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On the Search for High-Rate Quasi-Orthogonal Space–Time Block Code

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Abstract

A Quasi-Orthogonal Space–Time Block Code (QO-STBC) is attractive because it achieves higher code rate than orthogonal STBC and lower decoding complexity than non-orthogonal STBC. In this paper, we first derive the algebraic structure of QO-STBC, then we apply it in a novel graph-based search algorithm to find high-rate QO-STBCs with code rates greater than 1. From the four-antenna codes found using this approach, it is found that the maximum code rate is limited to 5/4 with symbolwise diversity level of four, and 4 with symbolwise diversity level of two. The maximum likelihood decoding of these high-rate QO-STBCs can be performed on two separate sub-groups of symbols. The rate-5/4 codes are the first known QO-STBCs with code rate greater 1 that has full symbolwise diversity level.

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ACKNOWLEDGMENTS

The authors would like to thank Hwor Shen Chong and Chee Muah Lim from Nanyang Technological University (Singapore) in assisting in part of this work. The authors would also like to thank the editor Dr Olav Tirkkonen and the anonymous reviewer for many constructive suggestions that help improve the quality of this paper.

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Authors and Affiliations

  1. School of Electrical and Electronic Engineering, Nanyang Technological University, Block S1-B1c-106, Nanyang Avenue, Singapore, 639798, Singapore

    Chau Yuen & Yong Liang Guan

  2. Institute for Infocomm Research, Singapore, Singapore

    Chau Yuen & Tjeng Thiang Tjhung

Authors
  1. Chau Yuen
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  2. Yong Liang Guan
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  3. Tjeng Thiang Tjhung
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Corresponding author

Correspondence to Chau Yuen.

Additional information

This is the work done when Chau Yuen was with the Nanyang Technological University

Appendices

Appendix A: Depth First Search (DFS) Algorithm

 

Table 2
Full size table

Appendix B: Modified Depth First Search (MDFS) Algorithm

 

Input: Graph, number of nodes in graph (N), number of groups required (G)

Output: a tree T with every branch as a valid solution

Begin

  Pick a node n, \(n \leq N\) as starting point;

  Assign node n to Group 1, i.e. g(n)=1;

  Assign node n as the root of tree T, i.e. \(T = T \cup \{n\}\);

  MDFS(n, T, g);

End

Procedure MDFS(n, T, g)

Begin

  for (v ∈{neighbor of n})

   if ( \(v \not\in \{\hbox{ancestor of } n\}\))

    Assume that node v is in Group p

    where p=g(n)+1 if \(g(n)+1 \leq G\),

       = 1 if \(g(n)+1 > G\);

    if (v possesses a link with all ancestors of n with

      different groups, i.e. g(ancestor of n) \(\ne p\))

       Assign node v to Group p, i.e. g(v)=p;

       Add node v to the tree T with n as parent, i.e. T=T ∪ {v};

       MDFS(v, T, g);

    end

    end

  end

End

Appendix C: Matrices Found by MDFS Using G=2, T=N t=4, Rank=4, Weight=2

Group 1

$$ \begin{array}{*{20}c} {{\mathbf{A}}_1 = \left[ {\begin{array}{*{20}c} 1 & 1 & 0 & 0 \\ 1 & { - 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & { - 1} \\ \end{array}} \right],} \quad & {{\mathbf{A}}_2 = \left[ {\begin{array}{*{20}c} 1 & 1 & 0 & 0 \\ 1 & { - 1} & 0 & 0 \\ 0 & 0 & { - 1} & { - 1} \\ 0 & 0 & { - 1} & 1 \\ \end{array}} \right],}\quad & {{\mathbf{A}}_3 = \left[ {\begin{array}{*{20}c} 1 & 1 & 0 & 0 \\ 1 & { - 1} & 0 & 0 \\ 0 & 0 & j & j \\ 0 & 0 & { - j} & j \\ \end{array}} \right],} \\ {{\mathbf{A}}_4 = \left[ {\begin{array}{*{20}c} 1 & 1 & 0 & 0 \\ 1 & { - 1} & 0 & 0 \\ 0 & 0 & j & { - j} \\ 0 & 0 & j & j \\ \end{array}} \right],}\quad & {{\mathbf{A}}_5 = \left[ {\begin{array}{*{20}c} 1 & 1 & 0 & 0 \\ 1 & { - 1} & 0 & 0 \\ 0 & 0 & { - j} & j \\ 0 & 0 & j & j \\ \end{array}} \right],}\quad & {{\mathbf{A}}_6 = \left[ {\begin{array}{*{20}c} 1 & 1 & 0 & 0 \\ 1 & { - 1} & 0 & 0 \\ 0 & 0 & { - j} & { - j} \\ 0 & 0 & j & j \\ \end{array}} \right],} \\ {{\mathbf{A}}_7 = \left[ {\begin{array}{*{20}c} 1 & 1 & 0 & 0 \\ 1 & { - 1} & 0 & 0 \\ 0 & 0 & { - j} & j \\ 0 & 0 & { - j} & { - j} \\ \end{array}} \right]}\quad & {{\mathbf{A}}_8 = \left[ {\begin{array}{*{20}c} 1 & 1 & 0 & 0 \\ 1 & { - 1} & 0 & 0 \\ 0 & 0 & j & { - j} \\ 0 & 0 & { - j} & { - j} \\ \end{array}} \right].} & {} \\ \end{array} $$

Group 2

$$ \begin{array}{*{20}c} {{\mathbf{A}}_9 = \left[ {\begin{array}{*{20}c} { - 1} & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & { - 1} & 1 \\ 0 & 0 & 1 & 1 \\ \end{array}} \right],}\quad & {{\mathbf{A}}_{10} = \left[ {\begin{array}{*{20}c} { - 1} & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & { - 1} \\ 0 & 0 & { - 1} & { - 1} \\ \end{array}} \right],}\quad & {{\mathbf{A}}_{11} = \left[ {\begin{array}{*{20}c} j & j & 0 & 0 \\ j & { - j} & 0 & 0 \\ 0 & 0 & {-1} & 1 \\ 0 & 0 & 1 & 1 \\ \end{array}} \right],} \\ {{\mathbf{A}}_{12} = \left[ {\begin{array}{*{20}c} j & j & 0 & 0 \\ { - j} & j & 0 & 0 \\ 0 & 0 & { - 1} & 1 \\ 0 & 0 & 1 & 1 \\ \end{array}} \right],}\quad & {{\mathbf{A}}_{13} = \left[ {\begin{array}{*{20}c} j & { - j} & 0 & 0 \\ j & j & 0 & 0 \\ 0 & 0 & { - 1} & 1 \\ 0 & 0 & 1 & 1 \\ \end{array}} \right],}\quad & {{\mathbf{A}}_{14} = \left[ {\begin{array}{*{20}c} j & j & 0 & 0 \\ j & { - j} & 0 & 0 \\ 0 & 0 & 1 & { - 1} \\ 0 & 0 & { - 1} & { - 1} \\ \end{array}} \right],} \\ {{\mathbf{A}}_{15} = \left[ {\begin{array}{*{20}c} j & j & 0 & 0 \\ { - j} & j & 0 & 0 \\ 0 & 0 & 1 & { - 1} \\ 0 & 0 & { - 1} & { - 1} \\ \end{array}} \right]}\quad & {{\mathbf{A}}_{16} = \left[ {\begin{array}{*{20}c} j & { - j} & 0 & 0 \\ j & j & 0 & 0 \\ 0 & 0 & 1 & { - 1} \\ 0 & 0 & { - 1} & { - 1} \\ \end{array}} \right].} & {} \\ \end{array} $$

Appendix D: Matrices Found by MDFS Using G=2, T=N t=4, Rank=2, Weight=2

Group 1

$$ \begin{array}{*{20}c} {{\mathbf{A}}_1 = \left[ {\begin{array}{*{20}c} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ \end{array}} \right]}\quad & {{\mathbf{A}}_2 = \left[ {\begin{array}{*{20}c} j & j & 0 & 0 \\ j & j & 0 & 0 \\ 0 & 0 & j & j \\ 0 & 0 & j & j \\ \end{array}} \right]}\quad & {{\mathbf{A}}_3 = \left[ {\begin{array}{*{20}c} 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ \end{array}} \right]}\quad & {{\mathbf{A}}_4 = \left[ {\begin{array}{*{20}c} 0 & 0 & j & j \\ 0 & 0 & j & j \\ j & j & 0 & 0 \\ j & j & 0 & 0 \\ \end{array}} \right]} \\ {{\mathbf{A}}_5 = \left[ {\begin{array}{*{20}c} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & { - 1} & { - 1} \\ 0 & 0 & { - 1} & { - 1} \\ \end{array}} \right]}\quad & {{\mathbf{A}}_6 = \left[ {\begin{array}{*{20}c} j & j & 0 & 0 \\ j & j & 0 & 0 \\ 0 & 0 & { - j} & { - j} \\ 0 & 0 & { - j} & { - j} \\ \end{array}} \right]} \quad & {{\mathbf{A}}_7 = \left[ {\begin{array}{*{20}c} 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ { - 1} & { - 1} & 0 & 0 \\ { - 1} & { - 1} & 0 & 0 \\ \end{array}} \right]}\quad & {{\mathbf{A}}_8 = \left[ {\begin{array}{*{20}c} 0 & 0 & j & j \\ 0 & 0 & j & j \\ { - j} & { - j} & 0 & 0 \\ { - j} & { - j} & 0 & 0 \\ \end{array}} \right]} \\ {{\mathbf{A}}_9 = \left[ {\begin{array}{*{20}c} { - 1} & 1 & 0 & 0 \\ { - 1} & 1 & 0 & 0 \\ 0 & 0 & { - 1} & 1 \\ 0 & 0 & { - 1} & 1 \\ \end{array}} \right]}\quad & {{\mathbf{A}}_{10} = \left[ {\begin{array}{*{20}c} { - j} & j & 0 & 0 \\ { - j} & j & 0 & 0 \\ 0 & 0 & { - j} & j \\ 0 & 0 & { - j} & j \\ \end{array}} \right]}\quad & {{\mathbf{A}}_{11} = \left[ {\begin{array}{*{20}c} 0 & 0 & { - 1} & 1 \\ 0 & 0 & { - 1} & 1 \\ 1 & { - 1} & 0 & 0 \\ 1 & { - 1} & 0 & 0 \\ \end{array}} \right]}\quad & {{\mathbf{A}}_{12} = \left[ {\begin{array}{*{20}c} 0 & 0 & j & { - j} \\ 0 & 0 & j & { - j} \\ { - j} & j & 0 & 0 \\ { - j} & j & 0 & 0 \\ \end{array}} \right]} \\ {{\mathbf{A}}_{13} = \left[ {\begin{array}{*{20}c} 1 & { - 1} & 0 & 0 \\ 1 & { - 1} & 0 & 0 \\ 0 & 0 & { - 1} & 1 \\ 0 & 0 & { - 1} & 1 \\ \end{array}} \right]}\quad & {{\mathbf{A}}_{14} = \left[ {\begin{array}{*{20}c} j & { - j} & 0 & 0 \\ j & { - j} & 0 & 0 \\ 0 & 0 & { - j} & j \\ 0 & 0 & { - j} & j \\ \end{array}} \right]}\quad & {{\mathbf{A}}_{15} = \left[ {\begin{array}{*{20}c} 0 & 0 & 1 & { - 1} \\ 0 & 0 & 1 & { - 1} \\ 1 & { - 1} & 0 & 0 \\ 1 & { - 1} & 0 & 0 \\ \end{array}} \right]} \quad & {{\mathbf{A}}_{16} = \left[ {\begin{array}{*{20}c} 0 & 0 & j & { - j} \\ 0 & 0 & j & { - j} \\ j & { - j} & 0 & 0 \\ j & { - j} & 0 & 0 \\ \end{array}} \right]} \\ \end{array} $$

Group 2

$$ \begin{array}{*{20}c} {{\mathbf{A}}_{17} = \left[ {\begin{array}{*{20}c} { - 1} & 1 & 0 & 0 \\ 1 & { - 1} & 0 & 0 \\ 0 & 0 & { - 1} & 1 \\ 0 & 0 & 1 & { - 1} \\ \end{array}} \right]}\quad & {{\mathbf{A}}_{18} = \left[ {\begin{array}{*{20}c} { - j} & j & 0 & 0 \\ j & { - j} & 0 & 0 \\ 0 & 0 & { - j} & j \\ 0 & 0 & j & { - j} \\ \end{array}} \right]}\quad & {{\mathbf{A}}_{19} = \left[ {\begin{array}{*{20}c} 0 & 0 & 1 & { - 1} \\ 0 & 0 & { - 1} & 1 \\ { - 1} & 1 & 0 & 0 \\ 1 & { - 1} & 0 & 0 \\ \end{array}} \right]}\quad & {{\mathbf{A}}_{20} = \left[ {\begin{array}{*{20}c} 0 & 0 & j & { - j} \\ 0 & 0 & { - j} & j \\ { - j} & j & 0 & 0 \\ j & { - j} & 0 & 0 \\ \end{array}} \right]} \\ {{\mathbf{A}}_{21} = \left[ {\begin{array}{*{20}c} { - 1} & { - 1} & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & { - 1} & { - 1} \\ 0 & 0 & 1 & 1 \\ \end{array}} \right]}\quad & {{\mathbf{A}}_{22} = \left[ {\begin{array}{*{20}c} { - j} & { - j} & 0 & 0 \\ j & j & 0 & 0 \\ 0 & 0 & { - j} & { - j} \\ 0 & 0 & j & j \\ \end{array}} \right]}\quad & {{\mathbf{A}}_{23} = \left[ {\begin{array}{*{20}c} 0 & 0 & { - 1} & { - 1} \\ 0 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ { - 1} & { - 1} & 0 & 0 \\ \end{array}} \right]}\quad & {{\mathbf{A}}_{24} = \left[ {\begin{array}{*{20}c} 0 & 0 & { - j} & { - j} \\ 0 & 0 & j & j \\ j & j & 0 & 0 \\ { - j} & { - j} & 0 & 0 \\ \end{array}} \right]} \\ {{\mathbf{A}}_{25} = \left[ {\begin{array}{*{20}c} 1 & { - 1} & 0 & 0 \\ { - 1} & 1 & 0 & 0 \\ 0 & 0 & { - 1} & 1 \\ 0 & 0 & 1 & { - 1} \\ \end{array}} \right]} \quad & {{\mathbf{A}}_{26} = \left[ {\begin{array}{*{20}c} j & { - j} & 0 & 0 \\ { - j} & j & 0 & 0 \\ 0 & 0 & { - j} & j \\ 0 & 0 & j & { - j} \\ \end{array}} \right]}\quad & {{\mathbf{A}}_{27} = \left[ {\begin{array}{*{20}c} 0 & 0 & 1 & { - 1} \\ 0 & 0 & { - 1} & 1 \\ 1 & { - 1} & 0 & 0 \\ { - 1} & 1 & 0 & 0 \\ \end{array}} \right]}\quad & {{\mathbf{A}}_{28} = \left[ {\begin{array}{*{20}c} 0 & 0 & j & { - j} \\ 0 & 0 & { - j} & j \\ j & { - j} & 0 & 0 \\ { - j} & j & 0 & 0 \\ \end{array}} \right]} \\ {{\mathbf{A}}_{25} = \left[ {\begin{array}{*{20}c} 1 & 1 & 0 & 0 \\ { - 1} & { - 1} & 0 & 0 \\ 0 & 0 & { - 1} & { - 1} \\ 0 & 0 & 1 & 1 \\ \end{array}} \right]}\quad & {{\mathbf{A}}_{26} = \left[ {\begin{array}{*{20}c} j & j & 0 & 0 \\ { - j} & { - j} & 0 & 0 \\ 0 & 0 & { - j} & { - j} \\ 0 & 0 & j & j \\ \end{array}} \right]}\quad & {{\mathbf{A}}_{27} = \left[ {\begin{array}{*{20}c} 0 & 0 & 1 & 1 \\ 0 & 0 & { - 1} & { - 1} \\ 1 & 1 & 0 & 0 \\ { - 1} & { - 1} & 0 & 0 \\ \end{array}} \right]}\quad & {{\mathbf{A}}_{28} = \left[ {\begin{array}{*{20}c} 0 & 0 & j & j \\ 0 & 0 & { - j} & { - j} \\ j & j & 0 & 0 \\ { - j} & { - j} & 0 & 0 \\ \end{array}} \right]} \\ \end{array} $$

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Yuen, C., Guan, Y.L. & Tjhung, T.T. On the Search for High-Rate Quasi-Orthogonal Space–Time Block Code. Int J Wireless Inf Networks 13, 329–340 (2006). https://doi.org/10.1007/s10776-006-0033-2

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