Instantaneous Frequency Research Articles

Joint extraction-based multivariate chirp mode decomposition and its application to fault diagnosis of rotating machinery under variable speed conditions

Vibration signals from rotating machinery during variable speed conditions exhibits non-stationary characteristics, posing a challenge for fault diagnosis. When a machine fails, the vibration signal of any component connected to the bearing carries valuable information about the equipment’s status. Processing multichannel variable speed signals at the same time avoids losing part of information and improves fault diagnosis. Therefore, this study put up a novel approach called joint extraction-based multivariate chirp mode decomposition (JMCMD) to analyze multichannel signals of rotating machinery with varying speeds. The approach comprises the joint extraction framework and the multichannel signal instantaneous frequency initialization (MSIFI). First, JMCMD employs a joint component extraction framework, which can precisely and simultaneously extract complicated characteristic components close to each other in multiple channels. Second, this study extends the parameterized demodulation (PD) technique to multiple channels to find the initialized instantaneous frequency of multichannel signals. Through the joint extraction strategy’s integration with MSIFI, JMCMD is capable of efficaciously extracting all significant characteristic components of multichannel signals. Additionally, the high-resolution time–frequency representation (TFR) constructed from JMCMD’s output can clearly reveal the time-varying fault characteristic frequency of the rotor-bearing system. Simulation analysis and actual cases demonstrate that the suggested approach is an effective way to identify fault feature components, making it highly applicable in signal decomposition and fault detection.

Amplitude and frequency modulation of EEG predicts Intraventricular hemorrhage in preterm infants

BackgroundIntraventricular hemorrhage (IVH) is a common and significant complication in premature infants. While cranial ultrasound is the golden standard for IVH detection, it may not identify lesions until hours or days after occurring, which limits early intervention. Predicting IVH in premature infants would be highly advantageous. Recent studies have shown that EEG data’s amplitude and frequency modulation features could offer predictive insights for neurological diseases in adults. MethodsTo investigate the association between IVH and EEG monitoring, a retrospective case-control study was conducted in preterm infants. All infants underwent amplitude integrated EEG monitoring for at least 3 days after birth. The study included 20 cases who had an IVH diagnosed on cranial ultrasound and had a negative ultrasound 24 h earlier, and 20 matched controls without IVH. Amplitude and frequency modulation features were extracted from single-channel EEG data, and various machine learning algorithms were evaluated to create a predictive model. ResultsCases had an average gestational age and birth weight of 26.4 weeks and 965 g, respectively. The best-performing algorithm was adaptive boosting. EEG data from 24 h before IVH detection proved predictive with an area under the receiver operating characteristic curve of 93 %, an accuracy of 91 %, and a Kappa value of 0.85. The most informative features were the slow varying instantaneous frequency and amplitude in the Delta frequency band. ConclusionAmplitude and frequency modulation features obtained from single-channel EEG signals in extremely preterm infants show promise for predicting IVH occurrence within 24 h before detection on cranial ultrasound.

Parameter estimation of the hyperbolic frequency-modulated bat calls using hyperbolic scale transform

Echolocating bats are known to vary their waveforms at the phases of searching, approaching, and capturing the prey. It is meaningful to estimate the parameters of the calls for bat species identification and the technological improvements of the synthetic systems, such as radar and sonar. The type of bat calls is species-related, and many calls can be modeled as hyperbolic frequency- modulated (HFM) signals. To obtain the parameters of the HFM-modeled bat calls, a reversible integral transform, i.e., hyperbolic scale transform (HST), is proposed to transform a call into two-dimensional peaks in the “delay-scale” domain, based on which harmonic separation and parameter estimation are realized. Compared with the methods based on time-frequency analysis, the HST-based method does not need to extract the instantaneous frequency of the bat calls, only searching for peaks. The verification results show that the HST is suitable for analyzing the HFM-modeled bat calls containing multiple harmonics with a large energy difference, and the estimated parameters imply that the use of the waveforms from the searching phase to the capturing phase is beneficial to reduce the ranging bias, and the trends in parameters may be useful for bat species identification.

Two-terminal traveling wave fault location approach based on frequency dependent electrical parameters of HVAC cable transmission lines

With the expansion of high-voltage alternating current transmission networks, the widely used two-terminal traveling wave method faces several challenges in cable fault location. On one hand, the frequency dependent characteristics of longer cable transmission lines can significantly impact the accuracy of traveling wave propagation velocity. On the other hand, the processing of traveling wave signals obtained synchronously is also a crucial factor that requires attention. To enhance the performance of cable fault location, the principles of two-terminal traveling wave method need to be updated. Firstly, the cable model derived from the theory of wave propagation in multi-conductor systems is presented, which reflects the frequency dependent characteristics of the electrical parameters. Whereupon, an innovative cable traveling wave fault location scheme primarily composed of arrival time identification and instantaneous frequency calculation is proposed. Eventually, based on electromagnetic transient simulation software PSCAD/EMTDC, the robustness of the innovative cable fault location method against fault type, fault location, fault resistance, fault initial angle and earth resistivity issues is demonstrated.

Influence of genetic mutations to atria vulnerability to atrial fibrillation: An in-silico 3D human atria study

Background and ObjectivePersonalized 3D computer models of atria have been extensively implemented in the last yearsas a tool to facilitate the understanding of the mechanisms underlying different forms of arrhythmia, such as atrial fibrillation (AF). Meanwhile, genetic mutations acting on potassium channel dynamics were demonstrated to induce fibrillatory episodes in asymptomatic patients. This research study aims at assessing the effects and the atrial susceptibility to AF of three gain-of-function mutations – namely, KCNH2 T895M, KCNH2 T436M, and KCNE3-V17M – associated with AF outbreaks, using highly detailed 3D atrial models with realistic wall thickness and heterogenous histological properties. MethodsThe 3D atrial model was generated by reconstructing segmented anatomical structures from CT scans of an AF patient. Modified versions of the Courtemanche human atrial myocyte model were used to reproduce the electrophysiological activity of the WT and of the three mutant cells. Ectopic foci (EF) were simulated in sixteen locations across the atrial mesh using an S1-S2 protocol with two S2 basic cycle lengths (BCL) and eleven coupling intervals in order to induce arrhythmias. ResultsThe three genetic mutations at 3D level reduced the APD90. The KCNE3-V17M mutation provoked the highest shortening (55 % in RA and LA with respect to WT), followed by KCNH2 T895M (14 % in RA and 18 % LA with respect to WT)and KCNH2 T436M (7 % in RA and 9 % LA with respect to WT). The KCNE3-V17M mutation led to arrhythmia in 67 % of the cases simulated and in 94 % of ectopic foci considered, at S2 BCL equal to 100 ms. The KCNH2 T436M and KCNH2 T895M mutations increased the vulnerability to AF in a similar way, leading to arrhythmic episodes in 7 % of the simulated conditions, at S2 BCL set to 160 ms. Overall, 60 % of the arrhythmic events generated arise in the left atrium. Spiral waves, multiple rotors and disordered electrical pattern were elicited in the presence of the KCNE3-V17M mutation, exhibiting an instantaneous mean frequency of 7.6 Hz with a mean standard deviation of 1.12 Hz. The scroll waves induced in the presence of the KCNH2 T436M and KCNH2 T895M mutations showed steadiness and regularity with an instantaneous mean frequencies in the range of 4.9 – 5.1 Hz and a mean standard deviation within 0.19 – 0.53 Hz. ConclusionsThe pro-arrhythmogenicity of the KCNE3-V17M, KCNH2 T895M and KCNH2 T436M mutations was studied and proved on personalized 3D cardiac models. The three genetic mutations were demonstrated to increase the predisposition of atrial tissue to the formation of AF-susceptible substrate in different ways based on their effects on electrophysiological properties of the atria.

The ambiguity of frequency determination in digital microwave frequency discriminators

Instantaneous frequency measurement devices are designated for very fast measurements of the current frequency value of microwave signals, even if they are very short in the time domain. Fast measurements of frequency temporary values may be based on the evaluation of the phase difference of signal propagating through the microwave transmission lines with unequal, but known, lengths. This paper presents the principle of determination of temporary values of the microwave signal frequency using the digitalized signals and the binary value of them eventually. In the purpose of increase the frequency discrimination resolution, additional tracks with lines with a larger length are proposed. For the system with elements with analytical model transmission characteristics it is typical that bands of ambiguity of frequency measurement occurs. To tackle this problem in addition to 4 x 4 Butler matrix implementation the method of using combination sine and cosine signals is proposed.

Attenuation of non-stationary random noise in ground penetrating radar data based on time-varying filtering

The attenuation of random noise is significant in the enhancement of the signal-to-noise ratio in ground penetrating radar (GPR) data. However, the non-stationary nature of random noise poses significant challenges to most conventional denoising methods. We propose a time-varying bandpass filter (TVBF) based on the local maxima multiple synchrosqueezing transform (LMMSST), referred to as TVBFLMMSST, for the attenuation of non-stationary random noise in GPR data. The LMMSST method is introduced to achieve highly compressed time–frequency representation (TFR). By integrating an instantaneous frequency trajectory detection algorithm, we design the TVBF to successfully separate useful signals from random noise. The efficacy of the TVBFLMMSST is validated through synthetic and field experiments, demonstrating its capability to effectively isolate components of useful signals and noise signals. Compared to classical denoising methods, TVBFLMMSST exhibits robust performance in attenuating random noise while preserving the amplitude of useful signals.

IDENTIFYING INNER RACE FAULTS IN DEEP GROOVE BALL BEARING USING NONLINEAR MODE DECOMPOSITION AND HILBERT TRANSFORM

Abstract This study focuses on the analysis of vibration-based signatures obtained from deep groove ball bearings with faults on the inner race. Various time-frequency-based methods are commonly used to diagnose faults in bearings. However, due to the nonself-adaptive nature of these methods and the nonlinear and nonstationary signals produced by the faults, mode decomposition techniques are seen as promising methods. This paper presents a novel approach based on Nonlinear Mode Decomposition (NMD), which decomposes the complex signal into nonlinear modes. The data is taken from an online database of deep groove ball bearing with inner race faults of different sizes. This data is then subjected to NMD to extract nonlinear modes. Statistical parameters are applied to select a subset of significant nonlinear modes from the complete set. Finally, the Fast Fourier Transform is applied to the Hilbert Transform (HT) of the selected modes to see fault frequency and its higher harmonics resulting from the nonlinearity. Additionally, the instantaneous frequency and instantaneous phase, two key parameters acquired from the HT are also plotted for normal and faulty bearings, and the results are discussed in the paper. The proposed method offers a valuable approach for accurately detecting and diagnosing deep groove ball-bearing faults.

Comprehensive degree of nonlinearity in the dynamic response of high-speed Maglev train

The aim of this study is to propose a method for evaluating the comprehensive degree of nonlinearity in the dynamic response of high-speed maglev train system through mathematical modeling and numerical simulation. A model of a typical high-speed maglev vehicle is established based on the principles of maglev vehicle dynamics, and control models for the suspension magnet and the guide magnet are developed using principles from the electromagnetism and automatic control theory. The vibration responses of the vehicle system, including acceleration and electromagnetic forces are calculated using a fast numerical integration method. This calculation is performed for various control parameters and track harmonic excitations. The Hilbert-Huang transform (HHT) is adopted to decompose the response signal and derive the local mean frequency (amplitude) and instantaneous frequency (amplitude) of the response, based on which the Degree of Nonlinearity (DN) of the vibration response of a single component is obtained by calculating the deviation between the instantaneous frequency and the mean frequency. The Comprehensive Degree of Nonlinearity (CDN) of the vibration response in the maglev train system is proposed and defined by incorporating the DNs of the vertical and lateral vibrations of the carbody, bogie, suspension magnet and guide magnet, and by quantitatively evaluating them. The effects of the electromagnet control parameters and harmonic irregularities (wavelength and amplitude) of the track on the DN of each component and the CDN of the entire vehicle system are analyzed in detail. The analysis results show that the DN and CDN can demonstrate the extent of instability in the vehicle system and can be applied for safety warning on maglev vehicles.

Speckle noise detection and correction for frequency-scanning interferometry in vibration measurement

Speckle noise significantly impairs the accuracy of vibration measurements for non-cooperative targets in frequency-scanning interferometry (FSI). To restore the real vibration, this paper proposes a comprehensive vibration demodulation algorithm based on the FSI. We established an FSI vibration model accounting for speckle noise. The extraction of instantaneous frequency from the interference signal is performed using a complex-shifted Morlet wavelet. An autoregressive differential model, incorporating a smooth weight function, is applied for bidirectional prediction to correct speckle noise. Finally, the true vibration of the target is restored through Kalman filtering. We validated the algorithm’s accuracy using a Monte Carlo strategy. The experimental results show that the surface vibration of black aluminum oxide alloy and the axial runout of the DC motor during operation was successfully captured, and the time–frequency analysis results showed that this method has strong robustness against dense speckle noise.

Accentuating the Reservoir Potentials of Balkassar Field by Delineating the Fracture Trends and Hydrocarbon Prospects with the Aid of Seismic Attributes.

Carbonates constitute a significant proportion of the world's hydrocarbon reserves, accounting for approximately 43%. Despite their substantial potential, accurately characterizing these reserves is a challenging task due to their complex and anisotropic nature. In the upper Indus basin of Pakistan, Eocene carbonates exhibit strong production capabilities. However, the Eocene reservoir (comprising the Chorgali and Sakesar formations) and the deeper Lockhart formation of the Paleocene age present a considerable challenge. The Chorgali formation is dolomitic in composition, featuring both primary and secondary porosity, while the Sakesar formation has only secondary porosity resulting from tectonic activity. The delineation of tectonically induced porosities is a highly demanding task that requires exceptional quality seismic and well data for reliable results. To address the complex heterogeneities present in the Eocene reservoir of the upper Indus basin, a variety of seismic attributes, such as sweetness, instantaneous frequency, amplitude, curvature, similarity variance, lateral continuity, and fault likelihood, have been employed in conjunction with fundamental interpreting techniques. These advanced seismic attributes greatly contribute to delineating fracture zones and identifying sweet spots with remarkable precision. They also enable a focus on high-frequency data content and differentiate between shale beds and reservoir zones based on frequency and amplitude. This helps in the concise marking of fractured zones to enhance our understanding of secondary porosity. Moreover, these attributes help delineate the continuity of reflectors and pinpoint disruptions caused by compression forces and tectonic activities. This study aims to capture the heterogeneity and complexity of reservoir zones to address a critical question regarding the Balkassar structure. The Western lobe of the field, which demonstrates promising oil production, outperforms the structurally higher Eastern part, which lags behind in production. This research seeks to recognize the geological elements contributing to the superior performance of the Western lobe and provide guidance for maximizing the potential of the Eastern lobe through advanced characterization techniques.

Extraction and evaluation of cable forces of a cable-stayed bridge based on amplitude and phase estimation method

In order to identify the time-varying frequency and amplitude of structural vibration based on the bridge structural health monitoring data and obtain the cable force of cable-stayed bridges in real time, a spectrum analysis method based on amplitude and phase estimation (APES) was proposed in this study. The amplitude spectrum of the acceleration data is first calculated by the APES method, the real-time spectrogram of the cable is obtained by the sliding window method. Then the modal frequency and amplitude are automatically extracted from the real-time spectrum by using a frequency extrusion post-processing technique, which can be regarded as the average value of the instantaneous frequency and amplitude respectively. Next, the fundamental frequency of the cable is extracted by using an automatic identification method, and the performance of the proposed method is verified. Finally, real-time scoring of cable forces and structural condition assessment is achieved with consideration of the moderation index model as well as the material strength. The results show that the APES method can use shorter calculation samples than the traditional Fast Fourier Transform (FFT) to obtain higher resolution and more accurate amplitude, which provides a theoretical basis for the real-time identification of fundamental frequency based on short-term monitoring data. The frequency extrusion post-processing-algorithm can reduce the spectrum recognition delay and improve timeliness of the cable force evaluation. The time-varying cable force with an interval of 10 s can be used to reflect the health status of the cable in time. The research results can provide technical support for the real-time extraction of cable force of long-span cable-stayed bridges, and can also provide an effective basis for component condition evaluation and bridge maintenance decision-making.

Improved spectral amplitude modulation for tacholess estimation of rotation speed

Rotation speed is key information for rotating machinery, whether it’s a negative feedback control method based on speed or equipment status monitoring and fault diagnosis. In order to accurately and efficiently estimate the rotation speed under tacholess condition, a novel method named improved spectral amplitude modulation (ISAM) is proposed in this paper. In ISAM, the vibration signal is first filtered through a band-pass filter to eliminate interference from irrelevant frequency components. Subsequently, an appropriate weight, which modifies the amplitude spectrum of the time–frequency representation (TFR) to decompose different components based on their powers, is selected to enhance the instantaneous frequency (IF) of rotation speed. The modified signal is then reconstructed based on the edited amplitude and original phase. Finally, the IF can be extracted from the TFR of the modified signal with the multiple time–frequency curve extraction (MTFCE) technique. Additionally, entropy is utilized to guide the selection of weight. The effectiveness of the proposed method is evaluated through simulation analysis and case studies. The results show that the ISAM can accurately estimate rotation speed, with the mean absolute error (MAE) at 0.13. Compared to an existing IF estimation method, ISAM can reduce computation time by 95%.

Multivariate iterative nonlinear chirp mode decomposition: principles and applications

Abstract The rotor-bearing system (RBS) of a hydroelectric unit includes multiple components such as the rotor, rotor shaft, coupling, turbine, and main shaft. There is a strong coupling relationship between the vibration signals of multiple bearing sections. At the same time, the vibration signal of the RBS contains rich frequency components, which can effectively reflect the operating state of the shaft system of the unit. Therefore, synchronous signal analysis across multiple rotor bearing sections is required to completely grasp the operating status of the unit. In this study, the Multivariate Iterative Nonlinear Chirp Mode Decomposition (MINCMD) is proposed to analyze signals on multiple sections simultaneously. MINCMD follows the step-by-step extraction idea of Adaptive Multivariate Chirp Mode Decomposition (AMCMD), but it adopts a new algorithm framework by modifying the objective function. Furthermore, AMCMD does not converge well without a good initial instantaneous frequency (IF). Therefore, a method based on parameterized demodulation (PD) is introduced in MINCMD to solve the multi-channel frequency initialization problem. The effectiveness of the proposed algorithm is verified through a simulation test and real data from hydroelectric units.

Self-compression of laser pulses induced by asymmetric self-phase modulation aided by backward Raman scattering in periodic density-modulated plasma.

Here a mechanism for self-compression of laser pulses is presented, based on period density-modulated plasma. In this setup, two pump beams intersect at a small angle within the plasma. This interaction is facilitated by the ponderomotive ion mechanism, which causes a modulation in the density of plasma with long wavelengths and low amplitude. This modulation enhances the backward Raman scattering of the probe pulse. The trailing edge of the probe experiences greater energy loss, resulting in a steeper intensity gradient. This, in turn, induces an asymmetric self-phase modulation, which elevates the instantaneous frequency. It is notable that the laser in plasma exhibits opposite group velocity dispersion compared to traditional solid-state media. This unique property allows laser pulses to undergo dispersion compensation while broadening the spectrum, ultimately leading to self-compression. The 2D-PIC simulations demonstrate these phenomena, highlighting how period density-modulated plasma contributes to an asymmetric spectral distribution. The intricate interplay among self-phase modulation, group velocity, and backward Raman scattering results in the self-compressing of the laser pulse. Specifically, the pulses are compressed from their Fourier transform limit duration of 50 fs to a significantly reduced duration of 8 fs at plasma densities below 1/4 critical density, without the transverse self-focusing effects.

Towards Analysis of Multivariate Time Series Using Topological Data Analysis

Topological data analysis (TDA) has proven to be a potent approach for extracting intricate topological structures from complex and high-dimensional data. In this paper, we propose a TDA-based processing pipeline for analyzing multi-channel scalp EEG data. The pipeline starts with extracting both frequency and temporal information from the signals via the Hilbert–Huang Transform. The sequences of instantaneous frequency and instantaneous amplitude across all electrode channels are treated as approximations of curves in the high-dimensional space. TDA features, which represent the local topological structure of the curves, are further extracted and used in the classification models. Three sets of scalp EEG data, including one collected in a lab and two Brain–computer Interface (BCI) competition data, were used to validate the proposed methods, and compare with other state-of-art TDA methods. The proposed TDA-based approach shows superior performance and outperform the winner of the BCI competition. Besides BCI, the proposed method can also be applied to spatial and temporal data in other domains such as computer vision, remote sensing, and medical imaging.

Integrated Ultra‐Wideband Dynamic Microwave Frequency Identification System in Lithium Niobate on Insulator

AbstractThe capability to identify the frequency of unknown microwave signals with an ultra‐wide measurement bandwidth is highly desirable in radar astronomy, satellite communication, and 6G networks. Compared to electronic solutions, the integrated photonic technology‐enabled dynamic instantaneous frequency measurement (DIFM) approach is attractive as it offers unique advantages, such as ultra‐wide frequency measurement bandwidth, high flexibility, and immunity to electromagnetic interference. However, so far the bandwidth of the reported DIFM systems based on integrated photonic technology is limited to below 30 GHz due to the finite bandwidth of electro‐optical modulators (EOMs), limiting their applications, particularly in the field of millimeter wave technology (30–300 GHz). Here, the first integrated dynamic microwave instantaneous frequency measurement system with a record‐breaking operation bandwidth (ranging from 5 to 65 GHz) and low root‐mean‐square (RMS) error (≈300 MHz) is presented on the lithium niobate on insulator (LNOI) integrated photonic platform. This demonstration paves the way for high‐performance millimeter wave photonic integrated devices using the LNOI platform.

Stochastic time series analysis of atrial fibrillatory electrograms in paroxysmal atrial fibrillation

Abstract Background/Introduction Regional dominant frequency and instantaneous frequency modulation of short fractionated atrial electrograms have been associated with ablation targets and the prognosis of atrial fibrillation (AF). However, inconsistent and not fully comprehensible results have been reported. Purpose To construct a simple computational algorithm for the analysis of continuous fibrillatory atrial electrograms and evaluate it in clinical practice, among patients with paroxysmal AF. Methods Before the first cryo-balloon ablation, 3-minute continuous local electrogram recordings during pace-induced AF were obtained by the circumferential catheter from the left atrial appendage (LAA) and each pulmonary vein: left superior (LSPV), left inferior (LIPV), right superior (RSPV) and right inferior (RIPV). Following time- and frequency-domain analysis, non-linear detrended fluctuation analysis (DFA α) with analysis for short-term (DFA α1) and long-term fluctuations (DFA α2) was performed. The signals were detrended by subtracting time series polynomial fit and badpass filtered in the MATLAB environment (cut-off voltage 0.1 mV to avoid scar and artifacts). Results Fifteen patients were studied. Among different atrial areas, higher mean peak rate (pr) along with lower standard deviation of between peaks intervals (sdpp) and root square of peak-to-peak differences (rmppd) were observed in the LAA and left veins (Fig. 1). In patients with longer AF history (> 2 yrs), lower α, α1 and α2 exponents of DFA were observed (Fig. 2). Conclusions This algorithm depicted dissimilar impact on atrial electrical activity between different atrial areas. Patients with longer AF history had less self-similarity in local LAA electrograms. This might indicate possible ablation targets and reveal patients with greater AF burden and worse arrhythmic prognosis.Leftheriotis - Figure 1Leftheriotis - Figure 2

Tacho-less spur gear condition monitoring at variable speed operation using the adaptive application of variational mode extraction

Effective condition monitoring of machine components contributes to a safer working environment for operators and assists in averting critical machinery shutdowns. In real-world industrial scenarios, fault detection must remain simplified, user-friendly and robust despite speed variations. The operational demands of machinery frequently result in speed fluctuations, leading to spectral smearing, thereby compromising the efficiency of the analysis methodology. Furthermore, the intricacies of parameter initialisation often increase the complexity of the analysis methods. This study introduces a fault diagnosis algorithm that requires minimal initialisation and operates independently of tacho pulses. The algorithm proposed in the study incorporates variational mode extraction with the maxima tracking algorithm for instantaneous frequency estimation. Hankel matrix-based selective spectral fusion is proposed to mitigate the impact of frequency tracking errors caused by transient noise. The results of spectral side-band-based fault severity analysis, conducted on an in-house spur gearbox test-bed with seeded tooth chips, underscore the superior performance of the proposed algorithm when compared to contemporary non-stationary analysis methods.

Radial vibration measurement of rotating shaft using constant density fringe pattern and line scan camera

The radial vibration signals of the rotor can provide abundant information about the health condition of the machine. In this paper, a simple vision-based measurement system is proposed to simultaneously measure two-dimensional displacements in radial directions for the rotating shaft, where the system consists of a constant density fringe pattern (CDFP), a line scan camera (LSC), and a lens. The CDFP should be installed around the surface of the rotating shaft to make the density of the fringe constant along the shaft axis, while the shaft axis is vertical to the optical axis of the LSC but not parallel to the line-array sensor of the LSC. Therefore, the density of the fringe imaged on the LSC is not constant because of the modulation of the circular surface of the shaft, and the distribution of the fringe density on the LSC is a U-shaped curve. Thus, the shaft centreline orbit can be tracked by the lowest point of the density distribution curve (DDC) of the fringe. Then, an efficient and accurate parameterized instantaneous frequency estimation method is employed to estimate the DDC of the fringe, because the variable density fringe can be regarded as an amplitude-modulated and frequency-modulated nonstationary signal whose instantaneous frequency function is equivalent to the DDC. Experimental results verify the feasibility and effectiveness of the proposed method by comparing it to the eddy current sensors.