Instantaneous Frequency Estimation Research Articles

A Fast Instantaneous Frequency Estimation for Underwater Acoustic Target Feature Extraction

Traditional auditory features merely present the amplitude characteristics of target signals in frequency domain. Such features are susceptible to environmental noise, resulting in significant degradation of recognition stability. Inspired by instantaneous information applied in speech signal processing field, this paper proposed a feature extraction method using sub-based instantaneous frequency. A fast instantaneous frequency information extraction algorithm is proposed with the normalized Gammatone filterbanks. Experiments confirm that the proposed feature extraction method can effectively maintain the recognition accuracy under low SNR conditions while reduce the computation cost.

A generalized stochastic resonance based instantaneous frequency estimation method under low SNR

In this work, the instantaneous frequency estimation for the frequency modulated (FM) signal embedded in strong noise is considered. For engineering signals, current instantaneous frequency estimation methods are less satisfactory under low signal-to-noise ratio (SNR). To address this problem, a generalized stochastic resonance (GSR) based instantaneous frequency estimation method is proposed. First, the generalized stochastic resonance (GSR) is proposed, which overcomes the low frequency limitation of classical stochastic resonance and can enhance weak FM signals with time varying frequency. The GSR transforms the FM signal to a low-constant frequency signal by multiplying the parameterized demodulation operator with polynomial phase and then enhances it by the stochastic resonance. Second, based on the maximization of the GSR output signal, the instantaneous frequency can be estimated by global optimization of the polynomial coefficients in the parametrized demodulation operator. For the multi-component FM signal, it works by enhancing the objective component and suppressing the noise and other irrelevant components, in which the instantaneous frequency of each component is estimated in the order of component strengths and each component can be extracted from the signal. Numerical simulations and experimental signals validate that the proposed method outperforms the state-of-art methods in estimation accuracy for both mono-component and multi-component FM signals under lower SNR.

Fractional Differential Equation-Based Instantaneous Frequency Estimation for Signal Reconstruction

In this paper, we propose a fractional differential equation (FDE)-based approach for the estimation of instantaneous frequencies for windowed signals as a part of signal reconstruction. This approach is based on modeling bandpass filter results around the peaks of a windowed signal as fractional differential equations and linking differ-integrator parameters, thereby determining the long-range dependence on estimated instantaneous frequencies. We investigated the performance of the proposed approach with two evaluation measures and compared it to a benchmark noniterative signal reconstruction method (SPSI). The comparison was provided with different overlap parameters to investigate the performance of the proposed model concerning resolution. An additional comparison was provided by applying the proposed method and benchmark method outputs to iterative signal reconstruction algorithms. The proposed FDE method received better evaluation results in high resolution for the noniterative case and comparable results with SPSI with an increasing iteration number of iterative methods, regardless of the overlap parameter.

RANSAC algorithm for instantaneous frequency estimation and reconstruction of frequency-modulated undersampled signals

Frequency modulated (FM) signals sampled below the Nyquist rate or with missing samples (nowadays part of wider compressive sensing (CS) framework) are considered. Recently proposed matching pursuit and greedy techniques are inefficient for signals with several phase parameters since they require a search over multidimensional space. An alternative is proposed here based on the random samples consensus algorithm (RANSAC) applied to the instantaneous frequency (IF) estimates obtained from the time-frequency (TF) representation of recordings (undersampled or signal with missing samples). The O’Shea refinement strategy is employed to refine results. The proposed technique is tested against third- and fifth-order polynomial phase signals (PPS) and also for signals corrupted by noise.

Motion Parameters Estimation of a Moving Target Based on Instantaneous Frequency Estimates Using a Passive Acoustic Sensor

This paper studies a method for estimating the motion parameters [Formula: see text] using the instantaneous frequency estimates of the acoustical signal emitted from a moving target in the air and received by an acoustic sensor placed on the ground. Here, [Formula: see text] is the target’s travel speed, [Formula: see text] is the closest point of approach slant range, [Formula: see text] is the source acoustical frequency, and [Formula: see text] is the time at the target get to the closest point of approach to the sensor. A novel relationship formula between the instantaneous frequency and motion parameters is derived and an algorithm for estimating the motion parameters based on the solution of quadratic nonlinear optimization from the instantaneous frequency estimates is proposed. The instantaneous frequency is estimated by searching the peak of a polynomial Wigner–Ville distribution. An approach for calculating the optimal window width in a polynomial Wigner–Ville distribution is proposed. Numerical examples are provided for validating our theoretic claims.

Analysis of an adaptive short-time Fourier transform-based multicomponent signal separation method derived from linear chirp local approximation

Recently, a direct method of the time–frequency approach, called the signal separation operator (SSO), which is based on sinusoidal signal approximation, was introduced to solving the inverse problem of multicomponent signal separation. In a very recent paper “Direct signal separation via extraction of local frequencies with adaptive time-varying parameters”, the authors obtained a more accurate component recovery formula derived from the linear chirp (also called linear frequency modulation signal) approximation at any local time. However the theoretical analysis of the recovery formula derived from linear chirp local approximation has not been studied there. In this paper, we carry out the analysis of SSO based on the adaptive short-time Fourier transform (STFT). We study both the sinusoidal signal-based model and the linear chirp-based model, and obtain the error bounds for the instantaneous frequency estimation and component recovery. The error bounds are derived by studying the approximation to the STFT of each component and by the assumption of the decrease of the Fourier transform of the window function for STFT. These results provide a mathematical guarantee to the proposed adaptive STFT-based non-stationary multicomponent signal separation method. In addition, experiments are provided to illustrate the general theory.

Second-order synchroextracting wavelet transform for nonstationary signal analysis of rotating machinery

Characterizing nonstationary signals with widely changing instantaneous frequency features is a challenging issue for time-frequency analysis (TFA) method. A novel TFA method termed second-order synchroextracting wavelet transform (SSEWT) is proposed for this purpose. The method extends the existing synchroextracting transform to the context of wavelet transform, and introduces the second-order instantaneous frequency estimate to improve its capability of processing nonstationary signals. The proposed method provides concentrated time-frequency representation with multiple resolutions, and retains good invertibility. The theories concerned and discrete implementation of the SSEWT are presented. Numerical simulations and experimental data collected from rotating machinery are employed to verify the competitive advantages of the SSEWT in revealing the local time-frequency features of nonstationary signals.

Enhancement of adaptive mode decomposition via angular resampling for nonstationary signal analysis of rotating machinery: Principle and applications

Vibration signal analysis provides an effective approach for condition monitoring and fault diagnosis of rotating machines. Under time-varying conditions, vibration signals feature nonstationarity, consist of multiple frequency components, and usually have spectral overlaps. Adaptive mode decomposition methods can extract mono-components from a given multi-component signal to meet the requirement for estimating instantaneous frequency. Among them, the recently proposed methods, including empirical wavelet transform, variational mode decomposition and Fourier decomposition method, outperform the classic empirical mode decomposition in terms of rigorous mathematical formulation. Nevertheless, for multi-component signals with spectral overlaps, these three methods are subject to mode mixing and/or integrity issues, and fail to extract true mono-components, because they essentially separate mono-components based on spectral segmentation. In this paper, we propose a framework by exploiting the capability of angular resampling to address the mono-component overlapping issue. This methodology can separate true mono-components, thus facilitating accurate estimation of instantaneous frequency through Hilbert transform and generating perfect time–frequency representations (TFRs). Firstly, angular resampling is employed to make the constituent mono-components well separable in the frequency domain. Then, true mono-components are separated through adaptive mode decomposition. Next, informative mono-components are selected for further processing based on the prominent order information, and the selected mono-components are mapped into the time domain according to the relationship between the equal time and equal angle sampling. Finally, the instantaneous frequency and amplitude envelope of the recovered mono-components are calculated via Hilbert transform, and the TFR of raw signal is obtained by superposing the TFRs of all recovered mono-components. By doing so, the TFR achieves a fine time–frequency resolution and is free of both outer and inner interferences. The proposed methodology is demonstrated by simulated signal analysis, and further validated using the vibration data sets of three typical rotating machines (including a planetary gearbox in a wind turbine drivetrain, a civil aircraft engine and a hydraulic turbine rotor). The analysis results show its excellent capability to reveal the time-varying features of rotating machinery nonstationary signals.

A Passive DOA Estimation Algorithm of Underwater Multipath Signals via Spatial Time-Frequency Distributions

Direction of arrival (DOA) estimation of underwater multipath signals has become an indispensable part of many military and civil applications in the oceanic field. However, DOAs of different paths of the multipath signals may be the same or too close to be separated in space, which brings a great challenge to the DOA estimation of multipath signals. Besides, the number of multipath is generally unknown and time-varying in passive scenarios which generally lead to an underdetermined case. Therefore, it is essential to study an efficient algorithm to realize the underdetermined DOA estimation of underwater multipath signals in passive scenarios. This paper proposes a passive DOA estimation algorithm of underwater multipath signals via spatial time-frequency distributions (STFDs). Firstly, we establish a multipath signals model and utilize the quadratic time-frequency (TF) analysis method to reduce cross-terms while keeping a good resolution. Then, we estimate the number of multipath and introduce a novel multi-ridge detection technique to extract the TF trajectories of multipath signals, which adopts the idea of smoothing forward and backward, and the edge TF point detection strategy. Lastly, we estimate the DOA of each path of the multipath signals separately and realize source association based on the clustering-based method. Simulation results and analysis demonstrate that this algorithm can effectively improve the estimation accuracy of instantaneous frequency (IF) and DOA estimation of underwater multipath signals in underdetermined cases and passive situations.

Scaled Wigner distribution using fractional instantaneous autocorrelation

The traditional Wigner distribution (WD) is extended to a novel one inspired by the definition of fractional bispectrum, giving rise to a scaled version with respect to the frequency variable, termed as the scaled Wigner distribution (SWD). A natural magnification effect characterized by a factor k on the frequency axis enables the SWD to have flexibility to be used in cross-term reduction. Some essential properties of this variation generalize very nicely and simply the classical results for the WD, and it enjoys a computational complexity in the order of O(kN2) comparable to O(N2) of the WD. The SWD does not use localization windows, but its resolutions depend on the factor k. Comparison of the detection performance (i.e. cross-term reduction, time–frequency resolution, angle or distance resolution) of SWD and WD for multicomponent linear frequency-modulated (LFM) signals processing is then observed by exploring the factor k selection results on two kinds of bi-component cases, including the one with a different frequency rate and the other one with the same frequency rate but having a different initial frequency. Correctness of the derived results and the SWD’s superiority in the instantaneous frequency estimation of noisy LFM signals compared with the WD are illustrated through simulations, and finally further discussions point out some promising future research directions and perspectives.

Multi-component instantaneous frequency estimation in mono-sensor and multi-sensor recordings with application to source localization

A computationally efficient method is proposed to estimate the instantaneous frequency of multi-component signals for both mono-sensor and multi-sensor recordings. The proposed scheme adopts a well known connected component linking method and extends it for signals with intersecting components. For mono-sensor recordings, the proposed method exploits the direction of ridges estimated through a gradient-based approach to estimate the IFs for crossing components. For multi-sensor recordings, the spatial diversity provided by a uniform linear array of sensors is exploited. The proposed multi-sensor IF estimation algorithm is applied for the estimation of the direction of arrival (DOA) of signal emitting sources. It is demonstrated that the DOA estimation method proposed in this paper is superior to the adaptive directional time-frequency distribution based method in terms of computational efficiency and performance.

Rule-Based EEG Classifier Utilizing Local Entropy of Time–Frequency Distributions

Electroencephalogram (EEG) signals are known to contain signatures of stimuli that induce brain activities. However, detecting these signatures to classify captured EEG waveforms is one of the most challenging tasks of EEG analysis. This paper proposes a novel time–frequency-based method for EEG analysis and characterization implemented in a computer-aided decision-support system that can be used to assist medical experts in interpreting EEG patterns. The computerized method utilizes EEG spectral non-stationarity, which is clearly revealed in the time–frequency distributions (TFDs) of multicomponent signals. The proposed algorithm, which is based on the modification of the Rényi entropy, called local or short-term Rényi entropy (STRE), was upgraded with a blind component separation procedure and instantaneous frequency (IF) estimation. The method was applied to EEGs of both forward and backward movements of the left and right hands, as well as to EEGs of imagined hand movements, which were captured by a 19-channel EEG recording system. The obtained results show that in a given virtual instrument, the proposed methods efficiently distinguish between real and imagined limb movements by considering their signatures in terms of the dominant EEG component’s IFs at the specified subset of EEG channels (namely, F3, F4, F7, F8, T3, and T4). Furthermore, computing the number of EEG signal components, their extraction, and IF estimation provide important information that shows potential to enhance existing clinical diagnostic techniques for detecting the intensity, location, and type of brain function abnormalities in patients with neurological motor control disorders.

FPGA based Identification of Frequency and Phase Modulated Signals by Time Domain Digital Techniques for ELINT Systems

In this paper, a decision tree algorithm based on time-domain digital technique is developed for the identification and classification of diverse radar intra-pulse modulated signals for the electronic intelligence system in real-time. This includes linear frequency modulation, non-linear frequency modulation, stepped frequency modulation and bi-phase modulation. The received signal is digitised and the instantaneous phase and high accuracy instantaneous frequency are estimated. The instantaneous amplitude is also estimated to get the start and stop of the pulse. Instantaneous parameters are estimated using a moving autocorrelation technique. The proposed algorithm is employed on the instantaneous frequency and the modulation is identified. The modulation type and modulation parameter are important for unique radar identification when similar radars are operating in a dense environment. Simulations are carried out at various SNR conditions and results are presented. The model for algorithm is developed using a system generator and implemented in FPGA. These results are compared when the proposed algorithm is used with the existing digital in-phase and quadrature-phase (DIQ) technique of instantaneous frequency and amplitude estimation.

Optimization of multicomponent signals entered to the system using estimation of instantaneous frequency

The frequency signal displays are not efficient for analyzing nonstationary signals because of their inability to represent frequency changes over time. In fact, because most of the signals are real, nonstationary, and time varying, analyzing the signals in the time–frequency domain to estimate the instantaneous frequency of a signal is inevitable. The methods of estimating the instantaneous frequency of the multicomponent signals are divided into three groups, which include the methods using signal phase derivatives that are sensitive to noise, methods that calculate the number of zero points of the signal and consider the signal frequency equal to half the frequency of the zero points and are suitable for signals that can be imagined as stationary, and methods based on time–frequency distributions and distributions such as Wigner for instantaneous frequency calculations and more for instantaneous frequency calculations on nonstationary noise signals that exhibit varied time–frequency distributions. In this article, a new hybrid algorithm is used to evaluate different distribution criteria and comparing their performance in investigating one or more features of time–frequency distributions, such as resolution and energy concentration.

A pde-Based Analysis of the Spectrogram Image for Instantaneous Frequency Estimation

Instantaneous frequency (IF) is a fundamental feature in multicomponent signals analysis and its estimation is required in many practical applications. This goal can be successfully reached for well separated components, while it still is an open problem in case of interfering modes. Most of the methods addressing this issue are parametric, that is, they apply to a specific IF class. Alternative approaches consist of non-parametric time filtering-based procedures, which do not show robustness to destructive interference—the most critical scenario in crossing modes. In this paper, a method for IF curves estimation is proposed. The case of amplitude and frequency modulated two-component signals is addressed by introducing a spectrogram time-frequency evolution law, whose coefficients depend on signal IFs time derivatives, that is, the chirp rates. The problem is then turned into the resolution of a two-dimensional linear system which provides signal chirp rates; IF curves are then obtained by a simple integration. The method is non-parametric and it results quite robust to destructive interference. An estimate of the estimation error, as well as a numerical study concerning method sensitivity and robustness to noise are also provided in the paper.

Lameness Detection in Cows Using Hierarchical Deep Learning and Synchrosqueezed Wavelet Transform

Objectives: Identification of cow lameness is important to farmers to improve and manage cattle health and welfare. No validated tools exist for automatic lameness detection. In this research, we aim to early detect the cow lameness by identifying the instantaneous fundamental gait harmonics from low frequency (16Hz) acceleration signals recorded using leg-worn sensors. Methods: A triaxial accelerometer has been worn on each cow leg. Synchrosqueezed wavelet transform (SSWT) has been applied to acceleration signals to generate the initial time-frequency spectrum related to the gait. This spectrum is given as an input to a designed deep neural network including time-frequency based long short-term memory (LSTM) to estimate instantaneous frequencies at each time point. An inverse SSWT (ISSWT) is then used to recover the gait harmonic and to estimate an enhanced spectrum. Results: Validation of instantaneous frequencies has been provided for each cow leg (combined signals from 23 cows) and the time-series cross validator across the three folds are provided. The average of mean squared errors in frequencies across 3 folds for each leg is obtained as 0.036, 0.033, 0.044 and 0.042 for left-front, right-front, right-back and left-back legs, respectively. Conclusion: Estimation of instantaneous gait frequencies is proved useful for identification of cow gait phases, lameness detection, accurate estimation of gait speed, coherency in movement among the legs and identification of non-gait episodes. Moreover, the proposed method can be used as a new frequency ridge estimation method exploiting SSWT for many other applications.

Matching Demodulation Synchrosqueezing S Transform and its Application in Seismic Time–Frequency Analysis

Time–frequency (TF) analysis is a powerful tool to describe the frequency response of subsurface rocks and hydrocarbon reservoirs. In this letter, the matching demodulation synchrosqueezing S transform (MDSST) that achieves a highly concentrated TF representation was introduced to seismic TF analysis. The MDSST includes two steps: the matching demodulation technique is preprocessing that provides a rough estimation of instantaneous frequency (IF) of the analyzed signal, and the synchrosqueezing transform (SST) is postprocessing that produces an accurate IF. Numerical experiments on synthetic and real signals demonstrate the superiority of the MDSST in seismic TF analysis.

Instantaneous frequency estimation for wheelset bearings weak fault signals using second-order synchrosqueezing S-transform with optimally weighted sliding window

The second-order synchrosqueezing S-transform (SSST2) is an important method for instantaneous frequency (IF) estimation of non-stationary signals. Based on the synchrosqueezing S-transform, the instantaneous frequency calculation method is modified using the second-order partial derivatives of time and frequency to achieve higher frequency resolution. However, weak multi-frequency signals with strong background noise are often drowned out during the transformation process. To achieve enhanced extraction of weak fault characteristic signals due to mechanical faults, this paper proposes an optimally weighted sliding window signal segmentation algorithm based on the SSST2. The results of simulations and experiments show that the time–frequency aggregation of the second-order synchrosqueezing S-transform based on the optimally weighted sliding window (OWSW-SSST2) is not only significantly higher than that of commonly used time–frequency transforms, but it also has better operational efficiency than the second-order synchrosqueezing S-transform. In this paper, the proposed algorithm is used to analyze fault signals from actual high-speed railway wheelset bearings. The results show that the OWSW-SSST2 algorithm greatly improves the spectral aggregation of the signal, and crucially, that high-precision IF estimates for signals can be obtained in low signal-to-noise ratio environments. This research is both of academic interest and significant for practical engineering use to ensure safe high-speed rail operations. It helps enable monitoring the status of wheelset bearings, correctly estimating the locations and causes of failures, and providing up-to-date systematic maintenance and system improvement strategies.

Parameterized Local Maximum Synchrosqueezing Transform and its Application in Engineering Vibration Signal Processing

The time-frequency (TF) analysis (TFA) method is an effective tool for analyzing the time-variant features of non-stationary signals. Synchrosqueezing transform (SST) is a promising TFA method that has recently shown its usefulness in a wide range of engineering signal processing applications. On the other hand, the SST method suffers from some drawbacks, one of which is that when processing the frequency-modulated (FM) signal, the TF representation will smear heavily, which hinders its application in engineering vibration signals. In this paper, we propose a new TFA method named parameterized local maximum synchrosqueezing transform (PLMSST) to study engineering vibration signals with FM characteristics. First, the limitation of SST in signal processing is discussed. Next, we demodulate the signal by parameterizing the short-time Fourier transform (STFT) to correct the deviation of instantaneous frequency (IF) estimation. Further, we detect the local maximum of the spectrogram in the frequency direction to get the accurate IF estimate, and then obtain the energy-concentrated TF representation. Finally, we introduce the reconstruction function of this method. The performance of the proposed method is validated by both the numerical and experimental signals including vibration signals of the rolling bearing and the bridge. The results show that the proposed method is more effective in processing engineering vibration signals than other TFA methods.

Local Maximum Synchrosqueezing Chirplet Transform: An Effective Tool for Strongly Nonstationary Signals of Gas Turbine

Time–frequency (TF) analysis (TFA) provides an effective tool to characterize nonstationary signals with time-varying features. However, the TFA of gas turbine’s vibration signals is a challenging topic due to high complexity and strong nonstationarity. There is an obstacle to generate more accurate and sharper TF results for such multicomponent signals. This article proposes a novel TFA technique, named local maximum synchrosqueezing chirplet transform (LMSSCT), to deal with this problem. This method can not only well match window function and modulated frequency but produce an unbiased instantaneous frequency (IF) estimator to correct the deviation caused by strong frequency modulation (FM) in TF results. We give the theoretical analysis that this method is an improvement of classical local maximum synchrosqueezing transform (LMSST), and we also prove that it allows for perfect signal reconstruction. The numerical validation shows that the proposed method can be employed to effectively address the multicomponent signals with complex FM laws, even those with heavy noise. The experimental analysis on the test-bench signal and the vibration signal of a dual-rotor gas turbine validates that this method can capture more detailed features that are helpful to identify the origins of abnormal vibration of gas turbine.