Cyclostationary Research Articles

Losengram: An effective demodulation frequency band selection method for rolling bearing fault diagnosis under complex interferences

Abstract Resonance demodulation is one of the most effective methods for rolling bearing fault diagnosis. However, the selection of the proper demodulation frequency band (DFB) has always been considered as a substantial challenge. Although many popular DFB selection methods have been developed, such as fast Kurtogram (FK), Protrugram, and Autogram, they would suffer unsatisfactory performance degradation when encountering random impulsive noise or cyclostationary noise. Therefore, this paper proposes a novel DFB selection method called Losengram to address this problem. In the proposed method, a robust sub-band indicator, localized square envelope spectrum kurtosis, is designed to evaluate the fault information in a sub-band. With this indicator, the interferences of random impulsive noise and cyclostationary noise could be suppressed well. Besides, in order to circumvent the various adverse effects incurred by the utilization of a multi-rate finite impulse response filter bank, a frequency-domain sub-band filtering strategy is presented to filter the divided sub-bands in a 1/3-binary tree structure. The effectiveness of the proposed method is tested on both simulated and experimental signals, and the results show that it has a superior performance than the FK, Protrugram, as well as Autogram.

Periodically correlated time series and the Variable Bandpass Periodic Block Bootstrap.

This research introduces a novel approach to resampling periodically correlated time series using bandpass filters for frequency separation called the Variable Bandpass Periodic Block Bootstrap and then examines the significant advantages of this new method. While bootstrapping allows estimation of a statistic's sampling distribution by resampling the original data with replacement, and block bootstrapping is a model-free resampling strategy for correlated time series data, both fail to preserve correlations in periodically correlated time series. Existing extensions of the block bootstrap help preserve the correlation structures of periodically correlated processes but suffer from flaws and inefficiencies. Analyses of time series data containing cyclic, seasonal, or periodically correlated principal components often seen in annual, daily, or other cyclostationary processes benefit from separating these components. The Variable Bandpass Periodic Block Bootstrap uses bandpass filters to separate a periodically correlated component from interference such as noise at other uncorrelated frequencies. A simulation study is presented, demonstrating near universal improvements obtained from the Variable Bandpass Periodic Block Bootstrap when compared with prior block bootstrapping methods for periodically correlated time series.

Fast computation of the spectral correlation via frequency-averaging

Spectral Correlation (SC) has become a critical analytical instrument for assessing the second-order cyclostationarity (CS) in signals, especially for mechanical vibration signals. Nonetheless, the computational demand for computing SC is considerable. Built upon the Smoothed Cyclic Periodogram (SCP), a consistent estimator named Fast Smoothed Cyclic Periodogram (FastSCP) is proposed to address this issue. FastSCP represents the computational process of SCP through the convolution of spectral components and incorporates the Overlap-Save algorithm for accelerating. Comparison results with other fast SC estimators, including Fast Spectral Correlation (FastSC) and Faster Spectral Correlation (FasterSC), indicate that the proposed method is competitive in terms of both computational cost and memory usage, particularly when dealing with wide cyclic frequency ranges.

Adaptive filter design for cyclostationary processes associated with sine-wave extraction operation

Cyclostationary signals have been generally used in condition monitoring and fault diagnosis of bearings, for which the adaptive filter is important in obtaining the signal-of-interest cyclostationary process from a disturbed signal. However, due to the nonstationarity of cyclostationary signals, their adaptive filters are usually time-varying and difficult to be implemented. In this study, a new optimum filter is obtained by converting the time-varying optimization function to be time-invariant using a sine-wave extraction (SE) operator, for which the adaptive implementation is designed and analyzed. The time-varying linear minimum mean-square error is converted to be time invariant using the SE operator, and its solution is time invariant but without explicit expression. Subsequently, an adaptive implementation, called the SE least mean square (SE-LMS), is introduced, for which the mean and weighted mean-square-deviation (MSD) are explicitly expressed. The scope of step size is determined to obtain a mean-convergent and mean-square-stable SE-LMS. The steady-state SE-operated excess mean-square-error is analyzed, and its explicit expression is obtained. The SE-LMS algorithm uses only a single cyclic frequency, based on which, the combined SE-LMS algorithm is introduced for a cyclostationary signal with multiple cyclic frequencies. Finally, several simulations are designed to verify the superiority of the proposed estimators over the time-average operator-based estimator.

Exploring and analyzing the role of hybrid spectrum sensing methods in 6G-based smart health care applications.

Researchers are focusing their emphasis on quick and real-time healthcare and monitoring systems because of the contemporary modern world's rapid technological improvements. One of the best options is smart healthcare, which uses a variety of on-body and off-body sensors and gadgets to monitor patients' health and exchange data with hospitals and healthcare professionals in real time. Utilizing the primary user (PU) spectrum, cognitive radio (CR) can be highly useful for efficient and intelligent healthcare systems to send and receive patient health data. In this work, we propose a method that combines energy detection (ED) and cyclostationary (CS) spectrum sensing (SS) algorithms. This method was used to test spectrum sensing in CR-based smart healthcare systems. The proposed ED-CS in cognitive radio systems improves the precision of the spectrum sensing. Owing to its straightforward implementation, ED is initially used to identify the idle spectrum. If the ED cannot find the idle spectrum, the signals are found using CS-SS, which uses the cyclic statistical properties of the signals to separate the main users from the interference. In the simulation analysis, the probability of detection (Pd), probability of a false alarm (Pfa), power spectral density (PSD), and bit error rate (BER) of the proposed ED-CS is compared to those of the traditional Matched Filter (MF), ED, and CS. The results indicate that the suggested strategy improves the performance of the framework, making it more appropriate for smart healthcare applications.

Theory, validation, and improvement of four enhancement algorithms for repetitive impulses

Analyzing vibration of rotating machinery signals is a popular methodology derived on the potent tools provided by cyclostationary process theory. Among them, the autocorrelation function (ACF), the Fourier transform (FT), the short-time Fourier transform (STFT), and the time synchronous average (TSA) are widely used for enhancing and analyze random cyclostationary components, i.e., repetitive impulses. This paper aims to thoroughly investigate these techniques through a comprehensive scientific research process. Initially, theoretical analyses are conducted to discuss their noise suppression principles, on the foundation of mean and variance analysis. Results indicate that for Gaussian random noise, the variance of the noise linked to the ACF, the FT, and TSA almost decreases linearly with signal length, while the noise variance of the short-time Fourier transform does so with window length. Therefore, longer signal lengths for the ACF, the FT, and the TSA, or extended window lengths for the STFT, improve their denoising performance. Subsequently, a series of simulated signals are generated to validate these findings, complemented by public datasets for further verification. Finally, the paper discusses and concludes with enhancement approaches derived from these four methodologies.

Exploring and analyzing the role of hybrid spectrum sensing methods in 6G-based smart health care applications

Background Researchers are focusing their emphasis on quick and real-time healthcare and monitoring systems because of the contemporary modern world’s rapid technological improvements. One of the best options is smart healthcare, which uses a variety of on-body and off-body sensors and gadgets to monitor patients’ health and exchange data with hospitals and healthcare professionals in real time. Utilizing the primary user (PU) spectrum, cognitive radio (CR) can be highly useful for efficient and intelligent healthcare systems to send and receive patient health data. Methods In this work, we propose a method that combines energy detection (ED) and cyclostationary (CS) spectrum sensing (SS) algorithms. This method was used to test spectrum sensing in CR-based smart healthcare systems. The proposed ED-CS in cognitive radio systems improves the precision of the spectrum sensing. Owing to its straightforward implementation, ED is initially used to identify the idle spectrum. If the ED cannot find the idle spectrum, the signals are found using CS-SS, which uses the cyclic statistical properties of the signals to separate the main users from the interference. Results In the simulation analysis, the probability of detection (Pd), probability of a false alarm (Pfa), power spectral density (PSD), and bit error rate (BER) of the proposed ED-CS is compared to those of the traditional Matched Filter (MF), ED, and CS. Conclusions The results indicate that the suggested strategy improves the performance of the framework, making it more appropriate for smart healthcare applications.

Generalized spectrum analysis of Chirp Cyclostationary signals associate with linear canonical transform

A chirp cyclostationary (CCS) process is a nonstationary stochastic process, whose statistics have been defined and utilized in radar and communication signal processing. However, spectrum analysis of CCS signals, which provides a spectral view to characterize a signal, has not been performed. Classical spectrum analysis is based on the Fourier transform, which is not suitable for CCS signals because of the multiplicative chirp signals in a CCS model. In this study, we propose a generalized spectral representation of chirp cyclostationary signals associated with a linear canonical transform. The proposed representation shows that a CCS signal can be represented by a linear combination of a series of jointly stationary and equal-bandwidth processes, which bridge the nonstationary and stationary processes. Furthermore, the relationships between the second-order statistics of the CCS process and its stationary representations are introduced. Finally, based on the above observations, a linear time-varying Wiener filter was designed for CCS signal denoising and time-varying channel estimation. Finally, the relationships between the nonstationarity of the input and output and time-varying properties of the systems are discussed, which are helpful for determining the nonstationarity of the outputs or the time-varying property of a system.

Combined Weighted Envelope Spectrum: An enhanced demodulation framework for extracting characteristic frequency of rotating machinery

Envelope spectrum plays a pivotal role in the surveillance and diagnostics of rotating machinery. Two prevailing techniques for constructing this spectral quantity are narrowband demodulation and cyclostationary analysis. However, the former is susceptible to failure under low Signal-to-Noise Ratio (SNR) conditions or in the presence of non-Gaussian noise, while the latter exhibits sensitivity to cyclostationary noise, such as electromagnetic interferences. Notably, specific tasks determine the characteristic frequencies expected to be detected. More precisely, the determination of target component and noise interference depends on the mechanical part under scrutiny. Consequently, the envelope spectra provided by classical methods often manifest deficiencies when confronted with intricate monitoring signals from rotating machinery. To address this challenge, this paper proposes an enhanced demodulation framework—Combined Weighted Envelope Spectrum (CWES), tailored for extracting specific characteristic frequencies. First, a cyclostationary model for rotating machinery is presented, demarcating the target component as solely the cyclostationary signal caused by the mechanical element of concern. Second, the weighting functions with/without prior knowledge of characteristic frequency are elaborately designed, and the Adaptive Combined Weighted Envelope Spectrum (ACWES) and the Optimal Combined Weighted Envelope Spectrum (OCWES) are constructed correspondingly. Moreover, the indicator Characteristic peak-to-Noise peak Ratio (CNR) is proposed to evaluate the demodulation performance of different envelope spectra. Ultimately, the effectiveness of ACWES and OCWES is validated by simulation signals and experimental data from rolling bearing, centrifugal pump, and Francis turbine, and the comparison with the state-of-the-art demodulation methods verifies the superiority of the proposed enhanced demodulation framework.

Cyclostationarity of Communication Signals in Underwater Acoustic Channels

The effect of underwater acoustic propagation on the cyclostationary features of communication signals is modeled and analyzed. Two kinds of channels are considered: the multiscale-multilag channel, over which mobile and wideband acoustic systems usually communicate, and the dispersive channel resulting from low-frequency modal propagation in shallow water. It is shown that multiscale-multilag channels transform cyclostationary signals into a sum of velocity and acceleration-dependent time-warped cyclostationary processes. This time-warping is carefully taken into account to efficiently recover the cyclostationary features. On the other hand, it is found that low-frequency dispersive channels preserve the original periodicity but attenuate the shorter cycles and spread the correlations. To illustrate the theoretical results, applications with simulated and real data are also presented. Specifically, the problem of estimating time-varying Doppler scales is addressed for multiscale-multilag channels as well as the detection of signals with unique cyclostationary signatures. The example of blind symbol-rate estimation applied to covert communications in dispersive channels is also discussed. Special attention is paid to phase-shift keying (PSK), quadrature amplitude modulation (QAM), orthogonal frequency-division multiplexing (OFDM), and direct-sequence spread-spectrum (DSSS) signals. Accompanying supplementary material provides the MATLAB code used for the estimation and detection examples.

Empirical study of periodic autoregressive models with additive noise – estimation and testing

Periodic autoregressive (PAR) time series with finite variance is considered as one of the most common models of second-order cyclostationary processes. However, in the real applications, the signals with periodic characteristics may be disturbed by additional noise related to measurement device disturbances or to other external sources. Thus, the known estimation techniques dedicated for PAR models may be inefficient for such cases. When the variance of the additive noise is relatively small, it can be ignored and the classical estimation techniques can be applied. However, for extreme cases, the additive noise can have a significant influence on the estimation results. In this paper, we propose four estimation techniques for the noise-corrupted PAR models with finite-variance distributions. The methodology is based on Yule-Walker equations utilizing the autocovariance function. It can be used for any type of the finite-variance additive noise. The presented simulation study clearly indicates the efficiency of the proposed techniques, also for extreme case, when the additive noise is a sum of the Gaussian additive noise and additive outliers. The proposed estimation techniques are also applied for testing if the data correspond to noise-corrupted PAR model. This issue is strongly related to the identification of informative component in the data in case when the model is disturbed by additive non-informative noise. The power of the test is studied for simulated data. Finally, the testing procedure is applied for two real time series describing particulate matter concentration in the air.

On the Capacity of Communication Channels With Memory and Sampled Additive Cyclostationary Gaussian Noise

In this work we study the capacity of interference-limited channels with memory. These channels model non-orthogonal communications scenarios, such as the non-orthogonal multiple access (NOMA) scenario and underlay cognitive communications, in which the interference from other communications signals is much stronger than the thermal noise. Interference-limited communications is expected to become a very common scenario in future wireless communications systems, such as 5G, WiFi6, and beyond. As communications signals are inherently cyclostationary in continuous time (CT), then after sampling at the receiver, the discrete-time (DT) received signal model contains the sampled desired information signal with additive sampled CT cyclostationary noise. The sampled noise can be modeled as either a DT cyclostationary process or a DT almost-cyclostationary process, where in the latter case the resulting channel is not information-stable. In a previous work we characterized the capacity of this model for the case in which the DT noise is memoryless. In the current work we come closer to practical scenarios by modelling the resulting DT noise as a finite-memory random process. The presence of memory requires the development of a new set of tools for analyzing the capacity of channels with additive non-stationary noise which has memory. Our results show, for the first time, the relationship between memory, sampling frequency synchronization and capacity, for interference-limited communications. The insights from our work provide a link between the analog and the digital time domains, which has been missing in most previous works on capacity analysis. Thus, our results can help improving spectral efficiency and suggest optimal transceiver designs for future communications paradigms.

Chirp cyclic moment for chirp cyclostationary processes: Definitions and estimators

In high-mobility wireless communications and radar systems, a higher carrier frequency and higher moving speed/acceleration of terminals enhance the impact of the time-varying Doppler effect. This effect alters the cyclostationarity of the transmitted signal and characterizes the received signal as a chirp cyclostationary (CCS) signal. The generalized cyclic statistics of CCS signals are defined based on Fourier analysis, which is either zero-value or non-bandlimited, and is thus inefficient. In this study, the linear canonical transform (LCT), a generalized form of the Fourier transform, was used to address these issues. The moment function is a k-dimensional variable comprising a single time variable and k−1 lag variables. The chirp cyclic moment and the time-varying canonical spectrum were designed as the LCT of the moment with respect to the time and lag variables, respectively. Furthermore, an estimator of the chirp cyclic moment was defined, which is demonstrated to be asymptotic unbiased, asymptotic mean-square-consistent and asymptotic complex normal. Applications of the chirp cyclic moment in radar parameter estimation and electrocardiogram signal characterization were verified using simulated and real-life data.

Algorithm for Estimating the Spectral Correlation Function Using the 2D Fast Fourier Transform

An algorithm for estimating the spectral correlation function of cyclostationary random processes on the basis of their finite-time realization is proposed. The main algorithm steps are described and the main operations performed at each of them are formalized. The presented estimates of the spectral correlation function in the presence of a correlated stationary noise overlapping the analyzed cyclostationary random process in the frequency domain confirm the efficiency of the algorithm with a low signal-to-noise ratio. The number of required computational operations and the amount of computer memory required for storing intermediate results are estimated.

Phase Noise Analysis for Stochastically Injected Oscillators

For energy-efficient low-noise clock generation, many multiplying delay-locked loops or injection-locked clock multipliers adopt probabilistic gating to calibrate circuit nonidealities. However, it has not yet been studied how this stochastic behavior affects the phase noise of the clock generator. To answer this question, this brief proposes a novel phase noise analysis methodology for stochastically injected oscillators. Specifically, the power spectral density of a non-wide-sense-stationary process, consisting of multiple cyclostationary processes occurring stochastically, is derived from time-averaged autocorrelation. In order to obtain the required expected value, a Markov-chain-based method to calculate the probability of each state is also presented. For validation, phase noise obtained from the proposed analysis is compared with time-domain simulation results, which demonstrates that the proposed analysis provides accurate estimation.

Informed sparsity-based blind filtering in the presence of second-order cyclostationary noise

This study investigates the potential to improve the fault detection capability of sparsity-based blind filtering. It optimizes a finite impulse response filter to maximize the sparsity of the squared envelope spectrum (SES) of vibration signals. However, the method is to be highly prone to fail optimization due to the immense number of non-fault-related second-order cyclostationary interferences. These interferences can skew the sparsity estimation of the SES and thus impair the sparsity-based blind filtering method. Even whitening or signal separation methods can fail to remove such cyclostationary components adequately. Hence, the technique is unlikely to function particularly on the signals measured on complex industrial machines. This paper extends the initial study introducing sparsity-based blind filtering in the literature. It proposes refining the filter optimization’s objective function by targeting narrow frequency bandwidths on the SES to avoid non-fault-related peaks. The narrow bandwidths are selected by exploiting the available engineering knowledge, such as an average shaft speed. The proof of concept is shown on simulated signals, and the performance tests are made on signals measured on an industrial rotating machine and a wind turbine gearbox. The study’s outcome demonstrates that targeting automatically selected narrow bandwidths bounded by shaft speed harmonics can significantly improve the detection capability of sparsity-based blind filtering.

PROPERTIES OF DISCRETE-TIME CONDITIONAL LINEAR CYCLOSTATIONARY RANDOM PROCESSES IN THE PROBLEMS OF ENERGY INFORMATICS

Modern challenges in the energy industry require comprehensive research in the field of energy informatics, which combines computer science, control systems, and energy management systems within a single methodology. An important area of energy informatics is the study of problems of systems and processes modeling in energy, including energy loads and consumption. Linear and conditional linear random processes (CLRP) are mathematical models of signals represented as the sum of a large number of random impulses occurring at random times. The energy consumption, vibration signals of energy objects, etc. can be modeled using this approach. A variant of the CLRP model with discrete time, taking into account the cyclic properties of energy consumption, has been investigated in the paper. The goal is to justify the conditions for the discrete-time CLRP to be a periodically correlated random process, as well as a cyclostationary process. It has been shown that the corresponding conditions depend on the periodicity of the probability distributions of the kernel and the generating white noise of the CLRP representation. To achieve the goal, the properties of mathematical expectation and covariance function of CLRP, as well as the method of characteristic functions, have been used. The paper proves that the discrete-time CLRP is a periodically correlated random sequence if the generating white noise has periodic mathematical expectation and variance, and the kernel is a periodically correlated random field. Based on the analysis of the multivariate characteristic function, it has been proven that the discrete-time CLRP is cyclostationary if the generating white noise is a cyclostationary process and the kernel is a cyclostationary random field. The properties of discrete-time conditional linear cyclostationary random processes are important for mathematical modeling, simulation, statistical analysis, and forecasting of energy consumption. Keywords: mathematical model, energy informatics, conditional linear random process, cyclostationary process, white noise, characteristic function.

A Detailed Model of Cyclostationary Noise in Switched-Resistor Circuits

The Switched-Resistor (S-R) technique is becoming more and more interesting to implement low-voltage low-power active- RC filters with high tuning range and front-end amplifiers for biomedical circuits and systems. This approach exploits MOS switches driven by a duty-cycle-controlled clock signal to achieve tunability of the equivalent resistance and, hence, of the RC time constant. Since S-R circuits can be considered as linear periodically time variant (LPTV) systems with sampled outputs, we exploit the theory of the adjoint (inter-reciprocal) network in order to develop a detailed model of the cyclostationary noise involved in this kind of circuits. The proposed model allows to gain insight into the different noise sources and transfer functions involved, highlighting the circuit parameters on which the cyclostationary noise depends. Validation of the proposed model has been carried out by comparing analytical results against periodic noise simulations referring to a commercial 130nm CMOS process. The validation activity has confirmed the good accuracy of the proposed noise model and provided some useful design guidelines to optimize the noise performance of S-R circuits.

Energy Harvesting in the UNB-PLC Spectrum: Hidden Opportunities for IoT Devices

This article focuses on the hidden benefits of harvesting the wasted energy from undesirable components of electric signals in electric power systems for powering the transceivers of Internet of Things (IoT) devices. These components, mainly harmonics and interharmonics, occupy the ultranarrowband power line communication spectrum (i.e., frequencies below 3 kHz). In this context, we introduce a mathematical formulation that allows us to quantify the number of transceivers of IoT devices that this kind of wasted energy can power. Based on a measurement campaign in a building facility, we show that the wasted energy can be modeled as a cyclostationary random process on weekdays and a stationary one on weekends, mimicking the energy consumption profile in a building facility. Numerical results also highlight achievable data rates obtained in the building facility if the wasted energy is reused. This analysis shows that this kind of harvested energy from wasted energy is suitable for powering numerous transceivers of IoT devices in practical scenarios.

Estimation of a Spectral Correlation Function Using a Time-Smoothing Cyclic Periodogram and FFT Interpolation—2N-FFT Algorithm

This article addresses the problem of estimating the spectral correlation function (SCF), which provides quantitative characterization in the frequency domain of wide-sense cyclostationary properties of random processes which are considered to be the theoretical models of observed time series or discrete-time signals. The theoretical framework behind the SCF estimation is briefly reviewed so that an important difference between the width of the resolution cell in bifrequency plane and the step between the centers of neighboring cells is highlighted. The outline of the proposed double-number fast Fourier transform algorithm (2N-FFT) is described in the paper as a sequence of steps directly leading to a digital signal processing technique. The 2N-FFT algorithm is derived from the time-smoothing approach to cyclic periodogram estimation where the spectral interpolation based on doubling the FFT base is employed. This guarantees that no cyclic frequency is left out of the coverage grid so that at least one resolution element intersects it. A numerical simulation involving two processes, a harmonic amplitude modulated by stationary noise and a binary-pulse amplitude-modulated train, demonstrated that their cyclic frequencies are estimated with a high accuracy, reaching the size of step between resolution cells. In addition, the SCF components estimated by the proposed algorithm are shown to be similar to the curves provided by the theoretical models of the observed processes. The comparison between the proposed algorithm and the well-known FFT accumulation method in terms of computational complexity and required memory size reveals the cases where the 2N-FFT algorithm offers a reasonable trade-off.