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Characterization of noncorrelated pattern sequences and correlation dimensions

  • * Corresponding author: Li Peng

    * Corresponding author: Li Peng 
This work was supported by the NSFC [11571127, 11431007].
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  • We consider the correlation functions of binary pattern sequences of degree 3 as well as those with general degrees and special patterns and obtain necessary and sufficient conditions to be noncorrelated. We also obtain the correlation dimensions for those with degree 2.

    Mathematics Subject Classification: Primary: 47B15; Secondary: 11B50.

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    \begin{equation} \\ \end{equation}
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  • Table 1.   

    $(a_1, a_2, a_3)$pattern setspectrum
    $(-1, 1, 1)$ $\{01, 10, 11\}$
    $a_1a_2a_3=-1$ $(1, -1, 1)$ $\{01\}$noncorrelated sequence
    $(1, 1, -1)$ $\{11\}$ $D_2=1$
    $(-1, -1, -1)$ $\{10\}$
    $a_1a_2a_3=1$ $(1, -1, -1)$ $\{01, 11\}$singular spectrum
    $a_2=-1$ $(-1, -1, 1)$ $\{10, 11\}$ $D_2=3-\log_2(1+\sqrt{17})$
    $a_1a_2a_3=1$ $(-1, 1, -1)$ $\{01, 10\}$periodic sequence
    $a_2=1$ $D_2=0$
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