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A Method of Construction of Perfect Sequences


E.I. Krengel

Perfect sequences are widely used in digital communication and radar engineering. By technical reasons the application of perfect polyphase sequences with small phase alphabet are more preferable. Unfortunately, the perfect polyphase sequences with small phase alphabet can be built only for few lengths. At the same time, perfect polyphase sequences with zeroes whose alphabet size doesn’t depend of their length have been discovered. Among them there are ternary Ipatov and Hoholdt-Justesen sequences, 4-phase Lee sequences, 8-phase Lüke sequences and etc. Recently, new perfect polyphase sequences of length 4N with zeroes and small phase alphabet are derived from pairs of perfect polyphase sequences with zeroes and odd-perfect ternary sequences of the same length N. In the paper, we present generalized construction method of new perfect sequences of length 4N based on the mixing of two perfect and odd-perfect sequences with length N and the same autocorrelation peak R. Also it is shown that the mixing of perfect polyphase sequence and odd-perfect ternary sequence with near autocorrelation peaks derives new almost-perfect polyphase sequence with small sidelobes and minimum alphabet size. The construction method of perfect sequences is following. Let sequences a={aj} and b={bj} be accordingly an arbitrary perfect sequence and an odd-perfect sequence of length N with the same autocorrelation peak R. Then by concatenation we form two sequences d=a•a and e=b of length 2N where . In result the sequence f={fk}, 0k <4N obtained by mixing of the sequences d and e is the perfect sequence of length 4N. To construct the reference pairs of sequences a and b we can use Even-odd transformation (EOT) which maps any odd-perfect sequence into a perfect sequence of the same length and on the contrary. Also, the relevant sequences can be connected by EOT indirectly through their decimations or cyclic shifts. However, there are cases when the relevant sequences are not connected by EOT. Let Mp and Mo be accordingly total number of shift-distinct perfect and odd-perfect sequences of the same length N with autocorrelation peak R. Obviously, distinct pairs of these sequences and their cyclic shifts form distinct perfect sequences of length 4N. Then the total number of new shift-distinct perfect sequence of length 4N is equal to M=NMpMo.

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