon_k(p)=q</math> denotes that <math>s_p</math> is (up to a sign difference), the <math>(k,q)</math> element of the coding matrix,
==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 3/4<sup>[[#References{{ref|8]]</sup>bounds}}. Codes have been designed<sup>[[#References|9]]</sup> 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|systematic}}
It has been [[conjecture]]d<sup>[[#References{{ref label|bounds|8]]</sup>|a}}, but not proven, that the highest rate any <math>n_T</math>-antenna code can achieve is
:<math>r_{\mathrm{max}} = \frac{n_0 + 1}{2n_0}</math>,
where <math>n_T = 2n_0</math> or <math>n_t = 2n_0 - 1</math>.
==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<sup>[[#References|10]]</sup>:{{ref|quasi}}
:<math>C_{4,1} =
\begin{bmatrix}
==References==
#'''"{{note|telatar}}{{journal reference|Author=I. Emre Telatar"''', [http://mars.bell-labs.com/papers/proof/proof.pdf "|Title=Capacity of multi-antenna gaussian channels"], ''|Journal=Technical Memorandum, Bell Laboratories'', Oct.|Date=October 1995|Pages=|URL=http://mars.bell-labs.com/papers/proof/proof.pdf}}
#'''{{note|limits}}{{journal reference|Author=Gerard. J. Foschini and Michael. J. Gans''', [http://www1.bell-labs.com/project/blast/wpc-v6n3.pdf "|Title=On limits of wireless communications in a fading environment when using multiple antennas"], ''|Journal=Wireless Personal Communications'', vol. 6, no. 3, pp. |Pages=311–335,|Volume=6|Issue=3|Date=January Jan1998|URL=http://www1.bell-labs.com/project/blast/wpc-v6n3.pdf|ID={{ISSN|0929-6212}}(paper), 1998{{ISSN|1572-834X}}(online) {{doi|10.1023/A:1008889222784}}}}
#'''{{note|alamouti}}{{note label|alamouti|3|a}}{{journal reference|Author=S.M. Alamouti''', [http://www.stanford.edu/~leipoo/ee359/alamouti_1.pdf "|Title=A simple transmit diversity schemetechnique for wireless communications"], ''[[|Journal=IEEE]] J.Journal Select.on Selected Areas Commun''., vol. 16, no. 8, pp.in Communications|Pages=1451–1458,|Volume=16|Issue=8|Date=Otober Oct1998|URL=http://www. 1998stanford.edu/~leipoo/ee359/alamouti_1.pdf|ID={{doi|10.1109/49.730453}}}}
#'''{{note|stbc}}{{note label|stbc|4|a}}{{note label|stbc|4|b}}{{journal reference|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 "Space–time block codes from orthogonal designs"], ''IEEE Trans|ID={{doi|10. Inform. Theory'', vol. 45, no. 5, pp 1456–1467, Jul. 19991109/18.771146}}}}
#'''{{note|sttc}}{{note label|sttc|5|a}}{{journal reference|Author=Vahid Tarokh, Nambi Seshadri, and A. R. Calderbank''', "Space–time|Title=Space–time codes for high data rate wireless communication: Performance analysis and code construction," |Journal=IEEE Trans.Transactions Inform.on Information Theory, vol. |Pages=744–765|Volume=44,|Issue=2|Date=March pp1998|ID={{doi|10. 744–765, Mar. 19981109/18.661517}}}}
#'''{{note|perform}}{{note label|perform|6|a}}{{journal reference|Author=Vahid Tarokh, Hamid Jafarkhani, and A. Robert Calderbank''', [http://www.mast.queensu.ca/~math800/W03/papers/TrkhJafarkCldb_JSAC99.pdf "|Title=Space–time block coding for wireless communications: performance results"],|Journal=IEEE Journal ''IEEEon J. Select.Selected Areas Commun''., vol. 17, pp.in Communications|Pages=451–460,|Volume=17|Issue=3|Date=March Mar1999|URL=http://www. 1999mast.queensu.ca/~math800/W03/papers/TrkhJafarkCldb_JSAC99.pdf|ID={{doi|10.1109/49.753730}}}}
#'''{{note|maxsnr}}{{journal reference|Author=G. Ganesan and P. Stoica''', "|Title=Space–time block codes: A maximum [[SNR]] approach", ''|Journal=IEEE Trans.Transactions Inform.on Information Theory'', pp. |Pages=1650–1656,|Volume=47|Issue=4|Date=May Apr2001|ID={{doi|10. 20011109/18.923754}}}}
#'''H.{{note|bounds}}{{note label|bounds|8|a}}{{journal reference|Author=Haiquan Wang and X.Xiang-G.Gen Xia''', "|Title=Upper bounds of rates of complex orthogonal space–time block codes|Journal=IEEE fromTransactions complexon [[orthogonal designs]]", ''IEEE Trans. Inform.Information Theory'', vol. 49, pp. |Pages=2788–2796,|Volume=49|Issue=10|Date=October Oct2003|ID={{doi|10.1109/TIT. 2003.817830}}}}
#'''{{note|systematic}}{{journal reference|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 Commun. Lett.'', vol. 8, no. 6, pp.Communications Letters|Pages=380–382,|Volume=8|Issue=6|Date=June Jun2004|ID={{doi|10.1109/LCOMM. 2004.827429}}}}
#'''{{note|quasi}}{{journal reference|Author=Hamid Jafarkhani''', "|Title=A quasi-orthogonal space–time block code", ''|Journal=IEEE Trans.Transactions Commun.", vol. 49, no. 1, pp.on Communications|Pages=1–4, Jan.|Volume=49|Issue=1|Date=January 2001|ID={{doi|0.1109/26.898239}}}}
[[Category:Wireless communications]]
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