Space–time block code: Difference between revisions

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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}}</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|format=PDF|doi=10.1109/18.771146|deadurl=yes|archiveurl=https://web.archive.org/web/20091229160912/http://www.mast.queensu.ca/~math800/W03/papers/TrkhJafarkCldb_IT99.pdf|archivedate=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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===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|last-author-amp=yes|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|doi=10.1109/49.753730|format=PDF|deadurl=yes|archiveurl=https://web.archive.org/web/20091229175920/http://www.mast.queensu.ca/~math800/W03/papers/TrkhJafarkCldb_JSAC99.pdf|archivedate=2009-12-29|df=}}</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====

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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 |lastauthoramp=yes |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|doi=10.1109/TIT.2003.817830|citeseerx=10.1.1.134.6261 }}</ref> Codes have been designed which achieve a good proportion of this, but they have very long block-length. This makes them unsuitable for practical use, because decoding cannot proceed until ''all'' transmissions in a block have been received, and 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|author1=Weifeng Su |author2=Xiang-Gen Xia |author3=K. J. Ray Liu |last-author-amp=yes |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|doi=10.1109/LCOMM.2004.827429|citeseerx=10.1.1.420.1452 }}</ref>

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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==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&ndash;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}