Periodic Correlation Function Research Articles

Correlation properties of complete code of binary spread spectrum phase-shift keyed signals with polarization coding

Problem statement. Investigation of correlation properties of complete code of binary spread spectrum phase-shift keyed signals as the limiting case of large systems is a topical task for building large systems of binary spread spectrum phase-shift keyed signals with polarization coding with optimal correlation properties. Also, this task is of independent academic interest due to the fact that the alphabet of symbols for these signals lacks multicasting properties unlike the familiar cases of complete codes of spread spectrum phase-shift keyed signals. Objective. To accurately define the distribution laws as well as the moments about zero and the mean for the n-th order of presented values of aperiodic and periodic correlation functions of the complete code of binary spread spectrum phase-shift keyed signals with polarization coding. To offer approximations of accurate distribution laws suitable for engineering applications. Results. The accurate distribution laws for presented values of aperiodic and periodic correlation functions of the complete code of binary spread spectrum phase-shift keyed signals with polarization coding were found in the form of the related distribution series. The analytic expressions for the n-th order of presented values of these functions for the complete code of the specified signals were assigned. The calculation example for the distribution series of presented values of aperiodic and periodic correlation functions for the complete code of the specified signals with the length of vector code sequences N=4 was given. The values of the relative frequencies of the correlation functions values for this case were found by the means of direct computing and their exact match with the corresponding predicted probability values was demonstrated. The approximations of the accurate distribution laws were given: for periodic correlation functions case – by Gaussian probability density, and for the aperiodic functions case – by the Pearson curve of VII type. Practical relevance. The found accurate laws for presented values of aperiodic and periodic correlation functions of the complete code of binary spread spectrum phase-shift keyed signals with polarization coding and their approximations allow to assess the correlation properties of large systems of such signals.

Rotor crack and bearing fault diagnosis based onVMD-ICA

Abstract The rotor-bearing system is vulnerable to rotor cracks, bearing failures, or even multiple failures at the same time during long-term service, and it is more difficult to diagnose the faults under rotor crack and bearing fault coupling conditions. This paper proposed a signal reconstruction method to classify the rotor crack, the bearing outer ring fault and their coupling faults based on Variational Mode Decomposition (VMD) and Independent Component Analysis (ICA). Firstly, the original vibration signal collected on a rotor-bearing test bench is decomposed with VMD, and several intrinsic mode functions (IMF) are obtained. Then, the correlation kurtosis CK M which introduces the characteristics of periodic correlation function is to evaluate the decomposed IMF component, and screen the rotor crack information component. Finally, the rotor crack information component is input into a virtual channel signal and separated with the original vibration signal with ICA, and then the separated two sets of signals are superimposed to obtain a vibration signal that simultaneously highlights the information of the rotor crack and the bearing outer ring fault. In addition, the time domain signal is directly input into a LightGBM to diagnose different faults. The research results show that this method can simultaneously diagnose the rotor crack, the bearing outer ring fault and the coupling fault.

МЕТОДИКА ВОССТАНОВЛЕНИЯ ИНФОРМАЦИИ КОНЦА OFDM СИМВОЛОВ, СУММИРУЕМЫХ С CAZAC ПОСЛЕДОВАТЕЛЬНОСТЯМИ ПРИ СИНХРОНИЗАЦИИ ПО ЦИКЛИЧЕСКОМУ ПРЕФИКСУ

In previous studies, it was proved to increase the synchronization efficiency by the periodic correlation function (CF) of the cyclic prefix (CP) OFDM symbols of LTE technology when using as a CAZAC CP the sequences of the Zadoff-Chu (ZC(2,9)) as part of the "empty" symbols of the central part of the frequency range of the LTE frame transmitted to DownLink (DL) direction. In this study, the symbols of the central part of the frequency range of the LTE frame are not "empty" and the information of the end of the OFDM symbols, on which elements of multiphase sequences are superimposed, is restored on the receiving side. A comparative analysis of the characteristics of cyclic CF sequences of Zadoff-Chu and Frank, used to fill the CP and the end of the characters in the sum with the elements of the binary information sequence, is carried out. The research was carried out in the MATLAB mathematical modeling system on a model of a communication channel close to ideal, with QPSK, 64QAM modulation options for frame information symbols. Analysis of the research results showed that the use of multiphase CAZAC sequences makes it possible to increase the efficiency of synchronization over the CP OFDM symbols of the LTE technology frame both for "empty" symbols and for symbols with the restoration of useful information elements of the end of the OFDM symbol from the sum of binary modulated codes and multiphase sequences. The modulus of the merit factor when filling the CP with the sum of modulated bit data and CAZAC sequences is higher by 0.425dB...1.404dB than when filling the CP with only modulated bit data. When summing sequences, the value of the peak factor of the transmitted symbols increases, but by an insignificant amount – no more than 0.175dB ... 0.418dB, depending on the type of modulation and the choice of the CAZAC sequence.

Performance of ZCZ codes in QS-DS-CDMA communication systems

Zero correlation zone (ZCZ) codes have been recently adopted for many communication systems based on code division multiple access (CDMA). In this paper, we focus on the use of ZCZ codes in quasi-synchronous direct-sequence (QS-DS) CDMA. To ensure Interference-free communication, spreading ZCZ codes must be carefully designed. Indeed, numerous ZCZ code constructions with various correlation properties have been proposed and studied in literature. However, most of them are not adequately designed for communication systems. Here, we aim at determining specific criteria in order to choose the suitable ZCZ code family. A new bit error rate (BER) for QS-DS-CDMA as a function of the code's even and odd periodic correlation functions (PCFs) is derived. It is shown that the adequate spreading codes are those with zero even and odd PCFs.

FPGA Implementation of a Novel Construction of Optical Zero-Correlation Zone Codes for OCDMA Systems

Abstract This paper proposes a novel method for constructing optical zero-correlation zone (OZCZ) codes based on the Mutually Orthogonal Complementary Sets (MOCS) matrix and the concatenation operation. The most important property of the obtained codes is that the periodic correlation function of the proposed codes is zero for the phase shifts within the zero-correlation zone. The procedure of construction is simple, flexible and offers increasing number of users M = 2 n + 1 \left( {M = {2^{n + 1}}} \right) . In the other hand, we have presented a fully reconfigurable hardware implementation based on very high description language of the proposed OZCZ codes for optical code-division multiple-access encoder system. The implementation is performed on an Altera field-programmable gate arrays device using ROM memory and an up counter. The simulation results have been found satisfactory and are in conformity with the theoretical observations.

A New Method to Construct Strictly Optimal Frequency Hopping Sequences With New Parameters

In order to evaluate the goodness of frequency hopping (FH) sequence design, the periodic Hamming correlation function is used as an important measure. Usually, the length of correlation window is shorter than the period of the chosen FH sequence, so the study of the partial Hamming correlation of FH sequence is particularly important. If an FH sequence or an FH sequence set has an optimal partial Hamming correlation with respect to the partial Hamming correlation bound for all length of correlation window, then the FH sequence or the FH sequence set is said to be strictly optimal. In this paper, we first prove that there is no strictly optimal FH sequence set of family size $M$ and sequence length $N$ over a frequency slot set of size $q$ with respect to the partial Hamming correlation bound derived by Niu et al. when $N>{q^{2}}/{M}$ and $q\geq 2$ , and that by Cai et al. when $N>{q^{2}}/{M}$ and $q\geq {2N}/({N-2})$ . Furthermore, we define a special partition-type difference packing (DP) called $[N,\nabla,H_{a}^{l}]$ PDP and give several classes of $[N,\nabla,H_{a}^{l}]$ PDPs. Then, we present a new construction of strictly optimal FH sequences. By choosing different PDPs, the FH sequences constructed can give new and flexible parameters. By utilizing this construction method recursively, we can obtain new $[N,\nabla,H_{a}^{l}]$ PDPs, which lead to infinitely many classes of strictly optimal FH sequences with new parameters. Moreover, based upon an $[N,\nabla,H_{a}^{l}]$ PDP, we present a construction of strictly optimal FH sequence sets. By preceding construction method and recursive construction, we can also obtain infinite classes of strictly optimal FH sequence sets which can give new and flexible parameters.

Optimization of Code Constructions of Binary Sequences Class on a Basis of Set-Theoretical Representation

In this paper there are solved two related problems: the development of the method and correspondent algorithm of code binary sequences synthesis with property of minimal level of side lobes of their aperiodic autocorrelation functions. The method is based on discrete representation of autocorrelation function in form of equations system considered on a set of integer numbers, set-theoretical representation of components of the sequence, their integer-number transformations, mutual properties and relations. There are obtained the expressions defining the dependences of absolute value of the sequence elements on a sum of levels of side lobes of aperiodic and periodic correlation functions; there are defined the necessary conditions of existence binary code sequences ; there are developed the mathematical models of binary code sequences components transformations. The efficiency criterion of each proposed algorithm is the relation of amount of all possible variants of binary code sequences of definite size l to amount of defined by proposed algorithms; the expressions for these quantities are obtained. The efficiency of proposed method and algorithms is verified by simulation results and it increases with increase of size l.

A New Construction of Optical Zero-Correlation Zone Codes

Abstract A new method for constructing optical zero-correlation zone (OZCZ) codes based on the Hadamard matrix and the concatenation operation is proposed. The most important property of the obtained codes is that the periodic correlation function of the proposed codes is zero for the phase shifts within the zero-correlation zone. The structure of the proposed OZCZ code is simple and their construction is more flexible. The proposed code is expected to be useful for reducing co-channel interference in optical code division multiple access system.

Correlation Property and Low Bound of Time-Hopping Codes for UWB Communications

In this paper, we analyze correlation property of time-hopping (TH) codes for time-hopping spread spectrum ultra wideband (THSS-UWB) communication systems. Several different definitions of TH periodic correlation function are compared and discussed. Based on the comparison, the relation between TH codes and frequency-hopping (FH) codes is obtained. Also, the averages of TH periodic correlation function values are investigated, and the relations between averages and four parameters of TH codes are introduced. Based on the result, low bound of maximal TH correlation function values is further given, where the expression of low bound of TH codes relates codes period, the number of time slots, TH codes family size and maximal TH correlation function values.

Sets of frequency hopping sequences under aperiodic Hamming correlation: Upper bound and optimal constructions

In order to evaluate the goodness of frequency hopping (FH) sequence design, the periodic Hamming correlation function is used as an important measure. Aperiodic Hamming correlation of FH sequences matters in real applications, while it received little attraction in the literature compared with periodic Hamming correlation. In this paper, an upper bound on the family size of FH sequences, with respect to the size of the frequency slot set, the sequence length, the maximum aperiodic Hamming correlation is established. Further, a construction of optimal FH sequence sets under aperiodic Hamming correlation from Reed-Solomon codes is presented, whose parameters meet the upper bound with equality. From generalized $m$ sequences (GM sequences) and generalized Gordon-Mills-Welch sequences (GGMW sequences), two classes of optimal FH sequence sets under aperiodic Hamming correlation are also presented, whose parameters meet the upper bound with equality.

Research on perfect binary correlation sequences based on higher-order cumulant

Current researches define the perfect signal by the inner product of sequence itself and its delay sequence. However, these traditional sequences based on second-order statistics cannot handle colored Gaussian measurement noise automatically. Because higher-order cumulant (HOC) is insensitive to the adding Gaussian noise and symmetrical non-Gaussian noise, a new kind of perfect binary signal with good periodic correlation function based on HOC is presented. This paper proposes a new concept of perfect binary-third-order cyclic autocorrelation sequences (PBTOCAS) and defines the quasi-perfect binary-third-order cyclic autocorrelation sequences (QPBTOCAS) and the almost perfect binary-third-order cyclic autocorrelation sequences (APBTOCAS). Then the properties of these binary sequences are studied, and we theoretically prove that binary-third-order cyclic autocorrelation sequences can effectively suppress colored Gaussian noise. Finally, some QPBTOCAS and APBTOCAS with short lengths by computer searching are listed. From the observation of the PBTOCAS, we can see that it can well suit engineering applications, remedying the defect of the conventional pseudo-noise code used in very low signal-noise-ratio environments.

Low-hit-zone frequency hopping sequence sets with optimal partial Hamming correlation properties

In order to evaluate the goodness of frequency hopping sequence (FHS) design, the periodic Hamming correlation function is used as an important measure. Usually, the length of correlation window is shorter than the period of the chosen FHS, so the study of the partial Hamming correlation of FHS is particularly important. In this paper, a class of low-hit-zone (LHZ) FHS sets with good partial Hamming correlation properties which has not yet been reported previously is constructed based on interleaving techniques. It is shown that new FHS sets are optimal with respect to the partial Hamming correlation bound of FHSs with LHZ.

Even periodic and odd periodic complementary sequence pairs from generalized Boolean functions

A pair of two sequences is called the even periodic (odd periodic)complementary sequence pair if the sum of their even periodic (oddperiodic) correlation function is a delta function. The well-knownGolay aperiodic complementary sequence pair (Golay pair) is aspecial case of even periodic (odd periodic) complementary sequencepair. In this paper, we presented several classes of even periodicand odd periodic complementary pairs based on the generalizedBoolean functions, but which do not form Gloay pairs. The proposedsequences could be used to design signal sets, which have beenapplied in direct sequence code division multiple (DS-CDMA) cellularcommunication systems.

A New Class of Binary Zero Correlation Zone Sequence Sets

This paper proposes a new class of binary zero correlation zone (ZCZ) sequence sets, in which the periodic correlation functions of the proposed sequence set is zero for the phase shifts within the zero- correlation zone. It is shown that the proposed zero correlation zone sequence set can reach the upper bound on the ZCZ codes.

Frequency/time hopping sequence sets with optimal partial Hamming correlation properties

Frequency hopping/time hopping(FH/TH) sequences have found wide applications in various modern FH/TH spread spectrum communications, such as digital cellular mobile communication, short-range wireless communication, personal communication, and multi-user radar, sonar systems. In order to evaluate the goodness of FH/TH sequences design, the periodic Hamming correlation function is used as an important measure. Usually, the length of correlation window is shorter than the period of the chosen FH/TH sequence, so the study of the partial Hamming correlation of FH/TH sequence is particularly important. In this paper, a new class of FH/TH sequence sets with good partial Hamming correlation properties is proposed and investigated. The construction of new FH/TH sequence sets is based upon the generalized m-sequence and generalized Gordon-Mills-Welch sequence over a polynomial residue class ring. It is shown that new FH/TH sequence sets are optimal with respect to the partial Hamming correlation bound.

A Ternary Zero-Correlation Zone Sequence Set Having Wide Inter-Subset Zero-Correlation Zone

The present paper introduces a novel construction of ternary sequences having a zero-correlation zone. The cross-correlation function and the side-lobe of the auto-correlation function of the proposed sequence set is zero for the phase shifts within the zero-correlation zone. The proposed sequence set consists of more than one subset having the same member size. The correlation function of the sequences of a pair of different subsets, referred to as the inter-subset correlation function, has a wider zero-correlation zone than that of the correlation function of sequences of the same subset (intra-subset correlation function). The wide inter-subset zero-correlation enables performance improvement during application of the proposed sequence set. The proposed sequence set has a zero-correlation zone for periodic, aperiodic, and odd correlation functions.

A Generalized Construction of Zero-Correlation Zone Sequence Set with Sequence Subsets

The present paper introduces a new approach to the construction of a sequence set with a zero-correlation zone for both periodic and aperiodic correlation functions. The proposed sequences can be constructed from a pair of Hadamard matrices of orders n0 and n1. The constructed sequence set consists of n0n0 ternary sequences, each of length n0(m+2)(n1+Δ), for a non-negative integer m and τ≥2. The zero-correlation zone of the proposed sequences is $|\\ au|\\le \\ablen^{m+1}-1$, where τ is the phase shift. The proposed sequence set consists of n0 subsets, each with a member size n1. The correlation function of the sequences of a pair of different subsets, referred to as the inter-subset correlation function, has a zero-correlation zone with a width that is approximately τ times that of the correlation function of sequences of the same subset (intra-subset correlation function). The inter-subset zero-correlation zone of the proposed sequences is $|\\ au|\\le \\slen\\ablen^{m+1}$, where τ is the phase shift. The wide inter-subset zero-correlation enables performance improvement during application of the proposed sequence set.

Lower Bounds on the Maximum Partial Correlations of Frequency Hopping Sequence Set with Low Hit Zone

In order to evaluate the goodness of frequency hopping sequence design, the periodic Hamming correlation function is used as an important measure. Usually, the length of correlation window is shorter than the period of the chosen frequency hopping sequence, so the study of the partial Hamming correlation of frequency hopping sequence is particularly important. In this paper, the maximum partial Hamming correlation lower bounds of frequency hopping sequences with low hit zone, with respect to the size of the frequency slot set, the length of correlation window, the family size, the low hit zone, the maximum partial Hamming autocorrelation and the maximum partial Hamming crosscorrelation are established. It is shown that the new bounds include the known Lempel-Greenberger bound, Peng-Fan bounds, Eun-Jin-Hong-Song bound and Peng-Fan-Lee bounds as special cases.

Fast aperiodic correlation algorithm for real‐valued shift‐orthogonal finite‐length sequence of length 2ν+1

AbstractReal‐valued shift‐orthogonal finite‐length sequences are sequences in which the side lobes of the aperiodic autocorrelation function become 0, except for the endpoints of the shift to both sides, and can be applied in pulse compression radar and spread spectrum communications. In this paper, a fast correlation algorithm for efficiently calculating the periodic correlation function is discussed for real‐valued shift‐orthogonal finite‐length sequences with length M=2ν+1. For input data, including a real‐valued shift‐orthogonal finite‐length sequence over a certain range, the value of the aperiodic correlation function is found in a certain shift range. Based on the synthesis of this sequence by the convolution of ν+1 partial sequences, the correlation processing is broken down into correlation processing of the ν+1 stages of partial sequences. As a result, the number of multiplications and the number of additions can be suppressed on the order Mlog2M. © 2007 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 90(9): 18– 30, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.20301

Ternary Sequence Set Having Periodic and Aperiodic Zero-Correlation Zone

A new class of ternary sequence with a zero-correlation zone is introduced. The proposed sequence sets have a zero-correlation zone for both periodic and aperiodic correlation functions. The proposed sequences can be constructed from a pair of Hadamard matrices of size n0 × n0 and a Hadamard matrix of size n1 × n1. The constructed sequence set consists of n0n1 ternary sequences, and the length of each sequence is n0(m+1)(n1 + 1) for a non-negative integer m. The zero-correlation zone of the proposed sequences is |τ|≤ n0m-1, where τ is the phase shift. The sequence member size of the proposed sequence set is equal to n1/n1+1 times that of the theoretical upper bound of the member size of a sequence set with a zero-correlation zone.