Periodic Autocorrelation Research Articles

A Quaternion-Based Approach to Construct Quaternary Periodic Complementary Pairs

Two arrays form a periodic complementary pair if the sum of their periodic autocorrelations is a delta function. Finding such pairs, however, is challenging for large arrays whose entries are constrained to a small alphabet. One such alphabet is the quaternary set which contains the complex fourth roots of unity. In this letter, we propose a technique to construct periodic complementary pairs defined over the quaternary set using perfect quaternion arrays. We show how new pairs of quaternary sequences, matrices, and four-dimensional arrays that satisfy a periodic complementary property can be constructed with our method.

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.

Combining PRF staggering with phase‐coded inter‐pulse overlay to combat delay–Doppler ambiguities

Pulse repetition frequency (PRF) staggering is a well-established technique in the pulse-Doppler radar, used in order to increase the unambiguous range and Doppler and to minimise blind spots. The study offers a new staggering dimension that can increase the number of different staggering choices. The new approach is to add an inter-pulse coherent phase-coded overlay. The code is shorter than the number of pulses in each PRF batch and is repeated an integer number of times within the batch. A required property of the phase code is perfect periodic autocorrelation, namely zero autocorrelation sidelobes. In order to minimise the added complexity, the overlay codes use a two-phase or three-phase alphabet. As in a conventional PRF staggering, there are P coherent batches, each using a different PRF, with its matched phase-coded overlay. The performance analysis uses a conventional binary integration rule that declares detection if there were H , or more, individual detections in the P batches.

Sequences with Perfect Periodic Auto and Cross Correlation Properties

Sequences with Perfect Periodic Auto and Cross Correlation Properties

Traffic Model Based on Autoregression for PPG Signals in Wearable Networks

Accurate data traffic models are crucial to characterize data flow and understand the network traffic behavior, playing paramount importance to network design and management. This letter presents an autoregression (AR) model to depict photoplethysmogram (PPG) signal-based traffic produced by wearable devices. We validate our model through empirical data using three public datasets and one local dataset. Results show that our AR model reproduces the pattern behavior, trend, and statistical characteristics of the original PPG signal. Also, the generated PPG samples have similar periodic autocorrelation peaks than the original ones, even under an error rate of 14%.

16-QAM Sequences with Good Periodic Autocorrelation Function

16-QAM Sequences with Good Periodic Autocorrelation Function

Optimization of the method for the synthesis of discrete folding signals in the most common bag-box-and-bag systems

Among the main areas of improvement of the performance indicators of information and communication systems (ICS), in particular, noise immunity, secrecy, and information security, it is possible to identify the areas associated with the use of phase-manipulated broadband signals (FM SHPS). Numerous ICS applications point to the importance of periodic autocorrelation properties of the signals used: continuous-beam systems with continuous radiation, pilot channel and channel of synchronization in digital data systems, radar and sonar systems, and others. In addition, a good periodic AKF signal indicates the ability to select signals with good aperiodic ACF. The minimization of the level of the side petals of the AKF is of greatest importance when designing a signal for such applications as measuring the lag time, time resolution, etc. In this paper, the problem of optimizing the synthesis of nonlinear discrete sequences, which have improved ensemble, structural and autocorrelation properties, is formulated and solved. The use of nonlinear discrete signals, which are formed on the basis of such sequences, will provide the necessary values of impedance protection, information and structural secrecy of the operation of the ICS.

Hybrid Zadoff‐Chu and multilateral piecewise exponential companding transform–based PAPR reduction technique in OFDM systems

SummaryThe orthogonal frequency‐division multiplexing (OFDM) is a multicarrier modulation system that is used to transmit the large volume of data to the receiver. Reducing the peak‐to‐average power ratio (PAPR) in OFDM system is one of the demanding and crucial task in recent days. For this reason, various precoding and companding mechanisms are developed in the traditional works, but it remains with the limitations of increased complexity, reduced performance, and nonlinear distortion. The reduction of PAPR is achieved by minimizing the companding distortion with the enhancement of the bit error rate (BER) performance significantly. Then, in order to avoid clipping in OFDM, a multilateral piecewise exponential companding transform (MPECT) method has been utilized rather than using piecewise exponential companding transform (PEC) where PAPR is getting reduced. The OFDM is very sensitive to synchronizing error. To overcome this sensitivity, employ the Zadoff‐Chu sequence to carrier frequency offsets. Zadoff‐Chu matrix transform (ZCMT) has numerous merits among the other ODFM systems such as the improvement in the performance of the channels that are fading away and provides an ideal periodic autocorrelation and a constant magnitude periodic cross correlation. Both of these techniques provide improvement in the ODFM systems. To get more efficiency, this paper aims to develop a hybrid technique by integrating the ZCMT and MPECT techniques for reducing the PAPR in OFDM systems. Further, convolutional encoding is applied for better BER and PAPR. The simulation results of the proposed ZCMT‐MPECT technique are evaluated and compared with the conventional OFDM and other precoding methods.

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.

Modular hyperbolas and Beatty sequences

Bounds for max⁡{m,m˜} subject to m,m˜∈Z∩[1,p), p prime, z indivisible by p, mm˜≡zmodp and m belonging to some fixed Beatty sequence {⌊nα+β⌋:n∈N} are obtained, assuming certain conditions on α. The proof uses a method due to Banks and Shparlinski. As an intermediate step, bounds for the discrete periodic autocorrelation of the finite sequence 0,ep(y1‾),ep(y2‾),…,ep(y(p−1‾)) on average are obtained, where ep(t)=exp⁡(2πit/p) and mm‾≡1modp. The latter is accomplished by adapting a method due to Kloosterman.

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.

Distributed Dynamic Strain Sensing via Perfect Periodic Coherent Codes and a Polarization Diversity Receiver

Rayleigh scattering-based dynamic strain sensing with high spatial resolution, fast update rate, and high sensitivity is highly desired for applications such as structural health monitoring and shape sensing. A key issue in dynamic strain sensing is the tradeoff between spatial resolution and the Signal-to-Noise Ratio (SNR). This tradeoff can be greatly relaxed with the use of coding. A sequence of optical pulses is injected into the fiber and the detected backscattered signal is cross correlated with the original signal. With the use of coding, SNR is indeed improved, but if the sequence is not well chosen, the resulting Peak to Sidelobe Ratio (PSR) can be rather low. An excellent choice of codes are biphase Legendre sequences which offer near Perfect Periodic Autocorrelation (PPA). Other common issues in Rayleigh scattering-based sensing techniques are signal fading and dynamic range. The former issue can occur due to destructive interference between lightwaves that are scattered from the same spatial resolution cell and, in coherent detection schemes, when the polarization states of the backscattered light and the reference light are mismatched. The latter issue is a concern in phase sensitive schemes which require signal jumps not to exceed 2π. In this paper, a biphase Legendre sequence with 6211 pulses is used in conjunction with polarization diversity scheme and a PM fiber. The setup provides two independent measurements of the sensing fiber complex profile and achieves highly sensitive, distributed dynamic strain sensing with very low probability of fading. In addition, the system can handle both very large perturbation signals and very small perturbation signals. The system operated at a scan rate of ~107 kHz and achieved spatial resolution of ~10 cm and sensitivity of ~1.1mrad/ √Hz. The ratio between the powers of the maximum and minimum excitations that can be measured by the system is 136 dB.

Linear Complexity of Geometric Sequences Defined by Cyclotomic Classes and Balanced Binary Sequences Constructed by the Geometric Sequences

Pseudorandom number generators are required to generate pseudorandom numbers which have good statistical properties as well as unpredictability in cryptography. An m-sequence is a linear feedback shift register sequence with maximal period over a finite field. M-sequences have good statistical properties, however we must nonlinearize m-sequences for cryptographic purposes. A geometric sequence is a sequence given by applying a nonlinear feedforward function to an m-sequence. Nogami, Tada and Uehara proposed a geometric sequence whose nonlinear feedforward function is given by the Legendre symbol, and showed the period, periodic autocorrelation and linear complexity of the sequence. Furthermore, Nogami et al. proposed a generalization of the sequence, and showed the period and periodic autocorrelation. In this paper, we first investigate linear complexity of the geometric sequences. In the case that the Chan--Games formula which describes linear complexity of geometric sequences does not hold, we show the new formula by considering the sequence of complement numbers, Hasse derivative and cyclotomic classes. Under some conditions, we can ensure that the geometric sequences have a large linear complexity from the results on linear complexity of Sidel'nikov sequences. The geometric sequences have a long period and large linear complexity under some conditions, however they do not have the balance property. In order to construct sequences that have the balance property, we propose interleaved sequences of the geometric sequence and its complement. Furthermore, we show the periodic autocorrelation and linear complexity of the proposed sequences. The proposed sequences have the balance property, and have a large linear complexity if the geometric sequences have a large one.

Perfect phase-coded pulse trains generated by Talbot effect

Abstract 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.

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.

Large Families of “Grey” Arrays with Perfect Auto-correlation and Optimal Cross-Correlation

Digital watermarking applications have a voracious demand for large sets of distinct 2D arrays of variable size that possess both strong auto-correlation and weak cross-correlation. We use the discrete Finite Radon Transform to construct “perfect” \(p \times p\) arrays, for p any prime. Here the array elements are comprised of the integers \(\{0,\pm 1,+2\}\). Each array exhibits perfect periodic auto-correlation, having peak correlation value \(p^2\), with all off-peak values being exactly zero. Each array, by design, contains just \(3(p-1)/2\) zero elements, the minimum number possible when using this “grey” alphabet. The grey alphabet and the low number of zero elements maximises the efficiency with which these perfect arrays can be embedded into discrete data. The most useful aspect of this work is that large families of such arrays can be constructed. Here the family size, M, is given by \(M = p^2-1\). Each of the \(M(M-1)/2\) intra-family periodic cross-correlations is guaranteed to have one of the three lowest possible merit factors for arrays with this alphabet. The merit factors here are given by \(v^2/(p^2-v^2)\), for \(v = 2\), 3 and 4. Whilst the strength of the auto-correlation rises with array size p as \(p^2\), the strength of the many (order \(p^4\)) cross-correlations between all M family members falls as \(1/p^2\).

Perfect Sequences of Odd Prime Length

A sequence is said to be perfect if it has an ideal periodic autocorrelation function. In addition, the degree of a sequence is defined as the number of distinct nonzero elements within each period of the sequence. This study presents a systematic method for constructing perfect sequences (PSs) of odd prime periods, where the general constraint equations for the sequence coefficients are derived, providing a solid theoretical foundation to the construction of PSs. The proposed scheme commences by partitioning a cyclic group $\mathbf {Z}_P=\lbrace 1,2, \ldots,P-1\rbrace$ into $K$ cosets of cardinality $M$ , where $P=K\cdot M+1$ is an odd prime. Based on this partition, the degree-( $K$ + 1) PSs are then constructed. Finally, case studies are presented to illustrate the proposed construction.

Code shift keying based joint radar and communications for EMCON applications

Due to the explosion in the demand for broadband wireless services by consumers, wireless network service providers require increased frequency allocations from an already crowded frequency spectrum. In an effort to satisfy the increased frequency spectrum requirements of wireless communications networks, several methods have been proposed to allow communication systems and radars to co-exist within the same bandwidth or adjacent frequency bands while minimizing or preventing mutual interference. Most methods explored in the literature focus on the use of cognitive sensing and dynamic spectrum allocation. Other proposed methods to prevent mutual interference between radar and communications systems focus on waveform design. In this manuscript, a dual function communication and radar system with EMCON applications is investigated that uses Gold or Kasami codes in a code shift keying digital modulation scheme. The use of both binary phase shift keying (BPSK) and quaternary phase shift keying (QPSK) to modulate the code bits is investigated. The communications signal is used as the radar signal to implement a pseudo-random BPSK or QPSK coded radar. The symbol error rate (SER) of the communication system is determined through Monte Carlo simulations and compared to an SER upper bound for M-ary frequency shift keying. The radar signal's periodic autocorrelation function (PACF) and periodic ambiguity function are also examined demonstrating the schemes potential for use in a low probability of intercept radar application. The radar signal's characteristics are compared when using pseudo-random BPSK and QPSK modulations.

Nearly perfect Gaussian integer sequences with arbitrary degree

Based on p-ary pseudorandom sequences, this study proposes a construction of degree-k Gaussian integer sequences of period N = k ( p m − 1 ) / ( p − 1 ) by utilising kth power residue symbol satisfying k | ( p − 1 ) , where p is an odd prime and positive integers m , k ≥ 2 . The periodic autocorrelation values are 0 at shifts τ ≢ 0 ( mod ( N / k ) ) of the resultant sequences. Specially, there is exactly one non-zero out-of-phase periodic autocorrelation value of the resultant sequences for k = 2 . The non-zero elements of the sequences are balanced and can be predefined flexibly. Moreover, the maximum energy efficiency of the proposed sequences is close to ( p − 1 ) / p for sufficiently large m.

On a Special Quaternionic Sequence

In this study, we investigate Fibonacci quaternions and their some important properties. Then, we define a special sequence using the elements of the Fibonacci quaternion sequence. Furthermore, we calculate the autocorrelation, right and left periodic autocorrelation values by using the elements of the newly defined sequence.