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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<ref>{{refcite journal|telatarauthor=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>{{refcite 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|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]]. InAn aalternative seminalapproach paper,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]], proposedthese space-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|id={{refdoi|sttc10.1109/18.661517}}}}</ref>(STCs) asachieve practicalsignificant schemes[[bit thaterror canrate|error approachrate]] theseimprovements theoreticalover gainssingle-antenna [[forward error correction|error-correcting codes]]. TheseTheir schemesoriginal werescheme was based on [[convolutional code|trellis codes.]] but the simpler [[block code]]s were utilised by [[Siavash Alamouti]]<ref name="alamouti">{{refcite journal|alamoutiauthor=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|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">{{refcite journal|stbc}}author=Vahid thenTarokh, proposedHamid space-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|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 [[multiple-inputfading]] multiple-outputand [[thermal noise]]. wirelessIn the case of STBC, the data systemsstream to achievebe significanttransmitted is encoded in [[bitblock error ratecode|error rateblocks]] improvement, which are thedistributed topicamong ofspaced thisantennas articleand 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. ▼STC involves the transmission of multiple redundant copies of data to compensate for [[Fade (radio)|fading]] and 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.
:<math>
\mbox{time-slots}
Here, <math>s_{ij}</math> is the [[modulation|modulated]] symbol to be transmitted in time slot <math>i</math> from antenna <math>j</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{{<ref label|name="stbc|4|a}}" />. If a block encodes <math>k</math> symbols, the code-rate is
:<math>r = \frac{k}{T} </math>.
===Orthogonality===
STBCs as originally introduced, and as usually studied, are [[orthogonal]]. This means that the STBC is designed such that the [[vector (spatial)|vector]]s representing any pair of ''columns'' taken from the coding matrix is orthogonal. The result of this is simple, [[linear]], [[optimization (mathematics)|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|Alamouti's code]]).
There are also '[[#Quasi-orthogonal STBCs|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 antheir earlier paper{{ on space–time trellis codes.<ref label|name="sttc|5|a}}." /> Orthogonal STBCs can be shown to achieve the maximum diversity allowed by this criterion.
===Diversity criterion===
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.
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 code|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 1998{{<ref label|name="alamouti|3|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, by computer search, a set of STBCs<ref name="stbc" /><ref name="perform">{{refcite labeljournal|stbcauthor=Vahid Tarokh, Hamid Jafarkhani, and A. Robert Calderbank|4title=Space–time block coding for wireless communications: performance results|b}}journal=IEEE Journal on Selected Areas in Communications|pages=451–460|volume=17|issue=3|date=March 1999|url=http://www.mast.queensu.ca/~math800/W03/papers/TrkhJafarkCldb_JSAC99.pdf|id={{refdoi|perform10.1109/49.753730}}}}</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.
</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>{{refcite journal|maxsnrauthor=G. Ganesan and P. Stoica|title=Space–time block codes: A maximum SNR approach|journal=IEEE Transactions on Information Theory|pages=1650–1656|volume=47|issue=4|date=May 2001|id={{doi|10.1109/18.923754}}}}</ref>
:<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{{<ref label|name="perform|6|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 3three antennas, the maximum possible rate is 3/4<ref name="bounds">{{refcite journal|author=Haiquan Wang and Xiang-Gen Xia|title=Upper bounds of rates of complex orthogonal space–time block codes|journal=IEEE Transactions on Information Theory|pages=2788–2796|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>{{refcite 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 label|name="bounds|8|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:<ref>{{refcite 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|id={{doi|10.1109/26.898239}}}}</ref>
:<math>C_{4,1} =
\begin{bmatrix}
==References==
<references />
#{{note|telatar}}{{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}}
#:Later published as {{cite journal|author=I. Emre Telatar|title=Capacity of multi-antenna gaussian channels|journal=European Transactions on Telecommunications|pages=585–595|volume=10|issue=6|date=November 1999|id=}}
#{{note|limits}}{{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|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}}}}
#{{note|alamouti}}{{note label|alamouti|3|a}}{{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|url=http://www.stanford.edu/~leipoo/ee359/alamouti_1.pdf|id={{doi|10.1109/49.730453}}}}
#{{note|stbc}}{{note label|stbc|4|a}}{{note label|stbc|4|b}}{{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|id={{doi|10.1109/18.771146}}}}
#{{note|sttc}}{{note label|sttc|5|a}}{{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|id={{doi|10.1109/18.661517}}}}
#{{note|perform}}{{note label|perform|6|a}}{{cite journal|author=Vahid Tarokh, Hamid Jafarkhani, and A. Robert Calderbank|title=Space–time block coding for wireless communications: performance results|journal=IEEE Journal on Selected Areas in Communications|pages=451–460|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}}}}
#{{note|maxsnr}}{{cite journal|author=G. Ganesan and P. Stoica|title=Space–time block codes: A maximum SNR approach|journal=IEEE Transactions on Information Theory|pages=1650–1656|volume=47|issue=4|date=May 2001|id={{doi|10.1109/18.923754}}}}
#{{note|bounds}}{{note label|bounds|8|a}}{{cite journal|author=Haiquan Wang and Xiang-Gen Xia|title=Upper bounds of rates of complex orthogonal space–time block codes|journal=IEEE Transactions on Information Theory|pages=2788–2796|volume=49|issue=10|date=October 2003|id={{doi|10.1109/TIT.2003.817830}}}}
#{{note|systematic}}{{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}}}}
#{{note|quasi}}{{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|id={{doi|10.1109/26.898239}}}}
[[Category:Wireless communications]]
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