25 research outputs found
A Novel Construction of Multi-group Decodable Space-Time Block Codes
Complex Orthogonal Design (COD) codes are known to have the lowest detection
complexity among Space-Time Block Codes (STBCs). However, the rate of square
COD codes decreases exponentially with the number of transmit antennas. The
Quasi-Orthogonal Design (QOD) codes emerged to provide a compromise between
rate and complexity as they offer higher rates compared to COD codes at the
expense of an increase of decoding complexity through partially relaxing the
orthogonality conditions. The QOD codes were then generalized with the so
called g-symbol and g-group decodable STBCs where the number of orthogonal
groups of symbols is no longer restricted to two as in the QOD case. However,
the adopted approach for the construction of such codes is based on sufficient
but not necessary conditions which may limit the achievable rates for any
number of orthogonal groups. In this paper, we limit ourselves to the case of
Unitary Weight (UW)-g-group decodable STBCs for 2^a transmit antennas where the
weight matrices are required to be single thread matrices with non-zero entries
in {1,-1,j,-j} and address the problem of finding the highest achievable rate
for any number of orthogonal groups. This special type of weight matrices
guarantees full symbol-wise diversity and subsumes a wide range of existing
codes in the literature. We show that in this case an exhaustive search can be
applied to find the maximum achievable rates for UW-g-group decodable STBCs
with g>1. For this purpose, we extend our previously proposed approach for
constructing UW-2-group decodable STBCs based on necessary and sufficient
conditions to the case of UW-g-group decodable STBCs in a recursive manner.Comment: 12 pages, and 5 tables, accepted for publication in IEEE transactions
on communication
STBCs from Representation of Extended Clifford Algebras
A set of sufficient conditions to construct -real symbol Maximum
Likelihood (ML) decodable STBCs have recently been provided by Karmakar et al.
STBCs satisfying these sufficient conditions were named as Clifford Unitary
Weight (CUW) codes. In this paper, the maximal rate (as measured in complex
symbols per channel use) of CUW codes for is
obtained using tools from representation theory. Two algebraic constructions of
codes achieving this maximal rate are also provided. One of the constructions
is obtained using linear representation of finite groups whereas the other
construction is based on the concept of right module algebra over
non-commutative rings. To the knowledge of the authors, this is the first paper
in which matrices over non-commutative rings is used to construct STBCs. An
algebraic explanation is provided for the 'ABBA' construction first proposed by
Tirkkonen et al and the tensor product construction proposed by Karmakar et al.
Furthermore, it is established that the 4 transmit antenna STBC originally
proposed by Tirkkonen et al based on the ABBA construction is actually a single
complex symbol ML decodable code if the design variables are permuted and
signal sets of appropriate dimensions are chosen.Comment: 5 pages, no figures, To appear in Proceedings of IEEE ISIT 2007,
Nice, Franc
Asymptotically-Optimal, Fast-Decodable, Full-Diversity STBCs
For a family/sequence of STBCs , with
increasing number of transmit antennas , with rates complex symbols
per channel use (cspcu), the asymptotic normalized rate is defined as . A family of STBCs is said to be
asymptotically-good if the asymptotic normalized rate is non-zero, i.e., when
the rate scales as a non-zero fraction of the number of transmit antennas, and
the family of STBCs is said to be asymptotically-optimal if the asymptotic
normalized rate is 1, which is the maximum possible value. In this paper, we
construct a new class of full-diversity STBCs that have the least ML decoding
complexity among all known codes for any number of transmit antennas and
rates cspcu. For a large set of pairs, the new codes
have lower ML decoding complexity than the codes already available in the
literature. Among the new codes, the class of full-rate codes () are
asymptotically-optimal and fast-decodable, and for have lower ML decoding
complexity than all other families of asymptotically-optimal, fast-decodable,
full-diversity STBCs available in the literature. The construction of the new
STBCs is facilitated by the following further contributions of this paper:(i)
For , we construct -group ML-decodable codes with rates greater than
one cspcu. These codes are asymptotically-good too. For , these are the
first instances of -group ML-decodable codes with rates greater than
cspcu presented in the literature. (ii) We construct a new class of
fast-group-decodable codes for all even number of transmit antennas and rates
.(iii) Given a design with full-rank linear dispersion
matrices, we show that a full-diversity STBC can be constructed from this
design by encoding the real symbols independently using only regular PAM
constellations.Comment: 16 pages, 3 tables. The title has been changed.The class of
asymptotically-good multigroup ML decodable codes has been extended to a
broader class of number of antennas. New fast-group-decodable codes and
asymptotically-optimal, fast-decodable codes have been include
Maximum Rate of Unitary-Weight, Single-Symbol Decodable STBCs
It is well known that the Space-time Block Codes (STBCs) from Complex
orthogonal designs (CODs) are single-symbol decodable/symbol-by-symbol
decodable (SSD). The weight matrices of the square CODs are all unitary and
obtainable from the unitary matrix representations of Clifford Algebras when
the number of transmit antennas is a power of 2. The rate of the square
CODs for has been shown to be complex symbols per
channel use. However, SSD codes having unitary-weight matrices need not be
CODs, an example being the Minimum-Decoding-Complexity STBCs from
Quasi-Orthogonal Designs. In this paper, an achievable upper bound on the rate
of any unitary-weight SSD code is derived to be complex
symbols per channel use for antennas, and this upper bound is larger than
that of the CODs. By way of code construction, the interrelationship between
the weight matrices of unitary-weight SSD codes is studied. Also, the coding
gain of all unitary-weight SSD codes is proved to be the same for QAM
constellations and conditions that are necessary for unitary-weight SSD codes
to achieve full transmit diversity and optimum coding gain are presented.Comment: accepted for publication in the IEEE Transactions on Information
Theory, 9 pages, 1 figure, 1 Tabl
Construction of Block Orthogonal STBCs and Reducing Their Sphere Decoding Complexity
Construction of high rate Space Time Block Codes (STBCs) with low decoding
complexity has been studied widely using techniques such as sphere decoding and
non Maximum-Likelihood (ML) decoders such as the QR decomposition decoder with
M paths (QRDM decoder). Recently Ren et al., presented a new class of STBCs
known as the block orthogonal STBCs (BOSTBCs), which could be exploited by the
QRDM decoders to achieve significant decoding complexity reduction without
performance loss. The block orthogonal property of the codes constructed was
however only shown via simulations. In this paper, we give analytical proofs
for the block orthogonal structure of various existing codes in literature
including the codes constructed in the paper by Ren et al. We show that codes
formed as the sum of Clifford Unitary Weight Designs (CUWDs) or Coordinate
Interleaved Orthogonal Designs (CIODs) exhibit block orthogonal structure. We
also provide new construction of block orthogonal codes from Cyclic Division
Algebras (CDAs) and Crossed-Product Algebras (CPAs). In addition, we show how
the block orthogonal property of the STBCs can be exploited to reduce the
decoding complexity of a sphere decoder using a depth first search approach.
Simulation results of the decoding complexity show a 30% reduction in the
number of floating point operations (FLOPS) of BOSTBCs as compared to STBCs
without the block orthogonal structure.Comment: 16 pages, 7 figures; Minor changes in lemmas and construction
Algebraic Distributed Space-Time Codes with Low ML Decoding Complexity
"Extended Clifford algebras" are introduced as a means to obtain low ML
decoding complexity space-time block codes. Using left regular matrix
representations of two specific classes of extended Clifford algebras, two
systematic algebraic constructions of full diversity Distributed Space-Time
Codes (DSTCs) are provided for any power of two number of relays. The left
regular matrix representation has been shown to naturally result in space-time
codes meeting the additional constraints required for DSTCs. The DSTCs so
constructed have the salient feature of reduced Maximum Likelihood (ML)
decoding complexity. In particular, the ML decoding of these codes can be
performed by applying the lattice decoder algorithm on a lattice of four times
lesser dimension than what is required in general. Moreover these codes have a
uniform distribution of power among the relays and in time, thus leading to a
low Peak to Average Power Ratio at the relays.Comment: 5 pages, no figures. To appear in Proceedings of IEEE ISIT 2007,
Nice, Franc
Generalized Silver Codes
For an transmit, receive antenna system (
system), a {\it{full-rate}} space time block code (STBC) transmits complex symbols per channel use. The well known Golden code is an
example of a full-rate, full-diversity STBC for 2 transmit antennas. Its
ML-decoding complexity is of the order of for square -QAM. The
Silver code for 2 transmit antennas has all the desirable properties of the
Golden code except its coding gain, but offers lower ML-decoding complexity of
the order of . Importantly, the slight loss in coding gain is negligible
compared to the advantage it offers in terms of lowering the ML-decoding
complexity. For higher number of transmit antennas, the best known codes are
the Perfect codes, which are full-rate, full-diversity, information lossless
codes (for ) but have a high ML-decoding complexity of the order
of (for , the punctured Perfect codes are
considered). In this paper, a scheme to obtain full-rate STBCs for
transmit antennas and any with reduced ML-decoding complexity of the
order of , is presented. The codes constructed are
also information lossless for , like the Perfect codes and allow
higher mutual information than the comparable punctured Perfect codes for . These codes are referred to as the {\it generalized Silver codes},
since they enjoy the same desirable properties as the comparable Perfect codes
(except possibly the coding gain) with lower ML-decoding complexity, analogous
to the Silver-Golden codes for 2 transmit antennas. Simulation results of the
symbol error rates for 4 and 8 transmit antennas show that the generalized
Silver codes match the punctured Perfect codes in error performance while
offering lower ML-decoding complexity.Comment: Accepted for publication in the IEEE Transactions on Information
Theory. This revised version has 30 pages, 7 figures and Section III has been
completely revise