Binary complementary sequences were first introduced by Marcel J. E. Golay in 1949.
Let (a0, a1, ..., aN − 1) and (b0, b1, ..., bN − 1) be a pair of bipolar sequences, meaning that a(k) and b(k) have values +1 or −1.
Or using Kronecker delta we can write: So we can say that the sum of autocorrelation functions of complementary sequences is a delta function, which is an ideal autocorrelation for many applications like radar pulse compression and spread spectrum telecommunications.
The complementarity property of the sequences is equivalent to the condition for all z on the unit circle, that is, |z| = 1.
Examples include the Shapiro polynomials, which give rise to complementary sequences of length a power of two.