AnOverview of the Adaptive Robust DFT
---
paper_title: Robust M-periodogram

paper_content:

A maximum likelihood-type M-periodogram is developed for observations contaminated by impulse random errors having unknown heavy-tailed error distributions. The periodogram is defined as being a squared amplitude of a harmonic signal fitting the observations with, a nonquadratic residual loss function given by the Huber's (1981) minimax robust statistics.
---
paper_title: Fundamentals of nonlinear digital filtering

paper_content:

Nonlinear Signal Processing Signal Processing Model Signal and Noise Models Fundamental Problems in Noise Removal Algorithms Statistical Preliminaries Random Variables and Distributions Signal and Noise Models Estimation Some Useful Distributions 1001 Solutions Trimmed Mean Filters Other Trimmed Mean Filters L-Filters C-Filters (Ll-Filters) Weighted Median Filters Ranked-Order and Weighted Order Statistic Filters Multistage Median Filters Median Hybrid Filters Edge-Enhancing Selective Filters Rank Selection Filters M-Filters R-Filters Weighted Majority with Minimum Range Filters Nonlinear Mean Filters Stack Filters Generalizations of Stack Filters Morphological Filters Soft Morphological Filters Polynomial Filters Data-Dependent Filters Decision-Based Filters Iterative, Cascaded, and Recursive Filters Some Numerical Measures of Nonlinear Filters Discussion Statistical Analysis and Optimization of Nonlinear Filters Methods Based on Order Statistics Stack Filters Multistage and Hybrid Filters Discussion Exercises Bibliography Index
---
paper_title: Optimum Reception in an Impulsive Interference Environment--Part I: Coherent Detection

paper_content:

Because communications systems are seldom significantly interfered with by classical white Gaussian noise, it is necessary to consider other, appropriate (and tractable) interference models, if realistic estimates of system performance are to be obtained for the general spectral-use environment. For this purpose, Middleton's recently developed canonical statistical-physical model of "impulsive" interference is applied to real-world communication channels. The principal features of this model are first summarized, including the statistical relations required for the solution of signal detection problems. [Excellent agreement of these model statistics with correspondingly measured statistics is also noted.] The model for narrow-band impulsive interference (Class A noise, a subset of the overall model) is next specifically applied to an important class of coherent signal detection problems. Algorithms for error probabilities in optimum detection are then obtained, along with performance bounds, for the same error probabilities. Since it is known that in order to gain significant improvement over current receivers, the number of (essentially) independent samples of the received interference waveform must be enlarged (i.e., large "processing gains"), the performance results here are given parametrically in the number of samples, or equivalently, in the time-bandwidth product. Performance of current suboptimum receivers is then obtained and compared to the optimum performance. It is shown that very substantial savings in signal power and/or spectrum space can usually be achieved by using the indicated optimal algorithms. Since physical realization of the completely optimum detection algorithms cannot, in general, be economically realized, the somewhat more conservative, corresponding locally optimum Bayes detection (LOBD) receivers are derived. In general, these LOBD structures require adaptive, highly non-linear filters, preceding the conventional correlation detector elements characteristic of optimum receivers for Gaussian interference. Performance for these non-linear, optimum threshold systems is then determined, specifically in Part I for coherent reception.
---
paper_title: Signal processing with alpha-stable distributions and applications

paper_content:

The Stable Distribution Symmetric Stable Random Processes Covariation and Conditional Expectation Parameter Estimates for Symmetric Stable Distributions Estimation of Covariations Parametric Models of Stable Processes Linear Theory of Stable Processes Symmetric Stable Models for Impulsive Noise Signal Detection in Stable Noise Current and Future Trends in Signal Processing with Alpha-Stable Distributions.
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paper_title: Robust processing of SAR hologram data to mitigate impulse noise impairments

paper_content:

The standard linear algorithm for synthetic aperture radar (SAR) image synthesis consists of fusing the real and imaginary components of the hologram data, and applying a matched filter to the radar returns from each point to generate the SAR image. This algorithm becomes inefficient when the SAR hologram data are corrupted by impulse noise due to data transmission and coding/decoding errors. The degradation of synthesized SAR images in the presence of impulse noise in the hologram data is considered and new nonlinear algorithms based on robust estimation are proposed. Experimental results for measured SAR data are presented. It is shown that the proposed algorithms efficiently reject impulse noise and substantially improve SAR image quality.
---
paper_title: Estimation of single-tone signal frequency by using the L-DFT

paper_content:

Frequency estimation of complex sinusoidal signal parameters for mixed Gaussian and impulse noise environment is considered. We assume that the sinusoid has constant amplitude. The first stage in the proposed algorithm is calculation of the L-DFT forms for various parameters. Then, an optimal value of the L-DFT parameter is estimated as a value minimizing the L-DFT energy. Position of the L-DFT maximum calculated for the optimal parameter is used as a coarse frequency estimate. Fine estimation is performed by a recently proposed iterative procedure. Numerical analysis confirms accuracy of the proposed technique.
---
paper_title: An algorithm for the Wigner distribution based instantaneous frequency estimation in a high noise environment

paper_content:

Estimation of the instantaneous frequency (IF) in a high noise environment, by using the Wigner distribution (WD), is considered. In this case the error is of impulse nature. An algorithm for the IF estimation, which combines the nonparametric method based on the WD maxima with the minimization of the IF variations between consecutive points, is proposed. The off-line and on-line realizations are presented. The on-line realization is an instance of the (generalized) Viterbi algorithm. Application of this algorithm on the monocomponent and multicomponent frequency modulated signals is demonstrated. For multicomponent signals, the algorithm is applied on other (reduced interference) distributions. Numerical examples, including statistical study of the algorithm performance, are given.
---
paper_title: Filtering of frequency modulated signals embedded in α-stable noise using robust DFT forms

paper_content:

A task of frequency modulated (FM) signal filtering by means of robust DFT is considered. It is shown that in the case of non-Gaussian noise with symmetric heavy-tailed pdf a standard DFT cannot efficiently suppress outliers that contaminate the signal. Three robust DFT forms are described and the procedure adaptive to tail behavior based on the myriad estimate for deriving real and imaginary spectral components is proposed. It is demonstrated that the use of the described and proposed approaches leads to considerable improvement of signal estimate in comparison with techniques based on standard DFT. Such improvement (MSE reduction) is provided for a wide range of parameters of noise with symmetric α-stable distribution.
---
paper_title: Myriad filter based form of the DFT

paper_content:

Estimation of the DFT coefficients for signals embedded in an α-stable noise is considered. The DFT form based on the myriad filter is proposed. The optimal value of the myriad filter parameter is determined as a function of the α-stable noise parameters. The proposed form of the DFT is compared with the other robust forms of the DFT in various noisy environments.
---
paper_title: Median filter based realizations of the robust time-frequency distributions

paper_content:

Recently, some new efficient tools for spectral analysis of signals corrupted by impulse noise have been introduced. They are based on the minimax robust M-estimation. The proposed robust distributions are calculated through an iterative procedure. Here we propose the robust short-time Fourier transform (STFT) and the robust Wigner distribution (WD), based on the simple median filter. Their efficiency in time-frequency analysis is demonstrated in examples.
---
paper_title: Robust DFT-based filtering of pulse-like FM signals corrupted by impulsive noise

paper_content:

Filtering of pulse-like FM signals with varying amplitude corrupted by impulse noise is considered. The robust DFT calculated for overlapped intervals is used for this aim. This technique is proposed in order to decrease amplitude distortion of output signals that can be introduced by the robust DFT calculated within a wide interval including possible zero-output. The proposed algorithm is realized through the following steps. In the first stage, the robust DFT is calculated for the intervals. Filtered signals from the intervals are obtained by applying the standard inverse DFT for the robust DFTs applied to input data. In the second stage, results for different overlapped intervals are combined using the appropriate order statistics. In addition, an algorithm inspired by the intersection of the confidence intervals rule is used for adaptive selection of the interval width in the robust DFT. Algorithm accuracy is tested on numerical examples. Computational complexity analysis is also provided.
---
paper_title: Spectral design of weighted median filters admitting negative weights

paper_content:

A closed-form spectral optimization method for the design of weighted median (WM) filters admitting negative weights is presented. The algorithm is a generalization of Mallows' (1980) theory for nonlinear smoothers that consists of first finding a set of positive weights for a WM filter whose sample selection probabilities are as close as possible to the coefficients of a corresponding finite impulse response (FIR) filter with the desired spectral response. The signs of the weights associated with general WM filtering structures are then coupled with the input samples prior to replication by the weight magnitudes. The spectral characteristics of these WM filters, designed under the proposed method, are shown to be very similar to those of the equivalent linear FIR filters and arbitrary spectral behavior can be achieved. Unlike their FIR filter counterparts, WM filters are robust to impulsive noise, as demonstrated by simulations.
---
paper_title: Robust DFT based on adaptive censored estimate for FM signal processing in non-Gaussian noise environment

paper_content:

A problem of denoising frequency-modulated signals embedded in non-Gaussian noise by means of robust DFT is considered. The requirements for robust estimation implied within robust DFT denoising framework are formulated. It is shown that several standard robust estimators do not meet these requirements well enough. A new adaptive robust estimator is proposed and tested. The robust DFT form based on the proposed estimator is compared to several known robust DFT-methods. Advantages of using the proposed adaptive censored estimator for the considered application are demonstrated via numerical experiments.
---
paper_title: Statistically-Efficient Filtering in Impulsive Environments : WeightedMyriad Filters

paper_content:

Linear filtering theory has been largely motivated by the characteristics of Gaussian signals. In the same manner, the proposed Myriad Filtering methods are motivated by the need for a flexible filter class with high statistical efficiency in non-Gaussian impulsive environments that can appear in practice. Myriad filters have a solid theoretical basis, are inherently more powerful than median filters, and are very general, subsuming traditional linear FIR filters. The foundation of the proposed filtering algorithms lies in the definition of the myriad as a tunable estimator of location derived from the theory of robust statistics. We prove several fundamental properties of this estimator and show its optimality in practical impulsive models such as the-stable and generalized-. We then extend the myriad estimation framework to allow the use of weights. In the same way as linear FIR filters become a powerful generalization of the mean filter, filters based on running myriads reach all of their potential when a weighting scheme is utilized. We derive the "normal" equations for the optimal myriad filter, and introduce a suboptimal methodology for filter tuning and design. The strong potential of myriad filtering and estimation in impulsive environments is illustrated with several examples.
---
paper_title: Adaptive Weighted Myriad Filter Algorithms for Robust Signal Processing in alpha-Stable Noise Environments

paper_content:

Stochastic gradient-based adaptive algorithms are developed for the optimization of weighted myriad filters (WMyFs). WMyFs form a class of nonlinear filters, motivated by the properties of /spl alpha/-stable distributions, that have been proposed for robust non-Gaussian signal processing in impulsive noise environments. The weighted myriad for an N-long data window is described by a set of nonnegative weights {w/sub i/}/sub i=l//sup N/ and the so-called linearity parameter K>0. In the limit, as K/spl rarr//spl infin/, the filter reduces to the familiar weighted mean filter (which is a constrained linear FIR filter). Necessary conditions are obtained for optimality of the filter weights under the mean absolute error criterion. An implicit formulation of the filter output is used to find an expression for the gradient of the cost function. Using instantaneous gradient estimates, an adaptive steepest-descent algorithm is then derived to optimize the weights. This algorithm involves a very simple update term that is computationally comparable to the update in the classical LMS algorithm. The robust performance of this adaptive algorithm is demonstrated through a computer simulation example involving lowpass filtering of a one-dimensional chirp-type signal in impulsive noise.
---
paper_title: REMOVAL OF α-STABLE NOISE IN FREQUENCY MODULATED SIGNALS USING ROBUST DFT FORMS

paper_content:

A task of filtering (denoising) of high-pass signals corrupted by heavy tail (α-stable) noise is considered. For this purpose, several techniques based on the robust DFT forms have been proposed recently. In this paper we study various approaches for design of the robust DFT and compare their performance for frequency-modulated (FM) signals. In particular, we consider the DFT estimates based on the linear combination of order statistics (L-estimate), myriad, Wilcoxon and Hodges-Lehman estimates. Possible strategies for selection of parameters in these robust estimation schemes are studied. Moreover, a new hard-switching adaptive robust estimate is proposed and tested.
---
paper_title: Robust time-frequency distributions based on the robust short time fourier transform

paper_content:

Design of time-frequency distributions (Tfds) that are robust to the impulse noise influence is considered. The robustTfds based on the robust short-time Fourier transform (Stft) are proposed. An efficient procedure to evaluate the robustStft is given. RobustTfds based on the robustStft have better energy concentration around the signal instantaneous frequency (If) than the robustStft itself. Also, theseTfds are more resistant to higher impulse noise than the robustTfds obtained using the local autocorrelation function (Laf) based minimization problem.
---
paper_title: Robust DFT with high breakdown point for complex-valued impulse noise environment

paper_content:

Modification of the robust discrete Fourier transform (DFT) is proposed in order to achieve a high breakdown point for signals corrupted by complex-valued impulse noise with independent real and imaginary parts. Obtained results are compared with existing robust DFT forms. In addition, an adaptive procedure for selection of the modified robust DFT form is developed.
---
paper_title: Robust M-periodogram

paper_content:

A maximum likelihood-type M-periodogram is developed for observations contaminated by impulse random errors having unknown heavy-tailed error distributions. The periodogram is defined as being a squared amplitude of a harmonic signal fitting the observations with, a nonquadratic residual loss function given by the Huber's (1981) minimax robust statistics.
---
paper_title: Optimum Reception in an Impulsive Interference Environment--Part I: Coherent Detection

paper_content:

Because communications systems are seldom significantly interfered with by classical white Gaussian noise, it is necessary to consider other, appropriate (and tractable) interference models, if realistic estimates of system performance are to be obtained for the general spectral-use environment. For this purpose, Middleton's recently developed canonical statistical-physical model of "impulsive" interference is applied to real-world communication channels. The principal features of this model are first summarized, including the statistical relations required for the solution of signal detection problems. [Excellent agreement of these model statistics with correspondingly measured statistics is also noted.] The model for narrow-band impulsive interference (Class A noise, a subset of the overall model) is next specifically applied to an important class of coherent signal detection problems. Algorithms for error probabilities in optimum detection are then obtained, along with performance bounds, for the same error probabilities. Since it is known that in order to gain significant improvement over current receivers, the number of (essentially) independent samples of the received interference waveform must be enlarged (i.e., large "processing gains"), the performance results here are given parametrically in the number of samples, or equivalently, in the time-bandwidth product. Performance of current suboptimum receivers is then obtained and compared to the optimum performance. It is shown that very substantial savings in signal power and/or spectrum space can usually be achieved by using the indicated optimal algorithms. Since physical realization of the completely optimum detection algorithms cannot, in general, be economically realized, the somewhat more conservative, corresponding locally optimum Bayes detection (LOBD) receivers are derived. In general, these LOBD structures require adaptive, highly non-linear filters, preceding the conventional correlation detector elements characteristic of optimum receivers for Gaussian interference. Performance for these non-linear, optimum threshold systems is then determined, specifically in Part I for coherent reception.
---
paper_title: Application of Bispectrum Estimation for Time-Frequency Analysis of Ground Surveillance Doppler Radar Echo Signals

paper_content:

A microwave coherent homodyne and polarimetric ground surveillance Doppler radar is employed for collecting the radar returns from moving objects. Nonstationary nonlinearly frequency-modulated and multicomponent backscattered signals are analyzed and described as a sum of Doppler frequency-shifted polynomial chirp-like components. Instantaneous frequencies corresponding to the radiation backscattered by the different parts of a moving spatially distributed object are extracted from the time-varying bimagnitude estimates of transient sample sequences separated from the total received signal by a sliding window and projected into the time-frequency (TF) domain. Experimental investigations demonstrate a clean recovery of evolutionary phase-coupled harmonics for such targets as a swinging metallic sphere or a walking human. The computed TF distributions can be used in radar automatic target recognition systems to retrieve new data for the classification and recognition of ground moving objects.
---
paper_title: Blind evaluation of noise variance in images using myriad operation

paper_content:

A typical problem in image processing is to evaluate the noise characteristics of an image at hand. It is often desirable to perform this operation automatically with appropriate accuracy, i.e. without errors due to user's subjective views. The proposed method is based on myriad operation at the final stage of the sequence of the traditional steps performed in blind evaluation of noise variance. Using it the accuracy of blind evaluation of noise variance can be considerably improved if the parameters of myriad operation are adjusted correctly. Recommendations concerning this adjustment are given. The proposed method is verified for both artificial and real images.
---
paper_title: Filtering of frequency modulated signals embedded in α-stable noise using robust DFT forms

paper_content:

A task of frequency modulated (FM) signal filtering by means of robust DFT is considered. It is shown that in the case of non-Gaussian noise with symmetric heavy-tailed pdf a standard DFT cannot efficiently suppress outliers that contaminate the signal. Three robust DFT forms are described and the procedure adaptive to tail behavior based on the myriad estimate for deriving real and imaginary spectral components is proposed. It is demonstrated that the use of the described and proposed approaches leads to considerable improvement of signal estimate in comparison with techniques based on standard DFT. Such improvement (MSE reduction) is provided for a wide range of parameters of noise with symmetric α-stable distribution.
---
paper_title: Myriad filter based form of the DFT

paper_content:

Estimation of the DFT coefficients for signals embedded in an α-stable noise is considered. The DFT form based on the myriad filter is proposed. The optimal value of the myriad filter parameter is determined as a function of the α-stable noise parameters. The proposed form of the DFT is compared with the other robust forms of the DFT in various noisy environments.
---
paper_title: Median filter based realizations of the robust time-frequency distributions

paper_content:

Recently, some new efficient tools for spectral analysis of signals corrupted by impulse noise have been introduced. They are based on the minimax robust M-estimation. The proposed robust distributions are calculated through an iterative procedure. Here we propose the robust short-time Fourier transform (STFT) and the robust Wigner distribution (WD), based on the simple median filter. Their efficiency in time-frequency analysis is demonstrated in examples.
---
paper_title: Robust DFT based on adaptive censored estimate for FM signal processing in non-Gaussian noise environment

paper_content:

A problem of denoising frequency-modulated signals embedded in non-Gaussian noise by means of robust DFT is considered. The requirements for robust estimation implied within robust DFT denoising framework are formulated. It is shown that several standard robust estimators do not meet these requirements well enough. A new adaptive robust estimator is proposed and tested. The robust DFT form based on the proposed estimator is compared to several known robust DFT-methods. Advantages of using the proposed adaptive censored estimator for the considered application are demonstrated via numerical experiments.
---
paper_title: REMOVAL OF α-STABLE NOISE IN FREQUENCY MODULATED SIGNALS USING ROBUST DFT FORMS

paper_content:

A task of filtering (denoising) of high-pass signals corrupted by heavy tail (α-stable) noise is considered. For this purpose, several techniques based on the robust DFT forms have been proposed recently. In this paper we study various approaches for design of the robust DFT and compare their performance for frequency-modulated (FM) signals. In particular, we consider the DFT estimates based on the linear combination of order statistics (L-estimate), myriad, Wilcoxon and Hodges-Lehman estimates. Possible strategies for selection of parameters in these robust estimation schemes are studied. Moreover, a new hard-switching adaptive robust estimate is proposed and tested.
---
paper_title: Robust DFT with high breakdown point for complex-valued impulse noise environment

paper_content:

Modification of the robust discrete Fourier transform (DFT) is proposed in order to achieve a high breakdown point for signals corrupted by complex-valued impulse noise with independent real and imaginary parts. Obtained results are compared with existing robust DFT forms. In addition, an adaptive procedure for selection of the modified robust DFT form is developed.
---
paper_title: Fundamentals of nonlinear digital filtering

paper_content:

Nonlinear Signal Processing Signal Processing Model Signal and Noise Models Fundamental Problems in Noise Removal Algorithms Statistical Preliminaries Random Variables and Distributions Signal and Noise Models Estimation Some Useful Distributions 1001 Solutions Trimmed Mean Filters Other Trimmed Mean Filters L-Filters C-Filters (Ll-Filters) Weighted Median Filters Ranked-Order and Weighted Order Statistic Filters Multistage Median Filters Median Hybrid Filters Edge-Enhancing Selective Filters Rank Selection Filters M-Filters R-Filters Weighted Majority with Minimum Range Filters Nonlinear Mean Filters Stack Filters Generalizations of Stack Filters Morphological Filters Soft Morphological Filters Polynomial Filters Data-Dependent Filters Decision-Based Filters Iterative, Cascaded, and Recursive Filters Some Numerical Measures of Nonlinear Filters Discussion Statistical Analysis and Optimization of Nonlinear Filters Methods Based on Order Statistics Stack Filters Multistage and Hybrid Filters Discussion Exercises Bibliography Index
---
paper_title: Estimates of Location Based on Rank Tests

paper_content:

A serious objection to many of the classical statistical methods based on linear models or normality assumptions is their vulnerability to gross errors. For certain testing problems this difficulty is suc-cessfully overcome by rank tests such as the two Wilcoxon tests or the Kruskal- Wallis H-test. Their power is more robust against gross errors than that of the t- and F-tests, and their efficiency loss is quite small even in the rare case in which the suspicion of the possibility of gross errors is unfounded.
---
paper_title: Blind evaluation of noise variance in images using myriad operation

paper_content:

A typical problem in image processing is to evaluate the noise characteristics of an image at hand. It is often desirable to perform this operation automatically with appropriate accuracy, i.e. without errors due to user's subjective views. The proposed method is based on myriad operation at the final stage of the sequence of the traditional steps performed in blind evaluation of noise variance. Using it the accuracy of blind evaluation of noise variance can be considerably improved if the parameters of myriad operation are adjusted correctly. Recommendations concerning this adjustment are given. The proposed method is verified for both artificial and real images.
---
paper_title: Use of minimal inter-quantile distance estimation in image processing

paper_content:

Nowadays multichannel (multi and hyperspectral) remote sensing (RS) is widely used in different areas. One of the basic factors that can deteriorate original image quality and prevent retrieval of useful information from RS data is noise. Thus, image filtering is a typical stage of multichannel image pre-processing. Among known filters, the most efficient ones commonly require a priori information concerning noise type and its statistical characteristics. This explains a great need in automatic (blind) methods for determination of noise type and its characteristics. Several such methods already exist, but majority of them do not perform appropriately well if analyzed images contain a large percentage of texture regions, details and edges. Besides, many blind methods are multistage where some preliminary and appropriately accurate estimate of noise variance is required for next stages. To get around aforementioned shortcomings, below we propose a new method based on using inter-quantile distance and its minimization for obtaining appropriately accurate estimates of noise variance. It is shown that mathematically this task can be formulated as finding a mode of contaminated asymmetric distribution. And this task can be met for other applications. The efficiency of the proposed method is studied for a wide set of model distribution parameters. Numerical simulation results that confirm applicability of the proposed approach are presented. They also allow evaluating the designed method accuracy. Recommendations on method parameter selection are given.
---
paper_title: Performance of FH SS radio networks with interference modeled as a mixture of Gaussian and alpha-stable noise

paper_content:

This paper considers the performance of FH SS radio networks in a Poisson field of interfering terminals using the same modulation and power. Assuming 1/r/sup m/ attenuation of signal strength over distance, the interference in the network is modeled as a mixture of Gaussian and circularly symmetric /spl alpha/-stable noise. The problem is relevant to mobile communication systems where the mobility of users requires random modeling of transmitter positions in the network. We derive formulas for the probability distributions of the interference in FH SS radio networks. We generalize the results of Sousa (see IEEE Trans. I.T., vol.38, no.6, p.1743-1754, 1992) where attention was limited to Cauchy random variables, a special subclass of stable distributions, and where the effect of a background (Gaussian) noise was neglected. Based on the formulas derived, we calculate the probability of symbol error for radio links in environments varying from urban settings to office buildings. The results obtained allow the prediction of the performance of wireless systems under a wide range of conditions.
---
paper_title: Optimality of the myriad filter in practical impulsive-noise environments

paper_content:

Linear filtering theory has been largely motivated by the characteristics of Gaussian signals. In the same manner, the proposed myriad filtering methods are motivated by the need for a flexible filter class with high statistical efficiency in non-Gaussian impulsive environments that can appear in practice. We introduce several important properties of the myriad filter and prove its optimality in the family of /spl alpha/-stable distributions.
---
paper_title: Bootstrap based adaptation of sample myriad to characteristics of SαS distribution data

paper_content:

In many practical applications, noise is non-Gaussian. Heavy-tailed symmetric α-stable (SαS) distributions have been shown to describe well many natural phenomena and interference in radio engineering, acoustics, communications, etc. For processing signals corrupted by such noise, robust estimators and the corresponding techniques are widely applied. The methods based on a sample myriad (SM) are considered quasi-optimal for removal of noise with SαS distributions. SM estimation implies setting a tunable parameter k, desirably in an adaptive manner. However, only quite approximate recommendations concerning the selection of k for a limited size samples exist. In this paper, we study some important properties of the SM estimator with application to SαS distributed data. Furthermore, we propose a novel bootstrap based approach to adaptation of the tunable parameter k. By simulations, we consider its accuracy and demonstrate the effectiveness of this approach for a wide range of SαS distribution characteristics. Practical recommendations on the selection of parameters for the bootstrap based approach are provided and justified.
---
paper_title: Adaptation of sample myriad tunable parameter to characteristics of SαS distribution

paper_content:

Processes obeying heavy-tailed symmetric alpha-stable (SalphaS) distributions have been shown to describe well many natural phenomena in radio engineering, acoustics, communications, etc. For processing such data, one needs to apply robust estimators and the methods based on a sample myriad are considered quasi-optimal ones. Sample myriad estimation requires setting a tunable parameter k. However, only quite approximate and impractical recommendations concerning its selection exist. Thus, in this paper we consider some approaches to deal with setting k in practical applications. It is shown that they are to be adaptive to data sample scale and tail heaviness.
---
paper_title: Robust DFT based on adaptive censored estimate for FM signal processing in non-Gaussian noise environment

paper_content:

A problem of denoising frequency-modulated signals embedded in non-Gaussian noise by means of robust DFT is considered. The requirements for robust estimation implied within robust DFT denoising framework are formulated. It is shown that several standard robust estimators do not meet these requirements well enough. A new adaptive robust estimator is proposed and tested. The robust DFT form based on the proposed estimator is compared to several known robust DFT-methods. Advantages of using the proposed adaptive censored estimator for the considered application are demonstrated via numerical experiments.
---
paper_title: REMOVAL OF α-STABLE NOISE IN FREQUENCY MODULATED SIGNALS USING ROBUST DFT FORMS

paper_content:

A task of filtering (denoising) of high-pass signals corrupted by heavy tail (α-stable) noise is considered. For this purpose, several techniques based on the robust DFT forms have been proposed recently. In this paper we study various approaches for design of the robust DFT and compare their performance for frequency-modulated (FM) signals. In particular, we consider the DFT estimates based on the linear combination of order statistics (L-estimate), myriad, Wilcoxon and Hodges-Lehman estimates. Possible strategies for selection of parameters in these robust estimation schemes are studied. Moreover, a new hard-switching adaptive robust estimate is proposed and tested.
---
paper_title: Estimation of single-tone signal frequency by using the L-DFT

paper_content:

Frequency estimation of complex sinusoidal signal parameters for mixed Gaussian and impulse noise environment is considered. We assume that the sinusoid has constant amplitude. The first stage in the proposed algorithm is calculation of the L-DFT forms for various parameters. Then, an optimal value of the L-DFT parameter is estimated as a value minimizing the L-DFT energy. Position of the L-DFT maximum calculated for the optimal parameter is used as a coarse frequency estimate. Fine estimation is performed by a recently proposed iterative procedure. Numerical analysis confirms accuracy of the proposed technique.
---
paper_title: An algorithm for the Wigner distribution based instantaneous frequency estimation in a high noise environment

paper_content:

Estimation of the instantaneous frequency (IF) in a high noise environment, by using the Wigner distribution (WD), is considered. In this case the error is of impulse nature. An algorithm for the IF estimation, which combines the nonparametric method based on the WD maxima with the minimization of the IF variations between consecutive points, is proposed. The off-line and on-line realizations are presented. The on-line realization is an instance of the (generalized) Viterbi algorithm. Application of this algorithm on the monocomponent and multicomponent frequency modulated signals is demonstrated. For multicomponent signals, the algorithm is applied on other (reduced interference) distributions. Numerical examples, including statistical study of the algorithm performance, are given.
---
paper_title: Robust DFT-based filtering of pulse-like FM signals corrupted by impulsive noise

paper_content:

Filtering of pulse-like FM signals with varying amplitude corrupted by impulse noise is considered. The robust DFT calculated for overlapped intervals is used for this aim. This technique is proposed in order to decrease amplitude distortion of output signals that can be introduced by the robust DFT calculated within a wide interval including possible zero-output. The proposed algorithm is realized through the following steps. In the first stage, the robust DFT is calculated for the intervals. Filtered signals from the intervals are obtained by applying the standard inverse DFT for the robust DFTs applied to input data. In the second stage, results for different overlapped intervals are combined using the appropriate order statistics. In addition, an algorithm inspired by the intersection of the confidence intervals rule is used for adaptive selection of the interval width in the robust DFT. Algorithm accuracy is tested on numerical examples. Computational complexity analysis is also provided.
---
paper_title: Accuracy improvement of the wigner distribution estimate in non-Gaussian noise environment by means of clipping technique application

paper_content:

Wigner distribution (WD) is known to be one of the most convenient time-frequency representations for FM signal analysis in Gaussian noise environment. However, in case of impulsive interferences, WD occurs to be non -robust to noise. For improving robustness of the WD, it is proposed to use filtering approach based on the clipping technique. It can be applied at one of the following two stages: for filtering input FM signal or for local autocorrelation function processing. Based on the fact that there are two robust estimators of FM signal amplitude, four new methods for obtaining WD are proposed. The comparative analysis has been performed for two test signals - harmonic and linear FM ones. It is shown that the proposed methods allow considerable improving the accuracy of WD both for impulse noise environments and Gaus sian noise.
---

Title: An Overview of the Adaptive Robust DFT

Section 1: Introduction
Description 1: Provide a general introduction to the topic, discussing the need for robust methods in environments with non-Gaussian noise and outlining the goals and structure of the paper.

Section 2: RDFT Framework and Requirements to Robust Estimators
Description 2: Describe the RDFT framework including the steps for RDFT-based signal processing and the requirements for robust estimators.

Section 3: Adaptive Robust Estimators and their Properties
Description 3: Explain the different types of adaptive robust estimators, their properties, and their suitability for various noise environments.

Section 4: Use of the Proposed Estimators in RDFT-Based Signal Processing
Description 4: Discuss the practical use and performance of the proposed adaptive robust estimators within the RDFT framework, focusing on their efficiency in signal processing tasks.

Section 5: Conclusions, Other Related Work, and Perspectives
Description 5: Summarize the findings, compare with related work, and discuss future perspectives and potential applications of RDFT-based methods.