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

Modeling and Forecasting Periodic Time Series data with Fourier Autoregressive Model

Most frequently used models for modeling and forecasting periodic climatic time series do not have the capability of handling periodic variability that characterizes it. In this paper, the Fourier Autoregressive model with abilities to analyze periodic variability is implemented. From the results, FAR(1), FAR(2) and FAR(2) models were chosen based on Periodic Autocorrelation function (PeACF) and Periodic Partial Autocorrelation function (PePACF). The coefficients of the tentative model were estimated using a Discrete Fourier transform estimation method. FAR(1) models were chosen as the optimal model based on the smallest values of Periodic Akaike (PAIC) and Bayesian Information criteria (PBIC). The residual of the fitted models was diagnosed to be white noise. The in-sample forecast showed a close reflection of the original rainfall series while the out-sample forecast exhibited a continuous periodic forecast from January 2019 to December 2020 with relatively small values of Periodic Root Mean Square Error (PRMSE), Periodic Mean Absolute Error (PMAE) and Periodic Mean Absolute Percentage Error (PMAPE). The comparison of FAR(1) model forecast with AR(3), ARMA(2,1), ARIMA(2,1,1) and SARIMA( 1,1,1)(1,1,1)12 model forecast indicated that FAR(1) outperformed the other models as it exhibited a continuous periodic forecast. The continuous monthly periodic rainfall forecast indicated that there will be rapid climate change in Nigeria in the coming yearly and Nigerian Government needs to put in place plans to curtail its effects.

QPSK-Modulation Modem Invariant to the Rotation of the Signal Constellation Plane

In order to increase the efficiency of dedicated frequency channels, i.e. to increase the specific data transfer rate, multipositional quadrature phase shift keying (QPSK, aka 4-PSK) should be used. The problems with QPSK signal demodulation is a rotation of the signal constellation plane by an angle multiple of 90° and a slow response of the carrier oscillation recovery scheme. The study considers the existing methods for eliminating the phase ambiguity of the recovered carrier frequency in typical QPSK modems, and identifies the shortcoming of a low-speed response oscillation recovery circuit. The authors propose a QPSK demodulator circuit with a fast adjustment of the reference oscillator, which is due to the fact that no loop filter is used in the feedback and that a digital calculator of the required phase shift is used. An algorithm for the frame synchronization restoration with the simultaneous elimination of the phase ambiguity multiple of 90° was also developed using synthesized binary sequences with an ideal non-periodic autocorrelation function (NACF) at even shifts that do not have the rotary symmetry property. The phase ambiguity elimination algorithm proposed in the article can be used as an alternative to standard modems with differential coding.

Analysis of Pseudo-Random Sequence Correlation Identification Parameters and Anti-Noise Performance

Using a pseudo-random sequence to encode the transmitted waveform can significantly improve the working efficiency and depth of detection of electromagnetic exploration. The selection of parameters of pseudo-random sequence plays an important role in correlation identification and noise suppression. A discrete cycle correlation identification method for extracting the earth impulse response is proposed. It can suppress the distortion in the early stage of the excitation field and the glitches of the cross correlation function by traditional method. This effectively improves the accuracy of correlation identification. The influence of the order and the cycles of m-series pseudo-random coding on its autocorrelation properties is studied. The numerical results show that, with the increase of the order of m-sequence, the maximum out-of-phase periodic autocorrelation function decreases rapidly. Therefore, it is very beneficial to achieve synchronization. The limited-cycle m-sequences have good autocorrelation properties. As the period of the m-sequence increases and the width of the symbol decreases, the overall autocorrelation becomes closer to the impact function. The discussion of the influence of symbol width and period of m-sequence on its frequency bandwidth and power spectral density shows that the narrower the symbol width, the wider its occupied band. The longer the period, the smaller the power spectral line spacing. The abilities of m-sequence to suppress DC (Direct-current) interference, Schumann frequency noise, and sine-wave noise are analyzed. Numerical results show that the m-sequence has excellent ability to suppress DC interference and Schumann frequency noise. However, for high-order harmonic noise, the correlation identification error appears severe oscillation in the middle and late stages of the impulse response. It indicates that the ability of m-sequence to suppress high-frequency sinusoidal noise is deteriorated. In practical applications, the parameters of the transmitted waveform should be reasonably selected in combination with factors including transmitter performance, hardware noise, and ambient noise level to achieve the best identification effect.

Study of the corrective ability of sync codes for the matched processing decoder

Increase of the operation reliability of the frame synchronization schemes of communication systems affected by interferences might be achieved by using sync sequences optimal according to the minimax criterion and having the best ratio of the maximum side-lobe level to the main peak of the non-periodic autocorrelation function. Obviously, the longer the minimax sequence, the higher its correction ability. However, with the increase in the length of the sync code, a conflict arises between the corrective ability and the speed of data transfer. This is caused by the decrease in the duration of the information section of the frame. At the same time, as the length of the code increases, the complexity of the technical implementation of the decoder synchronization scheme increases as well. This means that when developing an information transmission system, it is necessary to find a compromise between the complexity of the technical implementation of the decoder and the required noise immunity. To solve this problem, it is necessary to study the correcting ability of binary sync codes. The purpose of this work was to study the correcting ability of binary sequences with good non-periodic auto-correlation properties using a reception scheme with matched processing. This article discusses the possibility of using Barker sequences of lengths 7, 11, and 13 and M-sequences as corrective sync codes. The authors carry out an analysis of the correcting ability of the decoder model with matched processing for Barker codes of different lengths under the conditions of a one, two, and threefold error. Analytical expressions have been obtained to determine the decoding error probability of a binary minimax code depending on the length of the sync code. The dependence of the corrective ability of the code and the maximum level of the side lobe on the length of the sync code in the range from 4 to 128 with the best nonperiodic autocorrelation functions (NPAF) is obtained. It has been established that there are length ranges for minimax sequences within which their correction ability remains constant, that is, a sync code of shorter length has the same correction ability as a longer code. Thus, the research results allow us to optimize the complexity of the technical implementation of a sync signal decoder depending on the requirements for general noise immunity and information transfer rate in a communication system. The proposed approach can be successfully used to modify the existing synchronization schemes of communication systems with code division of channels.