Spring 2006Spring 2006EE228a Communication NetworksEE228a Communication Networks11MingMing--Yang ChenYang ChenDepartment of EECS, U.C. BerkeleyDepartment of EECS, U.C. BerkeleyConstellation Expansion Free QuasiConstellation Expansion Free Quasi--Orthogonal Orthogonal SpaceSpace--Time Block Codes with Full DiversityTime Block Codes with Full Diversity22Spring 2006Spring 2006EE228a Communication NetworksEE228a Communication NetworksTodayTodayIn a wireless relay network, the relay nodes encode In a wireless relay network, the relay nodes encode the signals they receive from the source node into the signals they receive from the source node into a distributed spacea distributed space--time block code (STBC) and time block code (STBC) and transmit the encoded signals to the sink node. It is transmit the encoded signals to the sink node. It is shown that at high SNR, the diversity gain shown that at high SNR, the diversity gain achieved by this scheme is asymptotically equal to achieved by this scheme is asymptotically equal to that of a multiplethat of a multiple--antenna system whose number antenna system whose number of of TxTxantennas is the same as the number of relay antennas is the same as the number of relay nodes in the network, which means that the relay nodes in the network, which means that the relay nodes work as if they can entirely cooperate and nodes work as if they can entirely cooperate and have full knowledge of the messages.have full knowledge of the messages.33Spring 2006Spring 2006EE228a Communication NetworksEE228a Communication NetworksTodayTodayNotions of Notions of orthogonalorthogonaland and quasiquasi--orthogonalorthogonalspacespace--time block codes and their differences.time block codes and their differences.Previous work and our improved schemes.Previous work and our improved schemes.A little bit details:A little bit details:constellation rotated method.constellation rotated method.construction of our new codes.construction of our new codes.new designs yield new advantages.new designs yield new advantages.Conclusions and open questions.Conclusions and open questions.Spring 2006Spring 2006EE228a Communication NetworksEE228a Communication Networks44STBC from Orthogonal DesignsSTBC from Orthogonal DesignsAn An orthogonal STBCorthogonal STBCSS(rate(rate--1 and complex) of size 1 and complex) of size nnis an is an nn××nnmatrix with entries the indeterminates matrix with entries the indeterminates ±±zz11, , ±±zz22,,, , ±±zznn, , their conjugations, or multiples of them by their conjugations, or multiples of them by iiwhere where ii2 2 = = −−1, 1, such thatsuch thatSSSS= (= (zz112 2 + + + + zznn22))IInn..For example, For example, AlamoutiAlamouti’’ss2 2 ××22matrix:matrix:Complex orthogonal designs with rateComplex orthogonal designs with rate--1 only exist as 1 only exist as nn= 2.= 2.Spring 2006Spring 2006EE228a Communication NetworksEE228a Communication Networks55QuasiQuasi--Orthogonal STBCOrthogonal STBCWhen When nn= 4, a quasi= 4, a quasi--orthogonal STBC is a 4 orthogonal STBC is a 4 ××4 STBC 4 STBC (rate(rate--1 and complex) that divides the four columns of 1 and complex) that divides the four columns of transmission matrix into two pairs, where the columns transmission matrix into two pairs, where the columns within each pair are not orthogonal but different pairs within each pair are not orthogonal but different pairs are orthogonal to each other.are orthogonal to each other.Most famous instances:Most famous instances:H. H. JafarkhaniJafarkhaniO. Tirkkonen, A. O. Tirkkonen, A. BoariuBoariu, , and A. and A. HottinenHottinen66Spring 2006Spring 2006EE228a Communication NetworksEE228a Communication NetworksThe ML decoder requires a time complexityThe ML decoder requires a time complexity--OO((nmnm22), where ), where mmis the cardinality of signal constellations.is the cardinality of signal constellations.Properties of QuasiProperties of Quasi--Orthogonal STBCOrthogonal STBC77Spring 2006Spring 2006EE228a Communication NetworksEE228a Communication NetworksHowever, the diversity (rank of the matrix) can be However, the diversity (rank of the matrix) can be either 2 or 4, instead of full diversity in the orthogonal either 2 or 4, instead of full diversity in the orthogonal cases becausecases becauseThis results in that the quasiThis results in that the quasi--orthogonal codes perform orthogonal codes perform better than the orthogonal codes with rates less than 1 better than the orthogonal codes with rates less than 1 at low SNR, but not at high SNR. at low SNR, but not at high SNR. Properties of QuasiProperties of Quasi--Orthogonal STBCOrthogonal STBC88Spring 2006Spring 2006EE228a Communication NetworksEE228a Communication NetworksPossible CompensationsPossible CompensationsRecently, Xia et. al. presented another scheme called Recently, Xia et. al. presented another scheme called constellation rotated methodconstellation rotated method..By rotating the signal constellations with an angle computed in By rotating the signal constellations with an angle computed in advance (may be constellationadvance (may be constellation--selective) for half the input selective) for half the input variables, the resulted system is proved to have full diversity.variables, the resulted system is proved to have full diversity.The modified system performs better than the orthogonal codes The modified system performs better than the orthogonal codes with rates less than 1 at all SNR.with rates less than 1 at all SNR.However, the tradeoff is that the size and shape of signal However, the tradeoff is that the size and shape of signal constellations become double and irregular since half the constellations become double and irregular since half the transmitted symbols come from the rotated constellation, transmitted symbols come from the rotated constellation, which is not contained in the original one due to that angle.which is not contained in the original one due to that angle.99Spring 2006Spring 2006EE228a Communication NetworksEE228a Communication NetworksSome Observations ......Some Observations ......Fact 1: Fact 1: Each of the examples previously illustrated constitutes a Each of the examples previously illustrated constitutes a ring as ring as zz11, , zz22,,, , zznnrun over therun over thefield of complex field of complex numbers. numbers. Fact 2 (main lemma): Fact 2 (main lemma): An STBC An
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