Cyclically orthogonal sequences with low multiplicity of values derived from quadratic residues

Abstract

AbstractOrthogonal cyclic sequences with low multiplicity of values are given for application to spread spectrum communications systems. Two‐, three‐, and five‐valued orthogonal cyclic sequences are obtained by representing real orthogonal cyclic sequences of length N by cosine series possessing phase constant of odd function, or even function with values {0, }, and by relating the phase constant to quadratic residues of N. We use the fact that an extended quadratic residue sequence taking values {0, +1, ‐1} becomes a sequence of odd function satisfying homogeneous equation of discrete sine transform when N is a prime number of the form N =4° ‐ 1 ( is a natural number). When N is a prime number of the form N =4° + 1, the sequence becomes that of even function satisfying a homogeneous equation of discrete cosine transform. Three‐valued orthogonal cyclic sequences are obtained by adding a dc component, to quadratic residue sequences of these odd and even functions. A two‐valued orthogonal cyclic sequence is obtained when a phase constant is given by a quadratic residue sequence of odd function multiplied by a constant. Moreover, when the phase constant is a monomial of odd power and is related to the quadratic residue sequence, a three‐valued orthogonal cyclic sequence is obtained for the prime N =4° ‐ 1, and a five‐valued orthogonal cyclic sequence is obtained for the prime N =4° + 1. Since N sequences obtained by shifting one of those sequences form an orthogonal system, cross‐talk does not occur when these sequences are applied to synchronous spread spectrum multiple access.