Sets of cyclically orthogonal multilevel sequences derived from monomial phase
Yoshihiro Tanada · Electronics and Communications in Japan (Part I Communications) · 1987
Abstract This paper derives a set of cyclically orthogonal sequences taking real multilevel values and having different waveforms, aiming at the application to the spread‐spectrum multiplex communication. The real cyclic orthogonal sequence is represented by the cosine function with the phase which is an odd function, and the cross‐correlation functions among sequences with the same or different lengths are discussed. By defining the convolution between sequences of the same length, a set of sequences with different waveforms and the same length can be derived. As examples of the set of sequences, the sets with length being prime number or compound number lengths are considered, which are obtained by forming convolutions of sequences with phases being monomial with odd power. Their properties are discussed. In the set of N sequences with the prime number length N, each of (N ‐ 1) sequence takes odd multilevels between 3 and N, and the cross‐correlation function between sequences is 1/√N times the sequence value. By forming the product of such sequences with different prime number lengths, a set of sequences with compound number lengths can be generated, which is a good compromise between the number of multilevels and the absolute maximum value of the cross‐correlation function. N ‐ 1 sequences of prime number lengths generated by the third‐order monomial takes 3 to N multilevels and their cross‐correlation function does not exceed 2/√N in the absolute value.