Three new lengths for cyclic Legendre pairs

Introduction: It is conjectured that the cyclic Legendre pairs of odd lengths >1 always exist. Such a pair consists of two functions a, b: G→Z, whose values are +1 or −1, and whose periodic autocorrelation function adds up to the constant value −2 (except at the origin). Here G is a finite cyclic group and Z is the ring of integers. These conditions are fundamental and the closely related structure of Hadamard matrices having a two circulant core and double border is incompletely described in literature, which makes its study especially relevant. Purpose: To describe the two-border two-circulant-core construction for Legendre pairs having three new lengths. Results: To construct new Legendre pairs we use the subsets X={x∈G: a(x)=–1} and Y={x∈G: b(x)=–1} of G. There are 20 odd integers v less than 200 for which the existence of Legendre pairs of length v is undecided. The smallest among them is v=77. We have constructed Legendre pairs of lengths 91, 93 and 123 reducing thereby the number of undecided cases to 17. In the last section of the paper we list some new examples of cyclic Legendre pairs for lengths v≤123. Practical relevance: Hadamard matrices are used extensively in the problems of error-free coding, and compression and masking of video information. Programs for search of Hadamard matrices and a library of constructed matrices are used in the mathematical network “mathscinet.ru” together with executable on-line algorithms

Stochastic Evaluation of Capacity and Demand Management of the Airline Industry

In the airline industry, the term load factor defined as the percentage of seats filled by revenue passengers and is used to measure efficiency and performance. This metric evaluates the airlines capacity and demand management. This paper applies stochastic models to analyse the load factor of the Association European Airlines (AEA) for flights of Europe - North Africa and Europe- Sub Saharan Africa. The estimation result prevails that the airlines have better demand management in the flights of Europe- Sub Saharan Africa than in the flight of Europe - North Africa. However, the capacity management of the airlines is poor for both regional flights. The autocorrelation structures for the load factor for both regional flights have both periodic and serial correlations. Consequently, the use of ordinal panel data models is inappropriate to capture the necessary variation of the load factor of the regional flights. Therefore, in order to control for the periodic autocorrelation, the author introduces dynamic time effects panel data regression model. Furthermore, in order to eliminate serial correlation the author applies the Prais–Winsten methodology to fit the model. Finally, the author builds realistic and robust forecasting model of the load factor of the Europe- North Africa and Europe-Sub Saharan Africa flights.

Aperiodic/periodic complementary sequence pairs over quaternions

<p style='text-indent:20px;'>Aperodic (or called Golay)/Periodic complementary pairs (GCPs/ PCPs) are pairs of sequences whose aperiodic/periodic autocorrelation sums are zero everywhere, except at the zero shift. In this paper, we introduce GCPs/PCPs over the quaternion group <inline-formula><tex-math id="M1">\begin{document}$ Q_8 $\end{document}</tex-math></inline-formula>, which is a generalization of quaternary GCPs/PCPs. Some basic properties of autocorrelations of <inline-formula><tex-math id="M2">\begin{document}$ Q_8 $\end{document}</tex-math></inline-formula>-sequences are also obtained. We present three types of constructions for GCPs and PCPs over <inline-formula><tex-math id="M3">\begin{document}$ Q_8 $\end{document}</tex-math></inline-formula>. The main ideas of these constructions are to consider pairs of a <inline-formula><tex-math id="M4">\begin{document}$ Q_8 $\end{document}</tex-math></inline-formula>-sequence and its reverse, pairs of interleaving of sequence, or pairs of Kronecker product of sequences. By choosing suitable sequences in these constructions, we obtain new parameters for GCPs and PCPs, which have not been reported before.</p>

A Phase-Coded Sequence Design Method for Active Sonar

Phase-coded sequences are widely studied as the transmitted signals of active sonars. Recently, several design methods have been developed to generate phased-coded sequences satisfying specific aperiodic or periodic autocorrelation sidelobe level metrics. In this paper, based on the majorization–minimization strategy and the squared iterative acceleration scheme, we propose a method to generate sequences with the periodic weighted integrated sidelobe level metric. Numerical simulations illustrate that the proposed method can effectively suppress the periodic autocorrelation sidelobe levels in specific time lags. Compared with other sequence design methods satisfying the periodic weighted integrated sidelobe level metric, our method improves the computational efficiency significantly. In addition, the proposed sequence demonstrates better matched filter performance in specific range intervals compared with its counterpart. The results suggest that the method could be applied as a valid and real-time design method for transmitted signals of active sonars.

Retrospective Review of Perfect Ternary Sequences and Their Generators

Introduction. Perfect polyphase unimodular sequences, i. e. sequences with ideal periodic autocorrelation and single amplitude of symbols are widely used in modern radio communications and radar. A special place among them is occupied by perfect ternary sequences (PTSs) with elements {–1, 0, 1}. In fact, these are binary sequences with the alphabet {–1, 1}, but with zero symbols in some positions. It is known that PTSs are quite numerous and their length in comparison with perfect binary sequences is not limited from above. The charge for this is a peak factor greater than one, which causes energy losses in the receiver. A large number of research papers and books are devoted to the design of PTSs and the study of their properties. In particular, the handbook on sequence design by Fan and Darnell (1996) which provides an overview of the then known PTS families has become widely famous. However, over the past two decades, numerous new PTS families were discovered, some theorems on their existence were obtained, and connections were established between them and circulant weighing matrices. Therefore, there is a need for a new review of existing PTSs.Objective. The article is devoted to a retrospective review of existing PTSs and their generation devices.Materials and methods. Considered and analyzed domestic and foreign sources of information (books, journal papers, conference proceedings, patents).Results. Along with solving an informational bibliographic problem, the review shows the relationship between PTSs obtained at different times, their connection with circulant weighing matrices, and also describes the block diagrams of generators of some PTS families.Conclusion. A brief retrospective review of PTSs for their almost 60 years history is presented and the generators of some PTS families are considered. The results of the study are relevant for use in modern radio communications and radar systems and in particular, in CW and LPI radars.

Perfect phase-coded pulse trains generated by Talbot effect

A perfect phase sequence is a finite and ordered set of constant-amplitude complex numbers whose periodic autocorrelation vanishes at any non-zero time shift. They find multiple applications in science an engineering as phase-coded waveforms, where the sequence defines the relative phases within a burst of electromagnetic or acoustic pulses. We show how a physical propagation effect, the so-called fractional Talbot phenomenon, can be used to generate pulse trains coded according to these sequences. The mathematical description of this effect is first reviewed and extended, showing its close relationship with Gauss perfect phase sequences. It is subsequently shown how it leads to a construction of Popović’s Generalized Chirp-Like (GCL) sequences. Essentially, a set of seed pulses with prescribed amplitude and phase levels, cyclically feeds a linear and dispersive medium. At particular values of the propagation length, multiple pulse-to-pulse interference induced by dispersion passively creates the sought-for pulse trains composed of GCL sequences, with the additional property that its repetition rate has been increased with respect to the seed pulses. This observation constitutes a novel representation of GCL sequences as the result of dispersive propagation of a seed sequence, and a new route for the practical implementation of perfect phase-coded pulse waveforms using Talbot effect.

Stochastic Evaluation of Capacity and Demand Management of the Airline Industry

In the airline industry, the term load factor defined as the percentage of seats filled by revenue passengers and is used to measure efficiency and performance. This metric evaluates the airlines capacity and demand management. This paper applies stochastic models to analyse the load factor of the Association European Airlines (AEA) for flights of Europe - North Africa and Europe- Sub Saharan Africa. The estimation result prevails that the airlines have better demand management in the flights of Europe- Sub Saharan Africa than in the flight of Europe - North Africa. However, the capacity management of the airlines is poor for both regional flights. The autocorrelation structures for the load factor for both regional flights have both periodic and serial correlations. Consequently, the use of ordinal panel data models is inappropriate to capture the necessary variation of the load factor of the regional flights. Therefore, in order to control for the periodic autocorrelation, the author introduces dynamic time effects panel data regression model. Furthermore, in order to eliminate serial correlation the author applies the Prais–Winsten methodology to fit the model. Finally, the author builds realistic and robust forecasting model of the load factor of the Europe- North Africa and Europe-Sub Saharan Africa flights.