| 摘要 |
A multicarrier direct-sequence code-division multiple-access (MC-DS/CDMA) communications system is provided. A code tree of two-dimensional orthogonal variable spreading factor (2D-OVSF) codes is then generated for the system. To generate the code tree, a set of existing M<SUB>1</SUB>xN<SUB>1 </SUB>2D-OVSF matrices, in the form of A<SUP>(i)</SUP><SUB>(M</SUB><SUB><SUB2>1</SUB2></SUB><SUB>xN</SUB><SUB><SUB2>1</SUB2></SUB><SUB>) </SUB>for i={1, 2, . . . , K<SUB>1</SUB>} is selected as seed matrices. M<SUB>1 </SUB>represents the number of available frequency carriers in the MC-DS/CDMA system, and N<SUB>1 </SUB>represents a spreading factor code length. Another set of existing M<SUB>2</SUB>xN<SUB>2 </SUB>2D-OVSF matrices, in the form of B<SUB>2</SUB><SUP>(i)</SUP><SUB>(M</SUB><SUB><SUB2>2</SUB2></SUB><SUB>xN</SUB><SUB><SUB2>2</SUB2></SUB><SUB>) </SUB>for i={1, 2, . . . , K<SUB>2</SUB>} is then selected as mapping matrices. The mapping matrices are used to generate corresponding children matrices. These second layer child matrices are M<SUB>1</SUB>M<SUB>2</SUB>xN<SUB>1</SUB>N<SUB>2 </SUB>matrices with cardinality K<SUB>1</SUB>K<SUB>2</SUB>, which are defined by reiterating the relationship: <maths id="MATH-US-00001" num="00001"> <MATH OVERFLOW="SCROLL"> <MROW> <MSUBSUP> <MI>C</MI> <MROW> <MO>(</MO> <MROW> <MSUB> <MI>M</MI> <MN>1</MN> </MSUB> <MO></MO> <MSUB> <MI>M</MI> <MN>2</MN> </MSUB> <MO>x</MO> <MSUB> <MI>N</MI> <MN>1</MN> </MSUB> <MO></MO> <MSUB> <MI>N</MI> <MN>2</MN> </MSUB> </MROW> <MO>)</MO> </MROW> <MROW> <MO>(</MO> <MROW> <MROW> <MROW> <MO>(</MO> <MROW> <MI>i</MI> <MO>-</MO> <MN>1</MN> </MROW> <MO>)</MO> </MROW> <MO></MO> <MSUB> <MI>K</MI> <MN>2</MN> </MSUB> </MROW> <MO>+</MO> <MN>1</MN> </MROW> <MO>)</MO> </MROW> </MSUBSUP> <MO>=</MO> <MROW> <MSUBSUP> <MI>B</MI> <MROW> <MN>2</MN> <MO></MO> <MROW> <MO>(</MO> <MROW> <MSUB> <MI>M</MI> <MN>2</MN> </MSUB> <MO>x</MO> <MSUB> <MI>N</MI> <MN>2</MN> </MSUB> </MROW> <MO>)</MO> </MROW> </MROW> <MROW> <MO>(</MO> <MN>1</MN> <MO>)</MO> </MROW> </MSUBSUP> <MO>⊕</MO> <MSUBSUP> <MI>A</MI> <MROW> <MO>(</MO> <MROW> <MSUB> <MI>M</MI> <MN>1</MN> </MSUB> <MO>x</MO> <MSUB> <MI>N</MI> <MN>1</MN> </MSUB> </MROW> <MO>)</MO> </MROW> <MROW> <MO>(</MO> <MI>i</MI> <MO>)</MO> </MROW> </MSUBSUP> </MROW> </MROW> </MATH> </MATHS> <maths id="MATH-US-00001-2" num="00001.2"> <MATH OVERFLOW="SCROLL"> <MROW> <MSUBSUP> <MI>C</MI> <MROW> <MO>(</MO> <MROW> <MSUB> <MI>M</MI> <MN>1</MN> </MSUB> <MO></MO> <MSUB> <MI>M</MI> <MN>2</MN> </MSUB> <MO>x</MO> <MSUB> <MI>N</MI> <MN>1</MN> </MSUB> <MO></MO> <MSUB> <MI>N</MI> <MN>2</MN> </MSUB> </MROW> <MO>)</MO> </MROW> <MROW> <MO>(</MO> <MROW> <MROW> <MROW> <MO>(</MO> <MROW> <MI>i</MI> <MO>-</MO> <MN>1</MN> </MROW> <MO>)</MO> </MROW> <MO></MO> <MSUB> <MI>K</MI> <MN>2</MN> </MSUB> </MROW> <MO>+</MO> <MN>2</MN> </MROW> <MO>)</MO> </MROW> </MSUBSUP> <MO>=</MO> <MROW> <MSUBSUP> <MI>B</MI> <MROW> <MN>2</MN> <MO></MO> <MROW> <MO>(</MO> <MROW> <MSUB> <MI>M</MI> <MN>2</MN> </MSUB> <MO>x</MO> <MSUB> <MI>N</MI> <MN>2</MN> </MSUB> </MROW> <MO>)</MO> </MROW> </MROW> <MROW> <MO>(</MO> <MN>2</MN> <MO>)</MO> </MROW> </MSUBSUP> <MO>⊕</MO> <MSUBSUP> <MI>A</MI> <MROW> <MO>(</MO> <MROW> <MSUB> <MI>M</MI> <MN>1</MN> </MSUB> <MO>x</MO> <MSUB> <MI>N</MI> <MN>1</MN> </MSUB> </MROW> <MO>)</MO> </MROW> <MROW> <MO>(</MO> <MI>i</MI> <MO>)</MO> </MROW> </MSUBSUP> </MROW> </MROW> </MATH> </MATHS> <maths id="MATH-US-00001-3" num="00001.3"> <MATH OVERFLOW="SCROLL"> <MROW> <MSTYLE> <mspace width="10.em" height="10.ex"/> </MSTYLE> <MO></MO> <MI>⋯</MI> </MROW> </MATH> </MATHS> <maths id="MATH-US-00001-4" num="00001.4"> <MATH OVERFLOW="SCROLL"> <MROW> <MSUBSUP> <MI>C</MI> <MROW> <MO>(</MO> <MROW> <MSUB> <MI>M</MI> <MN>1</MN> </MSUB> <MO></MO> <MSUB> <MI>M</MI> <MN>2</MN> </MSUB> <MO>x</MO> <MSUB> <MI>N</MI> <MN>1</MN> </MSUB> <MO></MO> <MSUB> <MI>N</MI> <MN>2</MN> </MSUB> </MROW> <MO>)</MO> </MROW> <MROW> <MO>(</MO> <MROW> <MROW> <MROW> <MO>(</MO> <MROW> <MI>i</MI> <MO>-</MO> <MN>1</MN> </MROW> <MO>)</MO> </MROW> <MO></MO> <MSUB> <MI>K</MI> <MN>2</MN> </MSUB> </MROW> <MO>+</MO> <MSUB> <MI>K</MI> <MN>2</MN> </MSUB> </MROW> <MO>)</MO> </MROW> </MSUBSUP> <MO>=</MO> <MROW> <MSUBSUP> <MI>B</MI> <MROW> <MN>2</MN> <MO></MO> <MROW> <MO>(</MO> <MROW> <MSUB> <MI>M</MI> <MN>2</MN> </MSUB> <MO>x</MO> <MSUB> <MI>N</MI> <MN>2</MN> </MSUB> </MROW> <MO>)</MO> </MROW> </MROW> <MROW> <MO>(</MO> <MSUB> <MI>K</MI> <MN>2</MN> </MSUB> <MO>)</MO> </MROW> </MSUBSUP> <MO>⊕</MO> <MSUBSUP> <MI>A</MI> <MROW> <MO>(</MO> <MROW> <MSUB> <MI>M</MI> <MN>1</MN> </MSUB> <MO>x</MO> <MSUB> <MI>N</MI> <MN>1</MN> </MSUB> </MROW> <MO>)</MO> </MROW> <MROW> <MO>(</MO> <MI>i</MI> <MO>)</MO> </MROW> </MSUBSUP> </MROW> </MROW> </MATH> </MATHS> where ⊕ indicates a Kronecker product, and i=1, 2, 3, 4, . . . , K<SUB>1</SUB>.
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