On ternary complementary sequences
Amnon Gavish, Abraham Lempel · IEEE Transactions on Information Theory · 1994
A pair of real-valued sequences A=(a/sub 1/,a/sub 2/,...,a/sub N/) and B=(b/sub 1/,b/sub 2/,...,b/sub N/) is called complementary if the sum R(/spl middot/) of their autocorrelation functions R/sub A/(/spl middot/) and R/sub B/(/spl middot/) satisfies R(/spl tau/)=R/sub A/(/spl tau/)+R/sub B/(/spl tau/)=/spl Sigma//sub i=1//sup N$/ -/sup /spl tau//a/sub i/a/sub i+/spl tau//+/spl Sigma//sub j=1//sup N-/spl tau//b/sub j/b/sub j+/spl tau//=0, /spl forall//spl tau//spl ne/0. In this paper we introduce a new family of complementary pairs of sequences over the alphabet /spl alpha//sub 3/=+{1,-1,0}. The inclusion of zero in the alphabet, which may correspond to a pause in transmission, leads both to a better understanding of the conventional binary case, where the alphabet is /spl alpha//sub 2/={+1,-1}, and to new nontrivial constructions over the ternary alphabet /spl alpha//sub 3/. For every length N, we derive restrictions on the location of the zero elements and on the form of the member sequences of the pair. We also derive a bound on the minimum number of zeros necessary for the existence of a complementary pair of length N over /spl alpha//sub 3/. The bound is tight, as it is met by some of the proposed constructions, for infinitely many lengths.>