Robust Wigner distribution with application to the instantaneous frequency estimation

Polynomial Wigner Ville distribution (PWVD) is considered as a tool for recognition of signals with an unknown frequency modulation law. This time-frequency distribution (TF) with multi-linear kernel is an optimal form for signals with a polynomial frequency modulation law. Instantaneous frequency (IF) is the important property of non-stationary signals, i.e. signals whose spectral contents vary with time. Then the recognition of unknown signals can be formulated as the problem of estimation of its instantaneous frequency. The studies of estimation of IF for noiseless signals can be found in literature. The noise embedded in a signal has large influence on time-frequency distribution and can change the true value of the IF. Time-frequency distribution strongly depends on the kind of noise. In this paper signals embedded in additive gaussian and impulse noise have been considered. Estimation of the instantaneous fiequency in impulsive noise background is different than in gaussian noise and requires robust forms of time-frequency distributions. In this paper the application of the robust form of the Wigner-Ville distribution (WVD_r) and the robust form of the polynomial Wigner-Ville distribution (PWVD_r) for IF estimation is presented. Maxima of these distributions are assumed as the estimates of true values of IF. The theoretical background of WVD_r and PWVD_r distributions is introduced and several numerical experiments of IF estimation are presented.

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.

Digital signals are often contaminated by noise during signal acquisition and transmission for sewage sensing signal treatment such as aeration volume, oxygen content and water transparency etc. Sometimes, noise is a mixed one of gaussian noise and impulse noise. Unfortunately, existing denoising algorithms are often designed for removing single gaussian noise or impulse noise. In this paper, an efficient algorithm for mixed noise removal in signal is proposed, including space impulse noise removal and wavelet Gaussian noise removal. An impulse noise detection algorithm based on median filter is given to filter impulse signal, also a lifting wavelet was constructed by lifting original wavelet. The threshold based on lifting wavelet transform for signal was applied to denoising gaussian noise. Simulations are conducted on the presented algorithm, and the simulation result shows that this algorithm can remove mixed Gaussian and impulse noise in signal efficiently.

The central goal of this dissertation is to design and model a smoothing filter based on the random single and mixed noise distribution that would attenuate the effect of noise while preserving edge details. Only then could robust, integrated and resilient edge detection methods be deployed to overcome the ubiquitous presence of random noise in images. Random noise effects are modeled as those that could emanate from impulse noise, Gaussian noise and speckle noise. In the first step, evaluation of methods is performed based on an exhaustive review on the different types of denoising methods which focus on impulse noise, Gaussian noise and their related denoising filters. These include spatial filters (linear, non-linear and a combination of them), transform domain filters, neural network-based filters, numerical-based filters, fuzzy based filters, morphological filters, statistical filters, and supervised learning-based filters. In the second step, switching adaptive median and fixed weighted mean filter (SAMFWMF) which is a combination of linear and non-linear filters, is introduced in order to detect and remove impulse noise. Then, a robust edge detection method is applied which relies on an integrated process including non-maximum suppression, maximum sequence, thresholding and morphological operations. The results are obtained on MRI and natural images. In the third step, a combination of transform domain-based filter which is a combination of dual tree – complex wavelet transform (DT-CWT) and total variation, is introduced in order to detect and remove Gaussian noise as well as mixed Gaussian and Speckle noise. Then, a robust edge detection is applied in order to track the true edges. The results are obtained on medical ultrasound and natural images. In the fourth step, a smoothing filter, which is a feed-forward convolutional network (CNN) is introduced to assume a deep architecture, and supported through a specific learning algorithm, l2 loss function minimization, a regularization method, and batch normalization all integrated in order to detect and remove impulse noise as well as mixed impulse and Gaussian noise. Then, a robust edge detection is applied in order to track the true edges. The results are obtained on natural images for both specific and non-specific noise-level.

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.

Chapter 10 - Instantaneous Frequency Estimation and Localization

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.

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.

Under diverse conditions, the vibration signals of complex rotating machinery exhibit non-stationary behavior, multi-component characteristics, closely spaced frequencies, and non-proportionality, posing challenges to conventional time-frequency analysis (TFA) methods. These limitations hinder accurate instantaneous frequency (IF) estimation and time-frequency representation (TFR) construction, directly impacting machinery fault diagnosis. As such, we propose the Local Entropy Selection Scaling-Extracting Chirplet Transform (LESSECT), which optimizes entropy-based chirp rate (CR) selection to match non-proportional fundamental frequencies. By adaptively selecting multiple CRs at the same time center, LESSECT enhances TFR resolution and energy concentration, leading accurate IF identification. Experimental validation on bat echolocation, bearing fault, and planetary gearbox signals shows its superior performance in resolving non-proportional, closely spaced IFs. This significantly improves state estimation and enhances machinery diagnostics, contributing to greater system reliability.

The synchrosqueezing transform is a time-frequency (TF) analysis tool to process non-stationary signals. Unfortunately, it does not produce accurate TF results when faced with LFM signals. In this paper, we propose synchrosqueezing-Hough transform (SS-HT) to address this problem. First, we introduce Hough transform to obtain more accurate instantaneous frequency (IF) estimation. Then, in order to improve the TF concentration, a new TF rearrangement operator is constructed. Furthermore, we demonstrated that SS-HT has the ability to reconstruct the signal. The experimental results prove that SSHT can not only obtain high TF concentration, but also improve the accuracy of IF estimation and parameters estimation of LFM signals.

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.

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

In this paper, we propose a frame-based iterative algorithm to restore images which are corrupted by mixed Gaussian and impulse noise, under the assumption that the image region corrupted by impulse noise is unknown. The removal of mixed Gaussian and impulse noise by our proposed algorithm is split into two subproblems which are solved alternatively and iteratively. With an initial guessed region of location for impulse noise, the first subproblem is to inpaint a corrupted image by solving a frame-based convex minimization scheme using the balanced approach, where sparse and redundant directional representations play a key role. Motivated by our recent work on frame-based image denoising and image inpainting, we shall employ the tight frame generated from the directional tensor product complex tight framelets in our balanced approach to remove the mixed Gaussian and impulse noise. Such tensor product complex tight framelets provide sparse directional representations for natural images and can capture the cartoon and texture parts of images very well. The second subproblem is to estimate the image region of locations where the pixels are corrupted by impulse noise. We solve the second subproblem using an $$l_0$$l0-minimization scheme. We consider both salt-and-pepper impulse noise and random-valued impulse noise. Numerical experiments show that our proposed algorithm compares favorably or often outperforms three well-known recent image-restoration methods employed for removing the mixed Gaussian and impulse noise.

In this paper, we improve the trilateral filter for removing the mix of Gaussian and impulse noise. The new algorithm incorporates Rank-Ordered Absolute Differences (ROAD) Statistic for detecting outliers in gradient domain and intensity domain of image with impulse noise. A switching mechanism is adopted in gradient bilateral filter and intensity bilateral filter for smoothing the gradients and intensities of impulse noise samples and impulse noise-free samples with different parameters. By introduce the impulse detector in both gradient domain and intensity domain, the proposed algorithm has demonstrated superior performance in suppressing noise, which include Gaussian, impulse, and mixed noise. Compared to most other nonlinear filters, the proposed algorithm consistently yields good results in a sharply-bounded, gradient piecewise-linear approximation which provides stronger noise reduction and better edge-limited smoothing behavior.

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.