General solution for orthogonal periodic real‐number sequences

Abstract

Abstract This paper shows the general solution for the real‐number periodic sequence with an autocorrelation function with zero side lobe, aiming at the application to the spread‐spectrum communication. The general solution is derived by representing the autocorrelation function by a Fourier series and utilizing the components. The solution contains the phase constant. In other words, the sequence and the phase constant are related through the discrete Fourier transform Under the condition that the seqeunce is real, the phase constant is an odd function. However, when the phase constant takes only the value in {O, π}, the phase constant is an even (= odd) function, and the sequence is an even function. From the general solution, it is shown that N sequences obtained by shifting a sequence of order N are orthogonal. An example of the sequence is shown for the case where the phase constant is fixed as φ k E {0, φ }, φk = 2πk 3 /N (k = 0, 1, ‥, N ‐ l), and for the case where it is continuous as Phiv;1=‐ φ N‐1= θ. The general solution provides a basis for calculating various kinds of pseudonoise sequences.