An adaptive Generalized S-transform for instantaneous frequency estimation

Chapter 10 - Instantaneous Frequency Estimation and Localization

The use of linear time-frequency (TF) representa-tions as instantaneous frequency (IF) estimators arises in various fundamental disciplines, mainly thanks to their simplicity and immunity against interfering cross-terms. In this paper, we derive the optimal window width for IF estimation of noisy multicomponent signals based on a family of linear TF transforms that use Gaussian windowing functions and the Fourier oscillatory kernel. Closed-form formulas concerning the estimation bias and the variance are presented, which, thanks to their generality, describe the statistical performance of IF estimators based on transforms with fixed, time-adaptive, frequency-adaptive, or time-frequency adaptive windows. The optimal window is dependent on the unknown first derivative of the IF; therefore, we employ a low-complexity procedure to infer the derivative and optimize the width accordingly. Two adaptive and fully automated TF representations (TFRs) are developed; the first employs a time-adaptive window that minimizes the sum of the mean-squared errors (MSEs) of the IF estimates at each time instant, while in the second TFR, the window is adaptive over time and frequency, minimizing the estimation MSE at each location in the TF domain. Examples using synthetic and real-world signals demonstrate that the proposed algorithms may outperform many popular state-of-the-art techniques, including those that are signal-adaptive, in terms of IF estimation.

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.

The Wigner distribution (WD) produces highly concentrated time-frequency (TF) representation of nonstationary signals. It may be used as an efficient signal analysis tool, including the cases of frequency modulated signals corrupted with the Gaussian noise. In some applications, a significant amount of impulse noise is present. Then, the WD fails to produce satisfactory results. The robust periodogram has been introduced for spectral estimation of this kind of noisy signals. It can produce good concentration for pure harmonic signals. However, it is not so efficient in the cases of signals with rapidly varying frequency. This is the motivation for introducing the robust WD. It is a reliable TF representation tool for wide class of nonstationary signals corrupted with impulse noise. This distribution produces good accuracy of the instantaneous frequency (IF) estimation. Using the Huber (1981) loss function, a generalization of the WD is presented. It includes both the standard and the robust WD as special cases. This distribution can be used for TF analysis of signals corrupted with a mixture of impulse and Gaussian noise. The presented theory is illustrated on examples, including applications on the IF estimation and time-varying filtering of signals corrupted with a mixture of the Gaussian and impulse noise. The case study analysis of the IF estimators' accuracy, based on the standard and the robust WD forms, is performed. In order to improve the IF estimation, a median filter is applied on the obtained IF estimate.

Instantaneous frequency (IF) estimation of multi-component signals with closely spaced and intersecting signal components of varying amplitudes is a challenging task. This paper presents a novel iterative time–frequency (TF) filtering approach to address this problem. The proposed algorithm first adopts a high-resolution time–frequency distribution to resolve close components in the TF domain. Then, IF of the strongest signal component is estimated by a new peak detection and tracking algorithm that takes into account both the amplitude and the direction of peaks in the TF domain. The estimated IF is used to remove the strongest component from the mixture, and this process is repeated till the IFs of all signal components are estimated. Experimental results show the superiority of the proposed method as compared to other state-of-the-art methods.

The paper presents a method for the estimation of instantaneous frequency (IF) of signals characterized by interfering IF trajectories. It would be a useful methodology for maintenance and troubleshooting of wireless networks sharing the same operating frequency interval or predictive maintenance of vibrating/rotating systems. Stemming from IF trajectories estimated through traditional or enhanced time-frequency representations, the method applies an agile digital signal processing approach in order to suitably merge different sections belonging to the same trajectory. Preliminary results show the efficacy of the method when signals characterized by multiple intersections in the time-frequency domain are experienced.

The estimation of the instantaneous frequency (IF) of a harmonic complex-valued signal with an additive noise using the Wigner distribution is considered. If the IF is a nonlinear function of time, the bias of the estimate depends on the window length. The optimal choice of the window length, based on the asymptotic formulae for the variance and bias, can be used in order to resolve the bias-variance tradeoff. However, the practical value of this solution is not significant because the optimal window length depends on the unknown smoothness of the IF. The goal of this paper is to develop an adaptive IF estimator with a time-varying and data-driven window length, which is able to provide quality close to what could be achieved if the smoothness of the IF were known in advance. The algorithm uses the asymptotic formula for the variance of the estimator only. Its value may be easily obtained in the case of white noise and relatively high sampling rate. Simulation shows good accuracy for the proposed adaptive algorithm.

Estimation of the instantaneous frequency (IF) in a high noise environment, by using the Wigner distribution (WD) and the Viterbi algorithm, is considered. The proposed algorithm combines nonparametric IF estimation based on the WD maxima with minimization of the IF variations between consecutive points. Algorithm realization is performed recursively using the (modified) Viterbi algorithm. Performances are compared with IF estimation based on the WD maxima.

In this work, we derive general formulae for the instantaneous frequency (IF) and instantaneous bandwidth (IB) according to matching pursuit (MP) signal decomposition, irrespective of the kind of dictionary used. We show that via MP decomposition, the IF is now exactly the weighted average IF of the decomposed signal and that it is also always real-valued. In addition, the IB is always positive for all dictionaries with Gaussian envelopes and arbitrary polynomial phase

In recent years, Doppler through-wall radar has been widely used in target localization. However, time–frequency aliasing occurs, which poses a great challenge to the estimation of target parameters. To solve this problem, we propose a novel time–frequency enhancement algorithm. In this paper, we roughly identify the time-frequency features of the interested target component from the time–frequency distribution of the local maximum synchrosqueezing transform and construct a series of discrete demodulation operators that approximate the true properties. Then an optimized window function with adaptive parameters (window width) is designed according to its time–frequency features, which can help effectively compress the echo energy from the selected target component. Finally, an instantaneous frequency (IF) estimator with a local maximum extraction criterion is designed to reduce the noise interference. The experimental results demonstrate that the presented algorithm successfully mitigates time–frequency aliasing, significantly enhancing the accuracy of IF estimation and target localization.

Synchrosqueezing transform (SST) benefits from an instantaneous frequency (IF) estimator in the time-frequency domain, providing an energy-concentrated time-frequency representation to describe the time-varying frequency of seismic signals. To enhance the concentration performance of SST, this paper theoretically proposes a statistical SST (SSST) by constructing a spectrum-weighted IF estimator in the short-time Fourier transform (STFT) domain. In this SSST, a linear chirp signal is introduced to better capture its chirp rate, and then the quadratic STFT spectra are weighted by a window-related weighting function. On this basis, two equations related to the IF and chirp rate are constructed to derive the spectrum-weighted IF estimator. Finally, the STFT coefficients are squeezed to the estimated IF trajectories by a frequency fixed-point iterative algorithm, thereby providing a more concentrated time-frequency representation for multi-component signals than existing advanced methods, while enabling to retrieve each component. Two synthetic examples and one field seismic data on thin interbeds are utilized to demonstrate the effectiveness of the proposed SSST and show its ability to highlight the time-varying frequency features of seismic signals, which is a promising seismic data analysis tool, such as characterizing thin interbed thickness variations.

This paper makes use of local polynomial Wigner-Ville distribution (LPWVD), originally designed for nonparametric instantaneous frequency (IF) estimation of transient signals, to propose a parametric IF estimation for polynomial phase signals (PPSs). Statistical performance such as asymptotic bias and variance of the LPWVD-based parametric IF estimator is derived in closed-form. Based on the analytical results, we extend the statistical efficiency of the Wigner-Ville distribution (WVD) for a second-order PPS only to that of the LP-WVD for an arbitrary order, when the IF is estimated at the middle of sample observations. Simulation results verify the analytical performance and comparisons with the polynomial Wigner-Ville distribution (PWVD) show that the LPWVD-based parametric IF estimator can provide better performance.

Nuclear magnetic resonance spectroscopy (NMRS) signals are modeled as a sum of decaying complex exponentials. The spectral analysis of these signals in order to detect their components and estimate their parameters is crucial to the biochemical analysis of the samples under examination. This paper presents a novel time frequency representation based on a Gabor filterbank/notch filtering instantaneous frequency estimator, in order to enable the detection of weaker and shorter lived exponentials. Building on prior work involving filterank-based instantaneous frequency (IF) estimation, this new approach is an iterative procedure where a Gabor filterbank is first employed in order to obtain a reliable estimate of the IF of the strongest component present. This component is then notch filtered in order to un-mask weaker components and the procedure repreated. The performance of this method was evaluated using an artificial signal and compared to the short time Fourier transform and the original Gabor filterbank approach. The results clearly demonstrate the superiority of the new method in uncovering weaker signals and resolving components that are very close to one another in frequency.

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

In order to solve the problem of calculating the instantaneous frequency by means of local mean decomposition, four methods were introduced such as Hilbert transform, Teager energy operator, direct quadrature and inverse cosine. By analysis of the advantages and disadvantages of various methods, the following conclusions could be made: first, the phenomenon of end swing by Hilbert transform was obvious; second, Teager energy operator method was easy to susceptible to signal form and noise, and the error was relatively large; third, inverse cosine method could be influenced by signal extreme points and brought in the mutations of instantaneous frequency. To solve the mutations of instantaneous frequency by inverse cosine method, an improved algorithm was proposed, which used cubic B-spline interpolation to obtain the instantaneous frequency of signal extreme points. It was shown from simulation and experimental results that the improved inverse cosine method could avoid the phenomenon of mutation and the error of instantaneous frequency was smaller.