General solution for orthogonal periodic real‐number sequences

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 = 2πk3/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.

A real‐valued periodic orthogonal sequence derived from a real‐valued shift‐orthogonal finite‐length sequence and its fast periodic correlation algorithm

“…Although a real-valued orthogonal periodic sequence of period N is normally represented by an amplitude spectrum that is 1 and a phase spectrum that is a discrete Fourier transformation of an odd function [3], here a proposal is shown that utilizes a real-valued shift-orthogonal finitelength sequence of length M = N + 1 to generate the sequence. At first, the following equation represents the aperiodic autocorrelation function of a real-valued shift-orthogonal finite-length sequence of length {a M,l ,i ; i = 0, 1, .…”

“…Conversely, if we form an orthogonal periodic sequence of a period N ≥ 8 using periodic partial sequences {b 2,i }, {b 2,i g }, {b 3 …”

“…This sequence is formed by convoluting partial sequences. In addition, this sequence is one sequence found by a common solution [3] of a real-valued orthogonal periodic sequence given by Tanada and since the proposal to form the sequence is different, it is difficult to directly derive it from a common solution. Next, we will propose an algorithm (fast correlation algorithm) that rapidly executes correlation processing for this sequence of period N = 2 n .…”

A generation method of an orthogonal set of real-valued periodic orthogonal sequences from Huffman sequences

“…In this paper, we propose a generation method of an orthogonal set of real-valued periodic orthogonal sequences of period N = 2 n with positive integer n derived from a realvalued Huffman sequences of length M = 2 ν + 1 with positive integer ν and ν ≥ n. This orthogonal set is included in the set derived from the general solution [4] of real-valued periodic orthogonal sequences, but it is very difficult to generate it by the general solution.…”

“…A binary periodic orthogonal sequence which takes 1 or −1 is only (1, 1, 1, −1) of period 4. On the other hand, the general solution of real-valued periodic orthogonal sequences take real numbers has already been proposed [4], and the periodic sequence of various periods can be generated. The real-valued periodic orthogonal sequence of period 2 ν with positive integer ν derived from a real-valued Huffman sequence (shift-orthogonal finite-length sequence) [5], [6] of length 2 ν +1 whose aperiodic autocorrelation function has no sidelobes except at left and right shift-ends exists, has fast correlation algorithm [7].…”