Instantaneous Frequency Estimation Research Articles

Time-frequency readability enhancement of compact support kernel-based distributions using image post-processing: Application to instantaneous frequency estimation of M-ary frequency shift keying signals

Time-frequency distributions (TFDs) based on time-lag kernels with compact support (KCS) have proved their high performance in terms of resolution and crossterms suppression. However, as for all kernel-based quadratic TFDs, these distributions suffer from spreading out signal terms. This is due to the unavoidable smoothing effects of the kernel in the ambiguity domain. The main objective of this paper is to improve concentration, interference rejection and so time-frequency localization of this representation class. The latter has the advantage of being tuned using a single parameter while external windows are no longer needed. The KCS-TFDs, referred to as KCSDs, are first optimized using a selection of the most used objective performance measures in the literature. Important signal features are extracted as well through analysis of time slice plots. The obtained TF diagrams are then enhanced using a specific method that includes two-dimensional Wiener filter, automatic binarization and morphological image processing techniques. The enhanced plots are compared to those obtained from the original TFDs using several tests on real-life and multicomponent frequency modulated (FM) signals including the noise effects. Moreover, a comparative study involving a selection of the best-performing reassignment time-frequency distributions is provided. The obtained results show a significant improvement of concentration, time-frequency localization of the autoterms as well as interference and noise suppression. As viable applications, the proposed approach is used first to instantaneous frequency (IF) estimation of several synthetic and real-life M-ary frequency shift keying (MFSK) signals. It is shown that the IF estimator from the enhanced plots performs better than smoothed pseudo Wigner-Ville distribution (SPWVD) and reassignment post-processing-based TFDs in terms of mainlobe width (MLW) and variance, respectively, even at low signal-to-noise ratio (SNR). On the other hand, time-frequency characterization of continuous wave linear frequency modulation (CW-LFM) and pulse linear FM (PLFM) radar signals is also investigated.

Automatic instantaneous frequency estimator for multicomponent signals with the variable number of components

A novel automatic instantaneous frequency (IF) estimator is proposed for multicomponent signals with the variable number of components in noise. To obtain maximum energy concentration along the IF, an extended synchrosqueezing transform is developed by combining fractional spectrograms. The IF estimation from the time-frequency representation is then achieved by using a novel automatic ridge extraction (AREA) algorithm. The AREA is implemented in three steps: extracting, merging, and reconstructing ridges. As the number of components may vary in time, mergeability criterion with the combined area-and-gap penalty merge two adjacent ridges to determine the IF for each component. The performance of the proposed IF estimator is demonstrated on synthetic and real signals.

A Method for Accurate Estimation of Instantaneous Frequency Considering Aliasing of Oscillation Components

A Method for Accurate Estimation of Instantaneous Frequency Considering Aliasing of Oscillation Components

Improved parametrized multiple window spectrogram with application in ship navigation systems

In analyzing non-stationary noisy signals with time-varying frequency content, it's convenient to use distribution methods in joint, time and frequency, domains. Besides different adaptive data-driven time-frequency (TF) representations, the approach with multiple orthogonal and optimally concentrated Hermite window functions is an effective solution to achieve a good trade-off between low variance and minimized stable bias estimates. In this paper, we propose a novel spectrogram method with multiple optimally parameterized Hermite window functions, with parameterization which includes a pair of free parameters to regulate the shape of the window functions. The computation is performed in the optimization process to minimize the variable projection (VP) functional problem. The proposed parametrized distribution method improves TF concentration and instantaneous frequency (IF) estimation accuracy, as shown in experimental results for synthetic signals and real-life ship motion response signals. With the optimization of nonlinear least-squares approximation of the ship response signals, the Hermite spectra are centralized, and only up to 15 basis functions are sufficient for concentration improvement in the TF domain.

Self-paced learning for instantaneous frequencies estimation in heavy noise environments

Estimating instantaneous frequencies (IFs) overlap in both time and frequency domains is desired in time-frequency (TF) analysis. In this paper, a simple method, called IFs estimation based on self-paced learning (IFsESPL) algorithm, is proposed to separate overlapped components of multi-component (MC) non-stationary signals in TF plane, and simultaneously estimate IFs. The proposed IFsESPL algorithm first detects TF ridge points from a TF representation, and then the TF ridge points are used to construct training dataset of self-paced learning (SPL). In the following, minimization of particular objective function of SPL yields iteratively selecting and classifying the samples in training dataset into clusters. Each cluster is corresponding to one component of a MC signal and hence component separation is achieved. In addition, together with the partitioning of the training dataset, the IF of each component is estimated by a predefined parameterized model. IFsESPL incorporates SPL regime into TF ridge points regrouping to sequentially include ridge points into IFs estimation from easy to hard, which not only can separate crossed components in TF plane, but also can prevent interference of noise and outliers. The effectiveness of the proposed method is verified through analytical results and numerical examples.

ADTFD-RANSAC For multi-component IF estimation

A RANSAC-based instantaneous frequency (IF) estimation algorithm for multi-component signals having overlapping time-frequency signatures is developed. The proposed algorithm first analyzes a multi-component signal using adaptive directional time-frequency distribution (ADTFD) to obtain a high-resolution signal representation. The algorithm iteratively selects a set of peaks from the ADTFD, fits the IF curve, and evaluates the IF curve using a novel objective function that takes into account both amplitude and smoothness of the curve. Among all the candidate IF curves, the one minimizing the objective function is selected. The IF information is used to remove the corresponding component from the mixture signal and this process is iterated till the IFs of all the components are estimated. Experimental results indicate the superiority of the proposed method as compared to the state of the art.

Multi-lifting synchrosqueezing transform for nonstationary signal analysis of rotating machinery

Blurry time–frequency representation (TFR) of most time–frequency analysis methods influenced by inaccurate instantaneous frequency estimation hinders the meaningful rotating machinery fault detection. To address the drawback, a novel time–frequency analysis method called multi-lifting synchrosqueezing transform (MLST) is proposed, which could enhance the energy concentration of TFR for fast-varying instantaneous frequency and strong noise signals of rotating machinery. Hereinto, the critical points are improved for nonstationary signal analysis: (1) construct a multisqueeze second-order lifting operator to accurately estimate the fast-varying instantaneous frequency; (2) propose a correction operator to correct the deviation in the process of assigning time–frequency energy; (3) put forward a time–frequency fault measure cluster (TFMC) to optimize and evaluate the result of MLST. Through such multi-lifting operations, the final TFR with a high readability could provide a promising support for rotating machinery fault diagnosis. The repeatable simulations and engineering applications are used to verify the effectiveness of the method.

Instantaneous Frequency Estimation of Nonlinear FM Radar Signal Based on Multi-Scale Chirplet Path

Instantaneous frequency is an important parameter to non-linear frequency modulated (NLFM) signal in low probability intercept (LPI) radar. For electronic intelligence, it is very important to accurately estimate instantaneous frequency of NLFM signal. A multi-scale chirplet path pursuit (MCPP) method and its improved method are proposed for electronic intelligence systems to estimate instantaneous frequency of NLFM radar signal in this paper. Firstly, signal duration is divided into a set of dynamic time interval, multi-scale chirplet basis function is established on each time interval simultaneously. And then, projection coefficient in each dynamic interval is calculated basing on chirplet basis functions. And then, chirplet basis functions which have the largest projection coefficient with the analysis signal in each time interval are connected by path pursuit algorithm. Rough estimation of instantaneous frequency will be achieved by connecting the linear frequency of those chirplet basis functions. At last, to solve the problem that instantaneous frequency curve is not smooth for the impact of noise and chirplet errors, least square fitting method is used to further improve estimation accuracy. Experimental results show that, proposed improved MCPP algorithm is suitable for the instantaneous frequency of the NLFM radar signal at low SNR. Compared with time-frequency analysis method, it has higher estimation accuracy. Proposed method can also be applied to the instantaneous frequencies estimation of other NLFM signal without prior knowledge, such as seismic signals and fault diagnosis signals.

Power spectrum slope confounds estimation of instantaneous oscillatory frequency

Oscillatory neural dynamics are highly non-stationary and require methods capable of quantifying time-resolved changes in oscillatory activity in order to understand neural function. Recently, a method termed ‘frequency sliding’ was introduced to estimate the instantaneous frequency of oscillatory activity, providing a means of tracking temporal changes in the dominant frequency within a sub-band of field potential recordings. Here, the ability of frequency sliding to recover ground-truth oscillatory frequency in simulated data is tested while the exponent (slope) of the 1/fx component of the signal power spectrum is systematically varied, mimicking real electrophysiological data. The results show that 1) in the presence of 1/f activity, frequency sliding systematically underestimates the true frequency of the signal, 2) the magnitude of underestimation is correlated with the steepness of the slope, suggesting that, if unaccounted for, slope changes could be misinterpreted as frequency changes, 3) the impact of slope on frequency estimates interacts with oscillation amplitude, indicating that changes in oscillation amplitude alone may also influence instantaneous frequency estimates in the presence of strong 1/f activity; and 4) analysis parameters such as filter bandwidth and location also mediate the influence of slope on estimated frequency, indicating that these settings should be considered when interpreting estimates obtained via frequency sliding. The origin of these biases resides in the output of the filtering step of frequency sliding, whose energy is biased towards lower frequencies precisely because of the 1/f structure of the data. We discuss several strategies to mitigate these biases and provide a proof-of-principle for a 1/f normalization strategy.

Second-order Synchrosqueezing Modified S Transform for wind turbine fault diagnosis

Under the working condition of variable speed, vibration signals of wind turbine gearboxes are characterized by instability coupled with noise interference, which has made it difficult to identify early fault characteristics through common Time-Frequency (TF) Analysis (TFA) methods. Stockwell Transform (ST) has such advantages as undergoing inverse transformation without loss and being free from cross-term interference when processing signals. Therefore, this paper has put forward Modified Stockwell Transform (MST), which enables the window width to change proportionally with the varying frequency. Moreover, the change caused by parameter adjustment is much more purposeful. It ensures that TF analysis windows are more flexible and changeable at different positions in the TF plane. Furthermore, to obtain much more accurate Instantaneous Frequency (IF) estimation of multi-component vibration signals, this paper uses the second-order partial derivative of phase function to modify it. Eventually, in order to achieve a concentrated Time-Frequency Representations (TFR) of wind turbines vibration signals, this paper proposes Second-order Synchrosqueezing Modified S Transform, namely SSMST2, through making use of the theoretical framework of Synchrosqueezing Transform (SST) and second-order IF estimations to extend MST. By reassigning the TF transform coefficients of MST, the proposed SSMST2 can better show the time-varying features. To verify this method is feasible, this paper will apply it to fault diagnosis of bearing test rig and to vibration signal processing of rolling bearing in 2 MW wind turbine. The result demonstrates that SSMST2 has yielded great outcome in TFR performance of variable speed vibration signals.

Analysis of Reassignment Operators Used in Synchrosqueezing Transforms: With an Application to Instantaneous Frequency Estimation

In this paper, our goal is first to investigate the behavior of reassignment operators used in synchrosqueezing transforms applied to multicomponent signals made of the superposition of amplitude and frequency modulated modes. Indeed, while these operators are associated with instantaneous frequency estimators very accurate on specific types of modes, the quality of the former worsens drastically when the modes depart from the ideal case they are designed for. We show in this paper that this particularly true when the modes interfere in the time-frequency plane or when some noise is present. Based on that analysis, we propose a novel instantaneous frequency estimator that only makes use of some specific points located on the ridges of synchrosqueezing transforms, and compare its performance with state-of-the-art techniques based on the same type of time-frequency representations.

Wideband Interference Suppression for SAR via Instantaneous Frequency Estimation and Regularized Time-Frequency Filtering

In complex electromagnetic environments, wideband interference (WBI) may severely affect the imaging quality of synthetic aperture radar (SAR). Because it occupies a large bandwidth, which overlaps with target echoes, the WBI is difficult to mitigate. The existing WBI suppression methods based on filtering or transform-domain analysis usually suffer from a model mismatch. To tackle this problem, a method combining instantaneous frequency (IF) estimation and regularized time-frequency filtering (RTFF) is proposed for WBI suppression and individual components extraction. First, the WBI-corrupted SAR echo is characterized in the time-frequency domain by short-time Fourier transform (STFT) with adaptive window width, determined by the proposed window width optimization method. Then, the IFs of the WBI components are estimated by ridge path detection and regrouping. Finally, the WBI is extracted by RTFF. Experimental results of measured SAR data corrupted by simulated and real WBIs have demonstrated the effectiveness and practicability of the proposed method.

Instantaneous Fundamental Frequency and Model Order Estimation of Non-Stationary Harmonic Signals

Instantaneous Fundamental Frequency and Model Order Estimation of Non-Stationary Harmonic Signals

Second-Order Horizontal Multi-Synchrosqueezing Transform for Hydrocarbon Reservoir Identification

Time-frequency (TF) analysis has received considerable attention in seismic spectral anomaly detection for its superiority in significantly revealing the frequency content of seismic signals changing with time variation. In this letter, a novel method termed second-order horizontal multi-synchrosqueezing transform (SHMSST) is proposed for hydrocarbon reservoir identification. First, a Gaussian-modulated linear chirp model is constructed in the frequency domain for describing the signals with slowly or rapidly linear-varying group delay. Then, the second-order local group delay estimation of the model is deduced in the TF domain through an approach similar to the instantaneous frequency estimation in the synchrosqueezing transform (SST). Finally, the rearrangement procedure is executed iteratively to concentrate the blurry TF energy into the estimated group delay, which provides an energy-concentrated TF representation for the signals with distinct nonlinear and nonstationary features. Synthetic signal and field seismic data illustrate the effectiveness of the proposed SHMSST in time localization and direct hydrocarbon detection.

Adaptive Photonic Microwave Instantaneous Frequency Estimation Using Machine Learning

Instantaneous microwave frequency estimation enables numerous essential applications in the commercial, defense, and civilian marketplace. The advancement of applications is hindered by the bottleneck in electronic-based frequency measurement systems including narrow bandwidth, high errors rate, and low dynamic range. Photonics-based frequency estimation approaches not only increase the operation frequency range and provide rapid measurement response, but also benefit from immunity to electromagnetic interference and enhancement in system adaptability. Despite the unique advantages offered by photonics-based frequency estimation approaches, it is challenging to obtain linear mapping between the unknown frequency and the measured optical characteristics due to the nonlinear response in electro-optical devices, which consequently results in degradation in measurement precision and a complex calibration relationship. Therefore, it is critical to mitigate the challenge to achieve dynamic, adaptive, and high-precision estimation of microwave frequency. To this end, this letter presents the design and demonstration of a high-precision photonic based instantaneous frequency estimation system driven by machine learning. A three-layer deep neural network is used to tackle device nonlinearity and system noise, resulting in absolute error of < 50 MHz and root mean square error of 1.1 MHz.

High-Order Synchroextracting Chirplet Transform for Accurate Instantaneous Frequency Estimation and Its Application in Fault Diagnosis of Rotary Machinery

Time-frequency analysis (TFA) technology is an effective tool for extracting time-varying characteristics of a signal, but for strong AM-FM signals, time-frequency representation (TFR) with high time-frequency (TF) resolution is often difficult to generate. The measured vibration signals of mechanical equipment often contain a large number of non-stationary features and are polluted by noise, which hinders the use of TFA technologies. In this paper, we propose a TFA method termed high-order synchroextracting chirplet transform (HSECT) to effectively process strong AM-FM signals. On the basis of Chirplet transform (CT), we first determine the best linear demodulation operator at each moment through the local maximum amplitude of the generated TFR. Subsequently, the amplitude and phase of the signal are subjected to high-order Taylor expansion, and a new frequency estimation operator termed high-order synchroextracting chirplet operator (HSECO) is obtained by the high-order approximation of the time-varying laws of the signal. HSECO can accurately extract the instantaneous frequency of strong AM-FM signal and the construction of HSECT is completed by Dirichlet function at last. HSECT can generate TFR with high energy concentration from which the time-varying law of strong AM-FM signal can be extracted accurately. Meanwhile, the signal reconstruction operation can be completed concisely. In addition, the proposed method is applied to the fault diagnosis of the rotor rub impact and variable speed equipment and compared with other TFA methods. Simulation and experiments can verify the effectiveness and competitiveness of the proposed method.

High-Precision and High-Resolution Synchrosqueezing Transform via Time-Frequency Instantaneous Phases

Synchrosqueezing transform (SST) can effectively improve time-frequency precision and resolution by squeezing time-frequency spectra via instantaneous frequencies, and it has been applied in many diverse disciplines; however, the precision of estimated instantaneous frequencies during SST is usually affected by the time-sample interval of the inputted signal; this usually leads to low-precision or inaccurate SST results and limits its further application. To obtain high-precision and high-resolution SST results with high efficiency, we propose a high-precision and high-resolution SST via time-frequency instantaneous phases (HSST); in HSST, time-frequency instantaneous phases with period-jumps removal are used for high-precision instantaneous frequencies estimation and SST. Two synthetic signal examples show that HSST can minimize the impact of the time-sample interval to achieve high-precision and high-resolution SST results with high efficiency. A real 3D seismic data application demonstrates that HSST has fantastic performance in time-frequency precision and resolution enhancement, and it can be widely used in digital signals processing and interpretation fields.

Reconstruction of gapped missing samples based on instantaneous frequency and instantaneous amplitude estimation

The recovery of missing samples in amplitude-modulation and frequency-modulation (AM-FM) signals is a challenging problem. This study proposes an effective approach to reconstruct missing samples from AM-FM signals through accurate estimation of instantaneous frequencies (IFs) and instantaneous amplitudes (IA) of signal components. Experimental results indicate that the proposed reconstruction method achieves better performance, in the terms of mean absolute error between the reconstructed signal and the original signal, as compared to the state of art methods including time-frequency filtering method and modified orthogonal matching pursuit algorithm.

Adaptive multiple second-order synchrosqueezing wavelet transform and its application in wind turbine gearbox fault diagnosis

Wind turbines usually operate in harsh environments and in working conditions of variable speed, which easily causes their key components such as gearboxes to fail. The gearbox vibration signal of a wind turbine has nonstationary characteristics, and the existing time-frequency (TF) analysis (TFA) methods have some problems such as insufficient concentration of TF energy. In order to obtain a more apparent and more congregated time-frequency representation (TFR), this paper proposes a new TFA method, namely adaptive multiple second-order synchrosqueezing wavelet transform (AMWSST2). Firstly, a short-time window is innovatively introduced on the foundation of classical continuous wavelet transform, and the window width is adaptively optimized by using the center frequency and scale factor. After that, a smoothing process is carried out between different segments to eliminate the discontinuity and thus adaptive wavelet transform is generated. Then, on the basis of the theoretical framework of synchrosqueezing transform and accurate instantaneous frequency estimation by the utilization of second-order local demodulation operator, adaptive second-order synchrosqueezing wavelet transform (AWSST2) is formed. Considering that the quality of actual TFA is greatly disturbed by noise components, through performing multiple synchrosqueezing operations, the congregation of TFR energy is further improved, and finally, the AMWSST2 algorithm studied in this paper is proposed. Since synchrosqueezing operations are performed only in the frequency direction, this method AMWSST2 allows the signal to be perfectly reconstructed. For the verification of its effectiveness, this paper applies it to the processing of the vibration signal of the gearbox of a 750 kW wind turbine.

Effectiveness of energy separation-based instantaneous frequency estimation for cochlear cepstral features for synthetic and voice-converted spoofed speech detection

In this article, we propose Cochlear Filter Cepstral Coefficient-Instantaneous Frequency feature set using Energy Separation Algorithm (CFCCIF-ESA) feature set to detect the speech synthesis (SS) and voice conversion (VC)-based spoofing attacks. The SS- and VC-based spoof generation techniques predominantly uses the magnitude spectrum information, neglecting the phase information. Hence, SS and VC generated speech signal possess the distorted phase in time or frequency-domain. In this work, we exploit this anomaly in phase to efficiently detect the spoofing attack. Here, instantaneous frequency (IF) is utilized to represent the phase information as IF is nothing but the derivative of unwrapped instantaneous (analytic) phase. The experiments are performed on ASVSpoof-2015 challenge dataset, which is specifically designed to do Spoof Speech Detection (SSD) task for SS and VC. In ASVSpoof-2015 challenge during INTERSPEECH 2015, SSD system designed using Cochlear Filter Cepstral Coefficient-Instantaneous Frequency (CFCCIF) feature set was the relatively best performing system. The CFCCIF feature set composed of the information obtained from the magnitude envelope derived using cochlear filterbank and instantaneous frequency (IF) which is derived from Hilbert transform-based approach. However, Hilbert transform-based estimation requires a speech segment of 10–30 ms and thus, it limits time resolution of IF estimation and hence, defeats the key objective of IF estimation to be able to fit the frequency of a sinusoid (corresponding to a monocomponent signal) locally and almost instantaneously. Energy Separation Algorithm (ESA) is known to accurately estimate the modulation patterns due to their relatively low computational complexity, high time resolution, and instantaneously adapting nature. To that effect, we exploit the ESA instead of Hilbert transform to estimate the IFs of the subband filtered signal using cochlear filterbank. The significant improvement in performance, in particular, relative reduction of 51.21% and 46.87% in EER is observed on development and evaluation subsets, respectively, for CFCCIF-ESA feature set over its CFCCIF counterpart, using Gaussian Mixture Model (GMM)-based classifier. This improvement in the performance indicates that the IFs estimated using ESA-based approach are able to efficiently capture the artefacts produced in the instantaneous phase by the SS- and VC-based spoof signals. Furthermore, experiments are also performed with convolutional neural network (CNN) classifier, which further enhances the performance.