Complex sequences with low periodic correlations

The importance of the upper and lower bound lies in the fact that they can be shown to agree for a large range of P, and M. R;(d), + . . ,R”-‘(d) were evaluated for M=3,4,5,... ,lO and M100 wit< P = 0.005, 0.05, and 0.5 using a computer program to find the m&mum of the convex function in (20) over the convex set 4. For these values of M and P,, it was found that co( R,‘, R,2,. . . , R/)(d) = co(R,‘, RzM)(d) so that the upper and Lwer bounds agree and R*(d)= co(R,‘, R/)(d). As an example, R,‘(d), Rz%O * . . ,Rz(d) are shown in Fig. 4 for M=5 and P, = O.Q5. The dashed line is z(R,‘, R:)(d) which can be seen to equal co(R,‘, Rz, * . . , R:)(d). We conjecture that R*(d) = co(RzTRzM)(d) for any M > 2 and P, satisfying Q < P, < l/2. When R*(d) = co(R,‘, Rf)(d), the optimum coding-decoding procedure is to have the decoder ignore side information and use only source information for low distortion; to have the decoder ignore source information and use only side information for distortion P,; and to time-share the two schemes for intermediate values of distortion. It is an interesting open problem to characterize the class of correlated sources for which the optimum performance is obtained by time-sharing strictly source information with strictly side information. In general, there exist correlated sources for which the optimum coding-decoding scheme is not a time-sharing scheme; for example, this is true when the source has p(xl y) = 1 for some x,y.