Gaussian Integer Sequences With Ideal Periodic Autocorrelation Functions

• Published in: IEEE transactions on signal processing Vol. 60; no. 11; pp. 6074 - 6079
• Main Authors: HU, Wei-Wen, WANG, Sen-Hung, LI, Chih-Peng
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.11.2012; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Autocorrelation functions; Computational efficiency; Construction; Correlation; Detection, estimation, filtering, equalization, prediction; Discrete Fourier transforms; Electrical engineering; Equivalence; Exact sciences and technology; Gaussian; Gaussian integer; Indexes; Information, signal and communications theory; Integers; Mathematical analysis; Peak to average power ratio; perfect sequence; periodic auto-correlation function (PACF); periodic cross-correlation function (PCCF); Signal and communications theory; Signal, noise; Studies; Sun; Telecommunications and information theory; Transaction processing; Vectors; Autocorrelation function; periodic auto-correlation function (PACF); Mathematical formula; perfect sequence; Energetic efficiency; Cross correlation; Signal processing; Correlation function; Periodic function; periodic cross-correlation function (PCCF); Gaussian integer; Periodic correlation
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2012.2210550

A Gaussian integer is a complex number whose real and imaginary parts are both integers. Meanwhile, a sequence is defined as perfect if and only if the out-of-phase value of the periodic autocorrelation function is equal to zero. This paper presents two novel classes of perfect sequences constructed using two groups of base sequences. The nonzero elements of these base sequences belong to the set {±1, ±j}. A perfect sequence can be obtained by linearly combining these base sequences or their cyclic shift equivalents with arbitrary nonzero complex coefficients of equal magnitudes. In general, the elements of the constructed sequences are not Gaussian integers. However, if the complex coefficients are Gaussian integers, then the resulting perfect sequences will be Gaussian integer perfect sequences (GIPSs). In addition, a periodic cross-correlation function is derived, which has the same mathematical expression as the investigated sequences. Finally, the maximal energy efficiency of the proposed GIPSs is investigated.