Instantaneous Frequency Estimation Research Articles

Multi-resolution dictionary learning for discrimination of hidden features: A case study of atmospheric gravity waves

In this work, we propose Multi-Resolution Dictionary Learning by promoting sparsity, and Instantaneous Frequency Estimation technique for the discrimination of measured parameter from the effect of unknown sources like noise, and natural phenomena that occur in background during the data acquisition. The Dictionary Learning technique using Multi-Resolution Analysis is performed in the Wavelet Analysis domain using Singular Value Decomposition. In this paper, we aim for the discrimination of contribution from unknown sources in the measured parameter in the amplitude domain and frequency domain by promoting sparsity and Instantaneous Frequency Estimation respectively. We implement the proposed technique and discuss its potential with a case study on the discriminating the Doppler shift in Atmospheric Gravity Waves that occur due to the horizontal wind during data acquisition.

Demodulated Multisynchrosqueezing S Transform for Fault Diagnosis of Rotating Machinery

Time–frequency analysis (TFA) plays an important role in fault diagnosis of rotating machinery. However, a time–frequency representation (TFR) with lower resolution is not conducive to extracting fault features of the equipment. In this article, a novel method, termed demodulated multisynchrosqueezing S transform (DMSSST), is proposed to achieve a highly energy-concentrated TFR, which combines demodulation and multireassignment (MT) techniques for S transform (ST). The proposed method integrates the adaptability of demodulated ST to the frequency with the iterative procedure of MT to optimize the instantaneous frequency (IF) estimation. Compared with the traditional approaches, the presented method has better robustness with respect to noise and can be well applied to the discrimination of several faults. Furthermore, the simulated signal shows that this method can effectively improve the resolution of a TFR and has excellent anti-noise ability. The real examples, including bearing fault and rotor rub-impact, further validate the effectiveness of the DMSSST method in fault diagnosis of rotating machinery.

A statistical instantaneous frequency estimator for high-concentration time-frequency representation

The instantaneous frequency (IF)-based post-processing methods, synchrosqueezing and synchroextracting, can accurately characterize the time-varying frequency and amplitude of multi-component nonstationary signals. However, the window's length needs to be a tradeoff between time-frequency (TF) localization and mode separation. To broaden the limitation of window's length on TF localization, this study theoretically proposes a statistical IF estimator (SIFE) to extract the IFs and concentrate the short-time Fourier transform (STFT) spectrum. First, we introduce a frequency filter associated with the window function and convolve it with STFTs to define an inner product space, in which the IF trajectories of multi-component signals can be better highlighted with low dependence on the window function. Then, the SIFE is derived from a macroscopic perspective in the inner product space rather than in the STFT space as the previous post-processing method does. Finally, the STFT energies are significantly concentrated by extracting only the TF coefficients at all IFs or reassigning the TF coefficients to the respective IFs. Numerical examples demonstrate that the SIFE is superior to previous methods in extracting IFs and concentrating TF energies.

Empirical Mode Decomposition-Based Feature Extraction for Environmental Sound Classification

In environment sound classification, log Mel band energies (MBEs) are considered as the most successful and commonly used features for classification. The underlying algorithm, fast Fourier transform (FFT), is valid under certain restrictions. In this study, we address these limitations of Fourier transform and propose a new method to extract log Mel band energies using amplitude modulation and frequency modulation. We present a comparative study between traditionally used log Mel band energy features extracted by Fourier transform and log Mel band energy features extracted by our new approach. This approach is based on extracting log Mel band energies from estimation of instantaneous frequency (IF) and instantaneous amplitude (IA), which are used to construct a spectrogram. The estimation of IA and IF is made by associating empirical mode decomposition (EMD) with the Teager–Kaiser energy operator (TKEO) and the discrete energy separation algorithm. Later, Mel filter bank is applied to the estimated spectrogram to generate EMD-TKEO-based MBEs, or simply, EMD-MBEs. In addition, we employ the EMD method to remove signal trends from the original signal and generate another type of MBE, called S-MBEs, using FFT and a Mel filter bank. Four different datasets were utilised and convolutional neural networks (CNN) were trained using features extracted from Fourier transform-based MBEs (FFT-MBEs), EMD-MBEs, and S-MBEs. In addition, CNNs were trained with an aggregation of all three feature extraction techniques and a combination of FFT-MBEs and EMD-MBEs. Individually, FFT-MBEs achieved higher accuracy compared to EMD-MBEs and S-MBEs. In general, the system trained with the combination of all three features performed slightly better compared to the system trained with the three features separately.

The second-order local polynomial Fourier transform as instantaneous frequency and chirp rate estimator

Statistical properties, bias and variance, of the second-order local polynomial Fourier transform (LPFT) in the instantaneous frequency (IF) and chirp rate (CR) estimation are considered. We have derived expressions based on the error linearization around the exact value of signal parameters as it is commonly done in this field. However, we have also demonstrated the limits of this methodology. Two particularly important frequency modulated (FM) signals are considered as test cases: quadratic and cubic phase signals. Approximative expression for the bias and variance in the latter case are also derived. It is demonstrated that the optimal design parameter of the LPFT, i.e., window width, must be optimized independently for the IF and CR estimation to produce the minimum mean squared error (MSE). The derived expressions are checked by simulations.

A robust method for instantaneous frequency estimation via inverse spectral decomposition

Instantaneous frequency is an important seismic attribute, which can indicate thin beds and lithofacies boundaries. However, instantaneous frequency attribute is susceptible to noise when it is obtained by the traditional Hilbert transform (HT) method. We propose a robust method for instantaneous frequency estimation. The method first obtains the time-frequency distribution of the seismic signal by inverse spectral decomposition (ISD) and then calculates the analytic signal through the window HT transform. Inverse spectral decomposition achieves high-resolution time-frequency distribution by adding sparse constraint to the corresponding inverse problem, and therefore the noise can be suppressed. The algorithm we choose to solve the mix ℓ2 − ℓ1 problem is the fast iterative shrinkage-thresholding algorithm (FISTA). Compared with the traditional iterative least squares (IRLS) algorithm, FISTA can achieve a better computational efficiency. We perform the method on a quadratic frequency modulation (QFM) signal, a synthetic data based on a wedge model and field data sets to demonstrate its performance, compared with the HT method and the time-frequency adaptive filtering method.

Polarimetric Direction of Arrival Estimations Based on Adaptive Linear Time-Frequency Transforms

A spatially polarized time-frequency distribution (SPTFD) based on dual-polarized double-fed antenna arrays is adapted to deal with polarization-unstable signals. A linear time-frequency (TF) representation was used for an instantaneous frequency (IF) estimate, primarily due to its simplicity and immunity to cross-interference. Using a set of linear TF transformations using Gaussian windows and Fourier oscillation kernels, the IF estimated window widths of multiple unstable signals are obtained. This paper introduces a new method for estimating the direction of arrival (DOA) of polarized waves using adaptive linear time-frequency transforms. In this paper, a narrowband far-field point source on the receiving array is analyzed. The source signal is split into two orthogonally polarized components. The optimal window is determined by the first derivative of the IF; for this purpose, we take a simple algorithm for solving the derivative and optimize it. In developing TF-adaptive and fully automatic TF display technology, the first method is to use the time-adaptive window for minimizing the IF estimate mean square error (MSE) sum at each moment, while the second procedure is to adjust according to time and frequency and minimize estimate MSE sum at each position in the TF region. Due to its combination with signal polarization, the spatial time-frequency distribution (STFD) gains more freedom and thus perfects the phonon space estimation of noise and signal. On the SPTFD platform, polarized time-frequency multiple signal classification (PTF-MUSIC) is used for the estimation of signal direction of arrival, which outperforms conventional time-frequency MUSIC. Using the example of a synthesized signal, this method outperforms conventional techniques in DOA estimation.

A Time–Frequency Representation Approach of Undersampled Signals With Multiple Periodic FM Components

Periodic frequency-modulated (FM) interference signals, commonly employed by jammers and radars, may lead to the performance degradation or even failure of global navigation satellite systems and wireless communications. Time–frequency (TF) representations play an essential role in FM signal analysis and instantaneous frequency (IF) estimation, which is the basis of many nonstationary interference suppression algorithms. This article provides a TF representation approach for signals with multiple periodic FM components in the presence of sample loss. The proposed approach utilizes the periodicity and the sparsity of signal components in the TF domain to suppress cross-terms and artifacts. We reconstruct TF distributions of each component by time-averaging Wigner–Ville distribution. Then, the residual artifacts are eliminated by a low-complexity sparse-based adaptive directional TF filtering method. Simulation results confirm that the proposed approach provides an outstanding suppression of cross-terms and artifacts. The proposed TF representation approach can effectively improve the accuracy of the IF estimation of each signal component.

Synchrosqueezing transform meets [formula omitted]-stable distribution: An adaptive fractional lower-order SST for instantaneous frequency estimation and non-stationary signal recovery

Most of the noises in practice are non-Gaussian and presented as impulsive phenomena. Impulsive noise brings enormous difficulties for signal processing, especially for multicomponent non-stationary signal representation and separation. The synchrosqueezing transform (SST) is a time-frequency (TF) analysis method with high energy concentration and inverse transform. Although SST and its variants have been used for instantaneous frequency estimation and modes recovery in many applications, they are unable to represent the signals clearly under impulsive noise. In this paper, we model the impulsive noise with the α-stable distribution, which satisfies the generalized central limit theorem. Based on the short-time Fourier transform (STFT), we proposed the fractional lower-order SST with adaptive order and adaptive window for multicomponent signal separation with single-channel observation, called the adaptive window and fractional lower-order SST (AWO-SST). Moreover, we consider the 2nd-order phase transformation for AWO-SST and provide a component recovery method under adaptive parameters. The proposed method can be easily extended to the cases of the continuous wavelet transform, the S-transform, and other linear TF analysis methods. The experimental results demonstrate that the proposed methods have the ability to suppress the α-stable distribution noise effectively, and are capable of recovering the component signal accurately.

Skeleton-based reassignment of nonstationary signals spectrogram

Many practical applications require ridge curves detection and enhancement as they allow for instantaneous frequency estimation of nonstationary signals. Unfortunately, the reconstruction of those curves becomes a difficult task whenever multicomponent signals having non-separable modes have to be analyzed. To address this issue, this paper introduces a novel binary model for the spectrogram of multicomponent signals, and applies skeleton to enhance IF curves. To further improve the skeleton-based reassignment, a convolution-deconvolution scheme, that is directly derived from a specific evolution law for signal time-frequency (TF) distribution, is also defined. Experimental results show that the skeleton-based approach can reproduce IF curves of separable modes accurately. In addition, it allows to outperform classical amplitude-based reassignment method in the case of non separable modes.

Synchro-Reassigning Transform for Instantaneous Frequency Estimation and Signal Reconstruction

Traditional signal postprocessing methods suffer from the repeated assignment problem (RAP), which can result in inaccurate instantaneous frequency (IF) estimation and signal recovery. In this article, to solve this problem, a novel time-frequency (TF) analysis (TFA) method called the synchro-reassigning transform (SRT) is proposed. This method aims to obtain the ideal TF representation (TFR) by utilizing derivatives of the constructed amplitude function and a three-step selection rule to adaptively extract the TF coefficients on the IF trajectories in the TF plane, and reassigning these TF coefficients into a new TFR. In this way, SRT can eliminate the RAP, thus obtaining an approximately ideal TFR that helps to realize more accurate IF estimation and signal reconstruction. Furthermore, SRT shows satisfactory performance on signals with high nonlinear IFs or relatively close IFs, even under strong noise conditions. Two simulated signal and three real-life signals were used to demonstrate the performance of SRT.

Velocity Synchronous Chirplet Extracting Transform: An Effective Tool for Fault Diagnosis of Variable-Speed Rotational Machinery

Bearing vibrations from actual sources always involve noise and other interferences, causing the poor time-frequency (TF) readability of time-frequency distribution (TFD) and the inaccurate estimation of instantaneous frequencies (IFs), thus bringing great challenges to the diagnosis of bearing failures. To this end, under the theoretical framework of synchroextracting transform (SET), this work studied a novel time-frequency analysis (TFA) technique named velocity synchronous chirplet extracting transform (VSCET). Compared with other traditional TFA techniques, the VSCET method can accurately depict the variation laws of time-varying features via a highly readable and energy-concentrated TFD. It actually introduced the basis function in velocity synchronous linear chirplet transform (VSLCT) that has the synchronous changing law of rotational speed into SET such that has better advantages in both energy concentration and noise robustness. In addition, a fault diagnosis scheme based on VSCET is further introduced to realize the identification of bearing faults. The validity and practicality of this technology are fully tested from the results of simulations and experiments.

A novel instantaneous frequency estimation method for operational time-varying systems using short-time multivariate variational mode decomposition

Operational modal identification of time-varying systems plays a crucial role in assessing the health condition and controlling the dynamic properties of engineering structures. However, only the response is measurable, making it challenging. Based on the variational mode decomposition (VMD) theory, this paper presents a short-time multivariate or multi-channel VMD (STMVMD) method for instantaneous frequency (IF) identification of time-varying structures in the case of output-only measurements. The idea of short-time windows overcomes the shortcoming of many VMD-based methods that employ the narrowband assumption of intrinsic mode functions (IMFs) and cannot decompose non-stationary signals involving closely-spaced wideband IMFs. After obtaining the multivariate IMFs by STMVMD, an average scheme is employed to estimate IFs, reducing the noise sensitivity of Hilbert Transform. Moreover, by tracking the center frequencies of STMVMD at different moments, another more noise-robust IF estimation method is also presented. A series of numerical and experimental examples illustrate the advantages of the proposal.

Replay spoof detection using energy separation based instantaneous frequency estimation from quadrature and in-phase components

Replay attacks in speech are becoming easier to mount with the advent of high quality of recording and playback devices. This makes these replay attacks a major concern for the security of Automatic Speaker Verification (ASV) systems and voice assistants. In the past, auditory transform-based as well as Instantaneous Frequency (IF)-based features have been proposed for replay spoofed speech detection (SSD). In this context, IF has been estimated either by derivative of analytic phase via Hilbert transform, or by using high temporal resolution Teager Energy Operator (TEO)-based Energy Separation Algorithm (ESA). However, excellent temporal resolution of ESA comes with lacking in using relative phase information and vice-versa. To that effect, we propose novel Cochlear Filter Cepstral Coefficients-based Instantaneous Frequency using Quadrature Energy Separation Algorithm (CFCCIF-QESA) features, with excellent temporal resolution as well as relative phase information. CFCCIF-QESA is designed by exploiting relative phase shift to estimate IF, without estimating phase explicitly from the signal. To motivate and validate effectiveness of proposed QESA approach for IF estimation, we have employed information-theoretic measures, such as Mutual Information (MI), Kullback–Leibler (KL) divergence, and Jensen–Shannon (JS) divergence. The proposed CFCCIF-QESA feature set is extensively evaluated on standard statistically meaningful ASVSpoof 2017 version2.0 dataset. When evaluated on the ASVSpoof 2017 v2.0 dataset, CFCCIF-QESA achieves improved performance as compared to CFCCIF-ESA and CQCC feature sets on GMM, CNN, and LCNN classifiers. Furthermore, in the case of cross-database evaluation using ASVSpoof 2017 v2.0 and VSDC, CFCCIF-QESA also performs relatively better as compared to CFCCIF-ESA and CQCC on GMM classifier. However, for the case of self-classification on the ASVSpoof 2019 PA data, CFCCIF-QESA only outperforms CFCCIF-ESA. Whereas, on BTAS 2016 dataset, it performs relatively close to CFCCIF-ESA. Finally, results are presented for the case when the ASV system is not under attack.

Self-matched extracting wavelet transform and signal reconstruction

Time-frequency (TF) analysis method provides a powerful tool to analyze non-stationary signals. However, for strongly time-varying signals, how to characterize the time-varying features of signals accurately, how to achieve a highly concentrated TF representation, and how to reconstruct signals with high accuracy are still challenging tasks. In this paper, in order to analyze strongly amplitude-modulated and frequency-modulated signals, we introduce an accurate instantaneous frequency (IF) estimator, further propose a high-resolution TF analysis method and three high-accuracy reconstruction methods. Firstly, we present signal parameter, chirp rate and IF estimation from the wavelet transform of Gaussian amplitude-modulated linear chirp model. Based on this, we introduce the self-matched extracting wavelet transform (SMEWT), which improves the energy concentration of TF representation, by only retaining TF information related to the IF of the signal. Furthermore, the signal parameter, chirp rate and IF estimation lead to accurate signal reconstruction formulas. In this regard, we provide a theoretical analysis and comparison of SMEWT reconstruction, and present two hybrid reconstruction methods of signal. It is well known that the reassignment method can improve the readability of the TF representation, but it cannot reconstruct the signal. It is worth mentioning that one of the proposed hybrid reconstruction methods is a combination of the reassignment method and deduced reconstruction formula, which not only captures the flavor and philosophy of the reassignment method, but also uses a hybrid way in signal reconstruction. Finally, simulated and real signals are employed to confirm the effectiveness of the proposed methods by comparing with some classical TF analysis methods.

Entropy-Based Concentration and Instantaneous Frequency of TFDs from Cohen's, Affine, and Reassigned Classes.

This paper explores three groups of time–frequency distributions: the Cohen’s, affine, and reassigned classes of time–frequency representations (TFRs). This study provides detailed insight into the theory behind the selected TFRs belonging to these classes. Extensive numerical simulations were performed with examples that illustrate the behavior of the analyzed TFR classes in the joint time–frequency domain. The methods were applied both on synthetic and real-life non-stationary signals. The obtained results were assessed with respect to time–frequency concentration (measured by the Rényi entropy), instantaneous frequency (IF) estimation accuracy, cross-term presence in the TFRs, and the computational cost of the TFRs. This study gives valuable insight into the advantages and limitations of the analyzed TFRs and assists in selecting the proper distribution when analyzing given non-stationary signals in the time–frequency domain.

Missing samples reconstruction using an efficient and robust instantaneous frequency estimation algorithm

In order to recover missing samples in a nonstationary signal, this paper employs a time-signal analysis and filtering method. The instantaneous frequency of a multicomponent signal is first estimated by employing a robust and computationally efficient method. Then the time-frequency filtering is performed using a dechirping operation to recover missing samples. These steps are repeated until convergence. The proposed method achieves better performance than the state of art methods both in terms of the accuracy of the recovered signal and computational efficiency.

The multitaper reassigned spectrogram for oscillating transients with Gaussian envelopes

Joint time-frequency representations are important tools when estimating the instantaneous frequency. The widely used spectrogram is known to have poor energy localisation, which the reassignment method improves. However, the reassignment method is sensitive to noise. In this paper we present a multitaper reassigned spectrogram (MTRS) that is robust to noise and tailored to short duration transient signals. The MTRS is designed for oscillating transients with Gaussian envelopes and uses the Hermite functions as windows. The novel reassignment coordinates needed for the MTRS are derived in this paper. The MTRS localises energy at the instantaneous frequencies of transients, while white Gaussian noise is made flatter by the use of multiple windows. It is shown that the MTRS is more noise robust compared to reassignment with just one window, and that it improves on the energy localisation of the spectrogram. Thus, our MTRS is sparse, easy to interpret, and well suited for instantaneous frequency estimation, even for signals with overlapping transients and low signal-to-noise ratios.

Instantaneous Frequency Estimation Method for the Vibration Signal of Rotating Machinery Based on STFTSC Algorithm

In this paper, an instantaneous frequency identification method known as STFTSC is developed by combining short-time Fourier transformation and the seam carving (SC) algorithm (widely used in image processing). In this method, the STFT is applied to analyze the time-frequency energy distribution of a vibration signal under variable-speed conditions. Subsequently, the energy gradient is calculated through the Sobel operator based on the time-frequency energy distribution. The energy gradient distribution contains multiple ridges, which are corresponding instantaneous frequencies of different orders. Finally, the targeted ridge extraction is transformed into an optimization problem, and the dynamic programming algorithm (DP) is used to search the targeted ridge with the minimum energy gradient to estimate the instantaneous frequency. The effectiveness of the proposed method is validated by a simulation experiment, moreover, a rotating machinery fault simulation test bench is employed to validate the method, which is then compared with polynomial chirplet transformation (PCT) and analyzed during the test process. The results show that the STFTSC instantaneous frequency estimation algorithm has a higher extraction accuracy and better application value in engineering problems than the PCT.

An Efficient and Accurate Multi-Sensor IF Estimator Based on DOA Information and Order of Fractional Fourier Transform.

Instantaneous frequency in multi-sensor recordings is an important parameter for estimation of direction of arrival estimation, source separation, and sparse reconstruction. The instantaneous frequency estimation problem becomes challenging when signal components have close or overlapping signatures and the number of sensors is less than the number of sources. In this study, we develop a computationally efficient method that exploits the direction of the IF curve in addition to the angle of arrival as additional features for the accurate tracking of IF curves. Experimental results show that the proposed scheme achieves better accuracy compared to the-state-of-art method in terms of mean square error (MSE) with a slight increase in the computational cost, i.e., the proposed method achieves MSE of −50 dB at the signal to noise ratio of 0 dB whereas the existing method achieves the MSE of −38 dB.