Periodic Autocorrelation Research Articles

Periodic INAR(1) model with Bell innovations distribution

Abstract In this paper, we introduce a class of periodic integer-valued autoregressive BL-PINAR(1) models with Bell innovations distribution based on the binomial thinning operator. The basic probabilistic and statistical properties of this class are studied. Indeed, the first and the second moment periodically stationary conditions are established. The closed forms of these moments are, under the obtained conditions, derived. Furthermore, the periodic autocovariance structure is also considered while providing the closed form of the periodic autocorrelation function. The conditional least squares (CLS), Yule–Walker (YW), weighted conditional least squares (WCLS), and conditional maximum likelihood (CML) methods are applied to estimate the underlying parameters. The asymptotic properties of the CLS and the YW estimators are obtained. The performances of these methods are compared through a simulation study. An application on a real data set is provided.

A Review Study of Prime Period Perfect Gaussian Integer Sequences

Prime period sequences can serve as the fundamental tool to construct arbitrary composite period sequences. This is a review study of the prime period perfect Gaussian integer sequence (PGIS). When cyclic group {1,2,…,N−1} can be partitioned into k cosets, where N=kf+1 is an odd prime number, the construction of a degree-(k + 1) PGIS can be derived from either matching the flat magnitude spectrum criterion or making the sequence with ideal periodic autocorrelation function (PACF). This is a systematic approach of prime period N=kf+1 PGIS construction, and is applied to construct PGISs with degrees 1, 2, 3 and 5. However, for degrees larger than 3, matching either the flat magnitude spectrum or achieving the ideal PACF encounters a great challenge of solving a system of nonlinear constraint equations. To deal with this problem, the correlation and convolution operations can be applied upon PGISs of lower degrees to generate new PGISs with a degree of 4 and other higher degrees, e.g., 6, 7, 10, 11, 12, 14, 20 and 21 in this paper. In this convolution-based scheme, both degree and pattern of a PGIS vary and can be indeterminate, which is rather nonsystematic compared with the systematic approach. The combination of systematic and nonsystematic schemes contributes great efficiency for constructing abundant PGISs with various degrees and patterns for the associated applications.

Periodic autocorrelation of sequences

Periodic autocorrelation of sequences

LPI Radar Detection Based on Deep Learning Approach with Periodic Autocorrelation Function.

In electronic warfare systems, detecting low-probability-of-intercept (LPI) radar signals poses a significant challenge due to the signal power being lower than the noise power. Techniques using statistical or deep learning models have been proposed for detecting low-power signals. However, as these methods overlook the inherent characteristics of radar signals, they possess limitations in radar signal detection performance. We introduce a deep learning-based detection model that capitalizes on the periodicity characteristic of radar signals. The periodic autocorrelation function (PACF) is an effective time-series data analysis method to capture the pulse repetition characteristic in the intercepted signal. Our detection model extracts radar signal features from PACF and then detects the signal using a neural network employing long short-term memory to effectively process time-series features. The simulation results show that our detection model outperforms existing deep learning-based models that use conventional autocorrelation function or spectrogram as an input. Furthermore, the robust feature extraction technique allows our proposed model to achieve high performance even with a shallow neural network architecture and provides a lighter model than existing models.

Channel estimation with Zadoff–Chu sequences in the presence of phase errors

AbstractDue to their perfect periodic autocorrelation property, Zadoff–Chu sequences are often used as stimulus signals in the measurement of radio channel responses. In this letter, the cross‐correlation of a linear shift‐invariant system's response to a Zadoff–Chu sequence is derived for the case that a multiplicative perturbation such as phase noise is present at its output. The perturbation signal is shown to incur a spurious discrete Fourier transform, thus degrading the correlation's peak to sidelobe ratio. The implications for the performance limits of channel measurement systems are discussed.

Sets of Nonbinary Sequences with a Low Level of Mutual Correlation for Systems of Digital Information Transmission

The sets of nonbinary pseudorandom sequences (NPSs) for periods N = p^S– 1 20 000 (p = 3, 5,7, and 11) generated in finite fields GF(pS) whose power is V = N + 1 and the maximum of the module of peaks of the periodic autocorrelation function (PACF) and periodic cross-correlation function (PCCF) satisfy the bounds obtained by Sidel’nikov. In addition to the minimal polynomials of elements α and α^2 , where α is a primitive element of field GF(p^S), the minimal polynomials of elements α and α^(i_d)(i_d is the decimation index) are determined on the basis of which the new sets of NPSs can be formed with the equivalent correlation properties. Sets of indices i_d 2 are determined for various combinations of parameters p and S. The cases of even and odd values of parameter S are considered, for which the values of the PACF and PCCF with the maximum module are obtained and the number and values of different levels of correlation functions are determined.

On maximal autocorrelations of Rudin–Shapiro sequences

In this paper, we present an alternative proof showing that the maximal aperiodic autocorrelation of the mth Rudin–Shapiro sequence is of the same order as λm, where λ is the real root of x3+x2−2x−4. This result was originally proven by Allouche et al. (2019) and Choi (2020) using a translation of the problem into linear algebra. Our approach simplifies this linear algebraic translation and provides another method of dealing with the computations given by Choi. Additionally, we prove an analogous result for the maximal periodic autocorrelation of the mth Rudin–Shapiro sequence. We conclude with a discussion on the connection between the proofs given and joint spectral radius theory, as well as a couple of conjectures on which autocorrelations are maximal.

Low-PAPR OFDM Waveform Design for Radar and Communication Systems

In this correspondence we address the problem of designing OFDM waveforms with low peak-to-average power ratio (PAPR) for joint radar and communication (JRC) applications. The design problem is formulated as the minimization of PAPR with the constraint that the integrated periodic autocorrelation sidelobe level is zero. We propose an iterative algorithm based on the majorization-minimization (MM) technique to solve this design problem. The proposed algorithm yields OFDM waveforms with PAPR values close to one. The algorithm can also be used to design waveforms with spectral nulls. In extensive numerical simulations we compare the performance of our algorithm with that of a recently proposed benchmark.

Novel Zero Circular Convolution Sequences for Detection and Channel Estimations

Multiuser communication systems or multiple access scheme systems favor sequences possessing the ideal periodic cross-correlation function (PCCF) property. In comparison, channel estimation, equalization, and synchronization applications favor sequences possessing the ideal periodic auto-correlation function (PACF) property. However, there is no set of sequences possessing both the ideal PCCF and PACF properties simultaneously, where auto-correlation and cross-correlation balance each other. In this work, sequences possessing the ideal PACF property are used as the base sequences. Then, a modulation technique is applied upon these base sequences to construct a set of zero circular convolution (ZCC) sequences within which an arbitrary pair of two sequences possesses the ideal PCCF property. Compared with least squares (LS) and minimum mean squared error (MMSE) algorithm, the simulation results show that the channel estimation performance of ZCC is better than MMSE and LS algorithms, and the computational complexity of the algorithm is the same as LS algorithm, but far lower than MMSE algorithm. This is the first study on ZCC sequences reported in the literature, in which their fundamental theorems, properties, construction, and applications are investigated. The advantage of possessing the desired PACF and the ideal PCCF properties allows the ZCC sequences to be used in a broader range of applications than other sets of sequences.

On Periodic Generalized Poisson INAR(p) Models

This paper deals with some probabilistic and statistical properties of periodic generalized Poisson integer-valued autoregressive processes of order p, Necessary and sufficient conditions for the periodic stationarity, both in mean and second order, are established. The closed-forms of the mean and the second moment are, under these conditions, obtained. Moreover, the Wold–Cramér expression of the underlying second-order periodically stationary process is then established. The autocovariance structure is studied, while providing the closed-form of the periodic autocorrelation function. The Yule–Walker (YW), the two-stage conditional least squares (CLS) and the conditional maximum likelihood (CML) estimation methods of the underlying parameters are obtained. An intensive simulation study and an application on a real count data consisting of the daily number of daytime road accidents in Schiphol area, in Netherlands are provided.

Doppler Shift Tolerance of Typical Pseudorandom Binary Sequences in PMCW Radar.

In the context of all-digital radar systems, phase-modulated continuous wave (PMCW) based on pseudorandom binary sequences (PRBSs) appears to be a prominent candidate modulation scheme for applications such as autonomous driving. Among the reasons for its candidacy are its simplified transmitter architecture and lower linearity requirements (e.g., compared to orthogonal-frequency division multiplexing radars), as well as its high velocity unambiguity and multiple-input multiple-output operation capability, all of which are characteristic of digital radars. For appropriate operation of a PMCW radar, choosing a PRBS whose periodic autocorrelation function (PACF) has low sidelobes and high robustness to Doppler shifts is paramount. In this sense, this article performs an analysis of Doppler shift tolerance of the PACFs of typically adopted PRBSs in PMCW radar systems supported by simulation and measurement results. To accurately measure the Doppler-shift-induced degradation of PACFs, peak power loss ratio (PPLR), peak sidelobe level ratio (PSLR), and integrated-sidelobe level ratio (ISLR) were used as metrics. Furthermore, to account for effects on targets whose ranges are not multiples of the range resolution, oversampled PACFs are analyzed.

Phase Noise Effects on Phase-Sensitive OTDR Sensors Using Optical Pulse Compression

We introduce a detailed theoretical, numerical, and experimental study of the effects of laser’s phase noise on the performance of phase-sensitive optical time-domain reflectometry ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\phi$</tex-math></inline-formula> -OTDR) sensors that use optical pulse compression (OPC). Pulse compression is a technique that can be used to improve the received signal amplitude by increasing the effective energy of the pulses that are launched into the fiber without degrading the spatial resolution of the measurements. Therefore, it is a valuable tool to extend the range of these sensors and mitigate fiber attenuation constraints. However, it has been observed that the limited coherence of the laser source has a degrading effect on the actual performance enhancement that this method can provide. Here, we derive a theoretical model that can be used to quantify this degradation for any type of OPC such as those based on either linear frequency modulation (LFM) pulses or perfect periodic autocorrelation (PPA) bipolar bit sequences. The model facilitates numerical estimation of the sensitivity of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\phi$</tex-math></inline-formula> -OTDR measurements. It also produces theoretical expressions for the mean and the variance of the phase-noise perturbed backscatter response. These results are validated via numerical simulations and experiments in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\phi$</tex-math></inline-formula> -OTDR setups using LFM as well as PPA OPC. Furthermore, we demonstrate the use of the model to investigate the basic trade-offs involved in the design of OPC <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\phi$</tex-math></inline-formula> -OTDR systems.

Ultrahigh scan-rate quasi-distributed acoustic sensing system using array match interrogation.

Inspired by compressed sensing techniques, a method for significantly enhancing the maximum allowable scan rate in quasi-distributed acoustic sensing (Q-DAS) is described and studied. Matching the scan parameters to the interrogated array facilitates orders of magnitude improvement in the scan rate and a corresponding increase in the maximum slew rate (SR) of differential phase variations which can be measured without ambiguity. The method is termed array matched interrogation (AMI). To improve the method's SNR, maximum number of sensing sections and maximum range, the interrogation pulse can be replaced by a perfect periodic autocorrelation (PPA) code. This version of the method is referred to as coded array matched interrogation (C-AMI). The implementation of C-AMI is not trivial and requires special design rules which are derived and tested experimentally. The design rules ensure that the 'folding' of the returning peaks of the Q-DAS array into a scan period, which is much shorter than the fiber's roundtrip time, will not lead to overlaps. The method demonstrated a scan rate of 20 times higher than the common limit and measurement of unprecedented slew-rate of 10.5 ×106 rad/s.

СИНТЕЗ ПОСЛЕДОВАТЕЛЬНОСТЕЙ С ИДЕАЛЬНЫМИ АВТОКОРРЕЛЯЦИОННЫМИ СВОЙСТВАМИ

Method of synthesis of the sequences, possessing ideal autocorrelation functions, has been developed. The review of existing approaches, analytical conclusion of requirements to ideal sequences with considering the necessity to provide low pick-factor of the sequence and low level of side petals of autocorrelation function has been given. Mathematical model and its program embodiment, allowing to generate periodical sequences and assess their properties, have been created on the basis of suggested method. As a result of pursued computational experiments, the sequences with number of symbols from 6 to 4096 have been obtained. It’s been clarified that such sequence can be formed at its any length which has ideal periodic autocorrelation function at pick-factor equal or close to 1. Because suggested sequences have ideal ACF they can be recommended for use in the systems of railway automatics and communication for highly reliable management of train traffic on high-speed trunks as well as in communication systems qua signals in navigating and locational systems, qua signals applied in ultrasound defectoscopy and so on.

M-ARY CDMA SCHEME WITH CCSK MODULATION AND PARALLEL INTERFERENCE CANCELLATION TECHNIQUE

In this paper, we tackle the problem of implementation a parallel interference cancellation scheme for multi-user detection (MUD) in M-ary CCSK-modulated CDMA system. Complexity remains the main obstacle to the practical implementation of MUD and a factor limiting, if not the very possibility, then the feasibility of using high order M-ary modulation formats for applications such as low-probability-of-intercept systems with a large spreading factor or, myriads of small self-powered IoT devices. The design of MUD algorithms for M-ary systems leads to a new round of growth in the receiver complexity comparing with the binary modulation. Demodulator has a major impact on the complexity of CDMA system. Cyclic code-shift keying (CCSK) is a modulation technique which is designed to reduce the complexity of M-ary signaling. In this, each symbol is a circularly shifted version of a single code sequence. Assuming synchronization, the receiver cyclically correlates the input signal plus noise with the base sequence and estimates the position of the correlation peak. The preliminary stage of the proposed scheme is a conventional multi-channel CDMA receiver. The preliminary estimates of the data is multiplied by user codes and amplitude estimates in the spreader, and then fed to the adder to generate the MAI estimates. After the latter are subtracted from the group signal, K-dimensional vector of cleared user signals is passed to the matched filter bank to form the refined estimates of the data. At each subsequent stage, the estimates of the output data of the previous stage are used as input data. The complexity gain is the ratio of the number of computations for finding cyclic convolution and, the amount of computations for finding M linear convolutions, i.e., M/log2M. Since the performance of the algorithm is sensitive to the reliability of the preliminary decisions, there are strict criteria for the selection of spreading codes: the length which must be an integer power of 2, the large family size, good periodic autocorrelation and good cross-correlation properties. To examine the proposed algorithm using computer simulation, we selected minimax periodic and odd-periodic complementary codes, the properties of which are close to the properties of the codes used in the CCSK scheme implemented in Link-16 protocol tactical data networks JTIDS and MIDS. Our study shows that the gain over the conventional receiver increases as the SNR increases, achieving 15 dB for BER 10-5. The system with complementary codes outperform system based on minimax PN-codes, achieving a target bit error rate at a lower SNR.

Identification of Widely Linear Systems Using Data-Dependent Superimposed Training

In this correspondence, we address the identification of widely linear (WL) systems using data-dependent superimposed training (DDST). The analysis shows that the nonlinear nature of WL systems can be exploited to decouple the finite impulse responses of the filters that constitute the system under identification. Unlike the DDST scheme considered for strictly linear systems, for the WL scenario two data-dependent sequences are required to distort the transmitted data and avoid interference with the training sequence during the estimation process. Closed form expressions for sequences with a complete second order characterization (good periodic autocorrelation and zero complementary periodic autocorrelation) are also provided. The performance of the method is compared with other techniques using numerical simulations.

Algorithm for Generating Extra-Long Gordon–Mills–Welch Sequences

The modification of an algorithm for determining polynomial multipliers hci(x) of test polynomial hGMWS(x), which is the main component of the method for synthesizing Gordon–Mills–Welch sequences (GMWSs), is set as a basis for developing the software implementation of the algorithm for generating extra-long GMWSs having a two-level periodic autocorrelation function and high structural latency and generated over the finite field with double extension GF(2S) = GF[(2m)n]. In the modified algorithm, the expressions for the number of operations in computing the vector of alternatives, from which the vector of decimation indices is formed, are derived for different values of parameter n. These expressions also serve as upper estimates for the number of summed sequences. To generate extra-long GMWSs with periods from N = 212 – 1 = 4095 to N = 220 – 1 = 1 048 575, the sets of vectors of decimation indices are obtained for admissible values of parameters m and n.

Three new lengths for cyclic Legendre pairs

Introduction: It is conjectured that the cyclic Legendre pairs of odd lengths &gt;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

Improved LPD Characteristics for QS-DS-CDMA Employing Randomization Techniques

Easily and flexibly deployable ad-hoc communication networks emerging in tactical military or even civilian contexts, frequently suffer from poor synchronization due to lack of coordinating infrastructure. In addition to synchronization issues, and especially in military settings, security from the aspect of detectability is also of crucial importance. Imperfect synchronization can be dealt with by making use of Quasi-Synchronous Code Division Multiple Access (QS-CDMA), relying on Loosely Synchronous Codes to maintain orthogonality in the presence of limited time delays. Security, in terms of low probability of detection (LPD) from the standpoint of a malicious adversary, can be improved (reduced detection) by employing randomization techniques that disrupt the inherent structure of the transmitted QS-CDMA signals. This is based on the fact that QS-CDMA signals are Cyclostationary, having (almost) periodic Auto-Correlation functions (ACF) due to eminent signal periodicities (such as spreading code repetition). In this paper, we propose techniques to disturb the ACF and equivalently the Spectral Correlation function, and reduce the Degree of Cyclostationarity (DCS), our LPD measure. Specifically, we investigate randomization via 1) random per symbol time dithering and 2) random selection of spreading sequences, as well as a hybrid approach combining time dithering and code randomization. In all proposed techniques knowledge of the randomization pattern is not required at the legitimate receiver. We derive the Spectral Correlation function of the QS-CDMA signal under the proposed randomization schemes and compare it to simulations. We show through analysis and extensive numerical simulations that the proposed technique can reduce the DCS by almost two orders of magnitude. We also show that enhanced LPD can be achieved using the proposed techniques while sacrificing a part of the reduced time synchronization requirement. We further analyze the implications of the friendly receiver not knowing the randomization pattern and present results on the resulting communication performance.

The sparsity of the response of a quasi-distributed fiber optic sensing system allows ‘overclocking’ its interrogation

The response of a quasi-distributed fiber optic sensing system to a short interrogation pulse can be considered as a sparse signal in the time domain. This attribute can be exploited to enhance its performance beyond the commonly accepted limits. The work described here is analogous to some compressed sensing systems, which can be adequately sampled at sub-Nyquist rates, based on knowledge of their sparsity in the spectral domain. Equivalently, it is shown here how the sparsity of the impulse-response of a Quasi-Distributed Acoustic Sensing (Q-DAS) system can be exploited to interrogate it at rates well above the limit dictated by the total duration of its impulse response. An additional improvement in performance is achieved by using a Perfect Periodic Auto-correlation (PPA) code, rather than a pulse, for interrogation of the system. Cross-correlating the raw response of the Q-DAS system with the original code retains the sparsity of the response and provides significant gain in SNR without losing spatial resolution. Experimentally, the method was utilized to implement a 76 km long acoustic sensing system and successfully detected 250 kHz ultrasound signal. This was achieved by interrogating the system at a rate of 971 kHz which is 737 times the inherent limit dictated by the duration of the impulse-response of the system.