Complementary series

A set of complementary series is defined as a pair of equally long, finite sequences of two kinds of elements which have the property that the number of pairs of like elements with any one given separation in one series is equal to the number of pairs of unlike elements with the same given separation in the other series. (For instance the two series, 1001010001 and 1000000110 have, respectively, three pairs of like and three pairs of unlike adjacent elements, four pairs of like and four pairs of unlike alternate elements, and so forth for all possible separations.) These series, which were originally conceived in connection with the optical problem of multislit spectrometry, also have possible applications in communication engineering, for when the two kinds of elements of these series are taken to be +1 and -1, it follows immediately from their definition that the sum of their two respective autocorrelation series is zero everywhere, except for the center term. Several propositions relative to these series, to their permissible number of elements, and to their synthesis are demonstrated.