On the Maximal Rate of Non-Square STBCs from Complex Orthogonal Designs

A linear processing complex orthogonal design (LPCOD) is a ptimesn matrix epsiv, (pgesn) in k complex indeterminates x₁,x₂,...,x_k such that (i) the entries of epsiv are complex linear combinations of 0, plusmnx_i, i = 1,...,k and their conjugates, (ii) epsiv^Hepsiv = D, where epsiv^H is the Hermitian (conjugate transpose) of epsiv and D is a diagonal matrix with the (i,i)-th diagonal element of the form l₁⁽ⁱ⁾|x₁|² + l₂⁽ⁱ⁾|x₂|² +...+ l_k⁽ⁱ⁾|x_k|² where l_j⁽ⁱ⁾,i = 1,2,...,n, j = 1,2,...,k are strictly positive real numbers and the condition l₁⁽ⁱ⁾ = l₂⁽ⁱ⁾ =...= l_k⁽ⁱ⁾, called the equal- weights condition, holds for all values of i. For square designs it is known that whenever a LPCOD exists without the equal-weights condition satisfied then there exists another LPCOD with identical parameters with l₁⁽ⁱ⁾ = l₂⁽ⁱ⁾ =...=l_k⁽ⁱ⁾ = 1. This implies that the maximum possible rate for square LPCODs without the equal-weights condition is the same as that of square LPCODs with equal-weights condition. In this paper, this result is extended to a subclass of non-square LPCODs. It is shown that, a set of sufficient conditions is identified such that whenever a non- square (p>n) LPCOD satisfies these sufficient conditions and do not satisfy the equal-weights condition, then there exists another LPCOD with the same parameters n, k and p in the same complex indeterminates with l₁⁽ⁱ⁾=l₂⁽ⁱ⁾ =...= l_k ⁽ⁱ⁾ = 1

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 $n$ is a power of 2. The rate of the square CODs for $n = 2^a$ has been shown to be $\frac{a+1}{2^a}$ 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 $\frac{a}{2^{a-1}}$ complex symbols per channel use for $2^a$ 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

Asymptotically-Optimal, Fast-Decodable, Full-Diversity STBCs

For a family/sequence of STBCs $\mathcal{C}_1,\mathcal{C}_2,\dots$, with increasing number of transmit antennas $N_i$, with rates $R_i$ complex symbols per channel use (cspcu), the asymptotic normalized rate is defined as $\lim_{i \to \infty}{\frac{R_i}{N_i}}$. 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 $N>1$ and rates $R>1$ cspcu. For a large set of $\left(R,N\right)$ 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 ($R=N$) are asymptotically-optimal and fast-decodable, and for $N>5$ 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 $g > 1$, we construct $g$-group ML-decodable codes with rates greater than one cspcu. These codes are asymptotically-good too. For $g>2$, these are the first instances of $g$-group ML-decodable codes with rates greater than $1$ 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 $1 < R \leq 5/4$.(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

STBCs from Representation of Extended Clifford Algebras

A set of sufficient conditions to construct $\lambda$-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 $\lambda=2^a,a\in\mathbb{N}$ 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

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