Aperiodic correlations and the merit factor of a class of binary sequences
نویسندگان
چکیده
and conjectured that F I 12.32, for all binary sequences, with the exception of the Barker sequence of length 13, for which F = 14.08. In a recent paper Golay [5] argued that the merit factor of Legendre sequences, shifted by one quarter of their lengths, has the highly probable asymptotic value 6, but he did not prove this. For maximal-length shift register sequences, one can see from [6] that if one considers the ensemble consisting of all cyclic shifts of a maximal-length sequence, then the average value of (2C~:~c~)/N* is approximately l/3. Thus, there exist maximallength sequences with merit factor of approximately 3. Skaug [7] has calculated the actual values of the aperiodic autocorrelations for a number of maximal-length shift register sequences, and from his calculations it seems possible that the magnitude of the largest correlation is of order fi. Based on results of Niederreiter [8], McEliece in [9] has proven a number of bounds, from which one can see that the magnitude of the largest aperiodic autocorrelation for maximal length shift register sequences of length N is bounded by fi log N. In this correspondence we consider sequences of length N = 2” defined recursively by 1, x2,2 I ( 1)‘+f(‘)X2,-,-1, OljI2’-1, i = O,l;. ‘,rn 1, (1.3) where f is any function mapping the set of natural numbers into (0, l}. For these sequences we prove a simple recursion, which gives all the aperiodic autocorrelations, and based on this we calculate the merit factor and prove that the asymptotic value is 3. Finally, we prove that the magnitude of maximal aperiodic autocorrelation is 0( N”,9). We note that if one chooses the function f as f(0) = f(2k 1) = 0 and f(2k) = 1, k > 0, then we get the first 2”’ elements of
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ورودعنوان ژورنال:
- IEEE Trans. Information Theory
دوره 31 شماره
صفحات -
تاریخ انتشار 1985