Distributed multiple-input and multiple-output synthetic aperture radar (MIMO-SAR) provides a new paradigm for radar imaging, which utilizes multiple distributed sensors to improve imaging performance. However, synchronization errors have a significant impact on imaging quality in these systems. The transmitted and received echo signals exhibit reciprocity, which can be exploited to estimate synchronization errors. By comparing echoes between different sensors, the synchronization errors could be estimated and compensated. This work presents a synchronization error-resistant imaging algorithm for distributed MIMO-SAR systems. First, the synchronization errors are estimated in the range domain by comparing the reciprocal echo signal pairs. Then, the errors are compensated during a fast back-projection (BP) based SAR imaging process. The effectiveness of the proposed algorithm has been verified by experiments.

Speckle noise poses a significant challenge in maintaining the quality of synthetic aperture radar (SAR) images. SAR despeckling techniques have drawn increasing attention. Despite the tremendous advancements of deep learning in SAR image despeckling, these methods still struggle to deal with large-scale SAR images. To address this problem, this paper introduces a novel despeckling approach termed Region Denoising Diffusion Probabilistic Model (R-DDPM) based on diffusion models. R-DDPM enables versatile despeckling of SAR images across various scales, accomplished within a single training session. Moreover, The artifacts in the fused SAR images can be avoided effectively with the utilization of region-guided inverse sampling. Experiments of our proposed R-DDPM on Sentinel-1 data demonstrates superior performance to existing methods.

Explainable artificial intelligence (XAI) holds immense significance in enhancing the deep neural network’s transparency and credibility, particularly in some risky and high-cost scenarios, like synthetic aperture radar (SAR). Shapley is a game-based explanation technique with robust mathematical foundations. However, Shapley assumes that model’s features are independent, rendering Shapley explanation invalid for high dimensional models. This study introduces a manifold-based Shapley method by projecting high-dimensional features into low-dimensional manifold features and subsequently obtaining Fusion-Shap, which aims at (1) addressing the issue of erroneous explanations encountered by traditional Shap; (2) resolving the challenge of interpretability that traditional Shap faces in SAR recognization tasks.

A recent study on the interpretability of real-valued convolutional neural networks (CNNs) [1] has revealed a direct and physically meaningful link with the task of finding features in data through matched filters. However, applying this paradigm to illuminate the interpretability of complex-valued CNNs meets a formidable obstacle: the extension of matched filtering to a general class of noncircular complex-valued data, referred to here as the widely linear matched filter (WLMF), has been only implicit in the literature. To this end, to establish the interpretability of the operation of complex-valued CNNs, we introduce a general WLMF paradigm, provide its solution and undertake analysis of its performance. For rigor, our WLMF solution is derived without imposing any assumption on the probability density of noise. The theoretical advantages of the WLMF over its standard strictly linear counterpart (SLMF) are provided in terms of their output signal-to-noise-ratios (SNRs), with WLMF consistently exhibiting enhanced SNR. Moreover, the lower bound on the SNR gain of WLMF is derived, together with condition to attain this bound. This serves to revisit the convolution-activation-pooling chain in complex-valued CNNs through the lens of matched filtering, which reveals the potential of WLMFs to provide physical interpretability and enhance explainability of general complex-valued CNNs. Simulations demonstrate the agreement between the theoretical and numerical results.

The matched filter is a crucial concept in both signal analysis and convolutional neural networks (CNNs). Previous work has addressed graph matched filtering principles for undirected graphs. This article expands upon the existing literature, by exploring matched filtering principles for signals on directed graphs. In such cases, the adjacency matrix is asymmetric, and commonly results in nonorthogonal eigenvectors. The presented concept is supported by a detailed analysis and numerical examples.

The utility of Graph Neural Networks (GNN) for the paradigm of forecasting short-term stock price movements is investigated. In particular, a finance-specific graph pooling operation, referred to as StockPool, is introduced to efficiently coarsen the stock graph. This is achieved by employing domain knowledge to cluster stocks, depending on some task-specific characteristics (e.g. industries, sub-industries, etc.). Unlike fully end-to-end learnable graph pooling strategies (e.g. differentiable pooling, MinCUT pooling, etc.), such a deterministic pooling operator is considerably more computationally efficient and thus scalable to larger stock graphs. Experimentations on the S&P500 stock index demonstrate that the StockPool operator outperforms existing graph pooling strategies on the prediction of price movements. Finally, different graph pooling methods are utilized to create a set of highly uncorrelated GNN models; these are used to construct a graph ensemble model with an improved performance.

Graph convolutional neural network (GCN) has drawn increasing attention and attained good performance in various computer vision tasks, however, there is a lack of a clear interpretation of GCN’s inner mechanism. For standard convolutional neural networks (CNNs), class activation mapping (CAM) methods are commonly used to visualize the connection between CNN’s decision and image region by generating a heatmap. Nonetheless, such heatmap usually exhibits semantic-chaos when these CAMs are applied to GCN directly. In this paper, we proposed a novel visualization method particularly applicable to GCN, Vertex Semantic Class Activation Mapping (VS-CAM). VS-CAM includes two independent pipelines to produce a set of semantic-probe maps and a semantic-base map, respectively. Semantic-probe maps are used to detect the semantic information from the semantic-base map to aggregate a semantic-aware heatmap. Qualitative results show that VS-CAM can obtain heatmaps where the highlighted regions match the objects much more precisely than CNN-based CAM. The quantitative evaluation further demonstrates the superiority of VS-CAM.

Deep neural networks (DNNs) and especially convolutional neural networks (CNNs) have revolutionized the way we approach the analysis of large quantities of data. However, the largely ad hoc fashion of their development, albeit one reason for their rapid success, has also brought to light the intrinsic limitations of CNNs—in particular, those related to their black box nature. In addition, the ability to “explain” both the way such systems behave and the results they produce is increasingly becoming an imperative in many practical applications. Therefore, it would be particularly useful to establish physically meaningful mechanisms underpinning the operation of CNNs, thus helping to resolve the issue of interpretability of the processing steps and explain their input-output relationship. To this end, we revisit the operation of CNNs from first principles and show that their very backbone—the convolution operation—represents a matched filter which examines the input for the presence of characteristic patterns in data. Our treatment is based on temporal signals, naturally generated by physical sensors, which admit rigorous analysis through systems science. This serves as a vehicle for a unifying account on the overall functionality of CNNs, whereby both the convolution-activation-pooling chain and learning strategies are shown to admit a compact and elegant interpretation under the umbrella of matched filtering. In addition to helping reveal the physical principles underpinning CNNs and providing an intuitive understanding of their operation, the treatment of CNNs from a matched filtering perspective is also shown to offer a platform to support further developments in this area.

Stock market returns are typically analyzed using standard regression models yet they reside on irregular domains, a natural scenario for graph signal processing. This motivates us to consider a market graph as an intuitive way to represent the relationships between financial assets. Traditional methods for estimating asset-return covariance operate under the assumption of statistical time-invariance, and are thus unable to appropriately infer the underlying structure of the market graph. To this end, this work introduces a class of graph spectral estimators which cater for the nonstationarity inherent to asset price movements, as a basis to represent the time-varying interactions between assets through a dynamic spectral market graph. Such an account of the time-varying nature of the asset-return covariance allows us to introduce the notion of dynamic spectral portfolio cuts, whereby the graph is partitioned into time-evolving clusters, thus allowing for robust and online asset allocation. The advantages of the proposed framework over traditional methods are demonstrated through numerical case studies using real-world price data.

The attention mechanism is at the core of modern Natural Language Processing (NLP) models, owing to its ability to focus on the most contextually relevant part of a sequence. However, current attention models rely on "flat-view" matrix methods to process tokens embedded in vector spaces; this results in exceedingly high parameter complexity which is prohibitive for practical applications. To this end, we introduce a novel Tensorized Graph Attention (TGA) mechanism, which leverages on the recent Graph Tensor Network (GTN) framework to efficiently process tensorized token embeddings via attention based graph filters. Such tensorized token embeddings are shown to effectively bypass the Curse of Dimensionality, reducing the parameter complexity of the attention mechanism from an exponential to a linear one in the embedding dimensions. The expressive power of the TGA framework is further enhanced by virtue of domain-aware graph convolution filters. Simulations across benchmark NLP paradigms verify the advantages of the proposed framework over existing attention models, at drastically lower parameter complexity.

Classical portfolio optimization methods typically determine an optimal capital allocation through the implicit, yet critical, assumption of statistical time-invariance. Such models are inadequate for real-world markets as they employ standard time-averaging based estimators which suffer significant information loss if the market observables are non-stationary. To this end, we reformulate the portfolio optimization problem in the spectral domain to cater for the nonstationarity inherent to asset price movements and, in this way, allow for optimal capital allocations to be time-varying. Unlike existing spectral portfolio techniques, the proposed framework employs augmented complex statistics in order to exploit the interactions between the real and imaginary parts of the complex spectral variables, which in turn allows for the modelling of both harmonics and cyclostationarity in the time domain. The advantages of the proposed framework over traditional methods are demonstrated through numerical simulations using real-world price data.

With its aim to reduce the amount of sensed data and to improve the energy efficiency, compressive sensing (CS) is recently witnessing a growing research interest in remote-sensing applications. The Fourier transform domain plays a significant role as a signal-processing tool and the sparsity domain for the CS-reconstruction methods. A generalized expression for the error in the reconstruction of nonuniformly sampled, approximately sparse, or nonsparse, noisy signals in the Fourier domain is presented in this letter. This expression holds for a wide range of practically important nonuniform signal-sampling strategies, covering the uniform and completely random sampling as the special cases. Additive noise and noise-folding effects are included in the analysis. Statistical examples and two real-world examples validate the presented theory.

Within the compressive sensing (CS) paradigm, sparse signals can be reconstructed based on a reduced set of measurements, whereby reliability of the solution is determined by its uniqueness. With its mathematically tractable and feasible calculation, the coherence index is one of very few CS uniqueness metrics with considerable practical importance. We propose an improvement of the coherence-based uniqueness relation for the matching pursuit algorithms. Starting from a simple and intuitive derivation of the standard uniqueness condition, based on the coherence index, we derive a less conservative coherence index-based lower bound for signal sparsity. The results are generalized to the uniqueness condition of the l0-norm minimization for a signal represented in two orthonormal bases.

Investment returns naturally reside on irregular domains, however, standard multivariate portfolio optimization methods are agnostic to data structure. To this end, we investigate ways for domain knowledge to be conveniently incorporated into the analysis, by means of graphs. Next, to relax the assumption of the completeness of graph topology and to equip the graph model with practically relevant physical intuition, we introduce the portfolio cut paradigm. Such a graph-theoretic portfolio partitioning technique is shown to allow the investor to devise robust and tractable asset allocation schemes, by virtue of a rigorous graph framework for considering smaller, computationally feasible, and economically meaningful clusters of assets, based on graph cuts. In turn, this makes it possible to fully utilize the asset returns covariance matrix for constructing the portfolio, even without the requirement for its inversion. The advantages of the proposed framework over traditional methods are demonstrated through numerical simulations based on real-world price data.

In many graph signal processing applications, finding the topology of a graph is part of the overall data processing problem rather than a priori knowledge. Most of the approaches to graph topology learning are based on the assumption of graph Laplacian sparsity, with various additional constraints, followed by variations of the edge weights in the graph domain or the eigenvalues in the graph spectral domain. These domains are high-dimensional, since their dimension is at least equal to the order of the number of vertices. In this paper, we propose a numerically efficient method for estimating of the normalized Laplacian through its eigenvalues estimation and by promoting its sparsity. The minimization problem is solved in quite a low-dimensional space, related to the polynomial order of the underlying system on a graph corresponding to the the observed data. The accuracy of the results is tested on numerical example.

The classical support uncertainty principle states that the signal and its discrete Fourier transform (DFT) cannot be localized simultaneously in an arbitrary small area in the time and the frequency domain. The product of the number of nonzero samples in the time domain and the frequency domain is greater or equal to the total number of signal samples. The support uncertainty principle has been extended to the arbitrary orthogonal pairs of signal basis and the graph signals, stating that the product of supports in the vertex domain and the spectral domain is greater than the reciprocal squared maximum absolute value of the basis functions. This form is then used in compressive sensing and sparse signal processing to define the reconstruction conditions. In this letter, we will revisit the graph signal uncertainty principle using the graph Rihaczek distribution as an analysis tool and derive an improved bound for the support uncertainty principle of graph signals.

Vertex-frequency analysis of graph signals is a challenging topic for research and applications. Counterparts of the short-time Fourier transform, the wavelet transform, and the Rihaczek distribution have recently been introduced to the graph-signal analysis. In this letter, we have extended the energy distributions to a general reduced interference distributions class. It can improve the vertex-frequency representation of a graph signal while preserving the marginal properties. This class is related to the spectrogram of graph signals as well. Efficiency of the proposed representations is illustrated in examples.

Sparse signals can be reconstructed from a reduced set of signal samples using compressive sensing (CS) methods. The discrete cosine transform (DCT) can provide highly concentrated representations of audio signals. This property implies the DCT as a good sparsity domain for the audio signals. In this paper, the DCT is studied within the context of sparse audio signal processing using the CS theory and methods. The DCT coefficients of a sparse signal, calculated with a reduced set of available samples, can be modeled as random variables. It has been shown that the statistical properties of these variables are closely related to the unique reconstruction conditions. The main result of this paper is in an exact formula for the mean-square reconstruction error in the case of approximately sparse and nonsparse noisy signals reconstructed under the sparsity assumption. Based on the presented analysis, a simple and computationally efficient reconstruction algorithm is proposed. The presented theoretical concepts and the efficiency of the reconstruction algorithm are verified numerically, including examples with synthetic and recorded audio signals with unavailable or corrupted samples. Random disturbances and disturbances simulating clicks or inpainting in audio signals are considered. Statistical verification is done on a dataset with experimental signals. Results are compared with some classical and recent methods used in similar signal and disturbance scenarios.

An analysis of errors in the reconstruction of approximately sparse and nonsparse noisy signals in the discrete Fourier transform domain is considered in this letter. Signal reconstruction is performed from a reduced set of data, using compressive sensing methods and the sparsity assumption. Random sampling positions in time are considered. Reconstruction results are compared with those obtained with a subset of uniformly sampled signals. A random subset of uniformly sampled data produces better reconstruction results. Theoretical results are statistically confirmed.

A novel approach for the parameter estimation of the polynomial phase signals (PPS) based on the short‐time Fourier transform (STFT) is proposed. Estimator accuracy is significantly higher compared to the high‐order ambiguity function (HAF), product HAF, and similar the non‐linear transforms based strategies. The proposed approach is more efficient than the maximum likelihood (ML) estimator for the high‐order PPS. One‐dimensional search is performed over a set of window widths in the STFT, whereas in the case of ML estimators, it has been done over the space of phase parameters. The proposed estimator is implemented in several steps: the instantaneous frequency (IF) estimation using the STFT for various window widths; a polynomial regression from the IF estimate producing a coarse signal coefficients estimate; refinement procedure, producing fine coefficients estimates; determination of the optimal window width in the STFT. The proposed technique is extended for the non‐parametric estimation with a 2D search over a set of polynomial orders and window widths. Good estimation results are achieved up to the SNR threshold of about SNR=0 dB for the parametric case and SNR=2 dB for the non‐parametric case.

Recovery of arbitrarily positioned samples that are missing in sparse signals recently attracted significant research interest. Sparse signals with heavily corrupted arbitrary positioned samples could be analysed in the same way as compressive sensed signals by omitting the corrupted samples and considering them as unavailable during the recovery process. The reconstruction of the missing samples is done by using one of the well‐known reconstruction algorithms. In this study, the authors will propose a very simple and efficient algorithm, applied directly to the concentration measures, without reformulating the reconstruction problem within the standard linear programming form. Direct application of the gradient approach to the non‐differentiable forms of measures lead us to introduce a variable step size algorithm. A criterion for changing the adaptive algorithm parameters is presented. The results are illustrated on the examples with sparse signals, including approximately sparse signals and noisy sparse signals.

An analysis of robust estimation theory in the light of sparse signals reconstruction is considered. This approach is motivated by compressive sensing (CS) concept which aims to recover a complete signal from its randomly chosen, small set of samples. In order to recover missing samples, the authors define a new reconstruction algorithm. It is based on the property that the sum of generalised deviations of estimation errors, obtained from robust transform formulations, has different behaviour at signal and non‐signal frequencies. Additionally, this algorithm establishes a connection between the robust estimation theory and CS. The effectiveness of the proposed approach is demonstrated on examples.

The L-estimate transforms and time-frequency representations are presented within the framework of compressive sensing. The goal is to recover signal or local auto-correlation function samples corrupted by impulse noise. The signal is assumed to be sparse in a transform domain or in a joint-variable representation. Unlike the standard L-statistics approach, which suffers from degraded spectral characteristics due to the omission of samples, the compressive sensing in combination with the L-estimate permits signal reconstruction that closely approximates the noise free signal representation.

In this paper, we present the S-method-based approach to real-time motion compensation, image formation and image enhancement of moving targets in ISAR and SAR. This approach performs better than the Fourier transform by drastically improving images of fast, maneuvering targets. These advantages are a result of the S-method's ability to automatically compensate for quadratic and all even higher-order terms in phase. Thus, targets with constant acceleration will undergo full motion compensation and their point-scatterers will each be localized. It should be noted that the source of the quadratic term can come from not only acceleration, but also non-uniform rotational motion and the cosine term in wide-angle imaging. The method is also computationally simple, requiring only slight modifications to the existing Fourier transform-based algorithm.

Instantaneous frequency (IF) is a fundamental concept that can be found in many disciplines such as communications, speech, and music processing. In this letter, analysis of an IF estimator, based on a time-frequency technique known as S-transform, is performed. The performance analysis is carried out in a white Gaussian noise environment, and expressions for the bias and the variance of the estimator are determined. The results show that the bias and the variance are signal dependent. This has been statistically confirmed through numerical simulations of several signal classes.

The instantaneous frequency law (IFL) is a very important item when the physical parameters of the corresponding signal have to be evaluated. Radar, sonar, mechanical diagnostic are just three domains where the signal's non-stationarity imposes the IFL estimation. There are several cases where the IFL is composed by fast variations. Digital phase modulations or signals emitted by electrical switches are typical examples of IFLs having fast transient parts. To deal with such kind of signals, we propose a new method based on the chirping of the phase transitions. Namely, the phase chirping operator (PCO) transforms a fast IFL variation in a chirp component. This chirp contains all the parameters about the initial variation : time, duration, covered bandwidth, etc. Results for some physical data will highlight the benefits of the PCO compared with wavelet transform and Wigner-Ville distribution

Synthetic aperture radar processors are generally made with the stationary targets in mind, hence commonly used technique for the SAR signal analysis is a two-dimensional Fourier transform. Moving targets induce Doppler-shift and Doppler spread in the returned signal, producing blurred or smeared images. Standard techniques for these kinds of the problems are motion compensation and time-frequency analysis application. Both of them are computationally intensive. Here, we will present a numerically simple S-method based approach, already applied in the ISAR imaging. This approach improves readability of the SAR images what will be analytically proved and demonstrated on the simulated SAR setup

The problem of signal detection, followed by a characterization stage, is considered in this paper. The main difficulties arising in the detection stage are caused by noise, which acts in a real environment, and by multiple time-frequency (TF) structures of the signal. In this paper a detection method based on the adaptive grouping of the TF information provided by a Gabor filter bank is proposed. A Viterbi-type algorithm is used as a tool for grouping of TF components. The results obtained for real data prove the capability of the proposed approach for accurately detect and characterize signals with a complex TF behavior.

An analytical form of the instantaneous frequency law (IFL) is very important when the physical parameters have to be evaluated from the time-frequency content of a signal. Generally, the analyzed signals are composed of several time-frequency components characterized by various nonlinear IFL. To deal with such signals, we propose a new method based on the warping operators (WO) and on the complex time distribution (CTD). Using parallel structures composed of several WO, some time-frequency components of the analyzed signal are linearized. This linearization is highlighted by using the CTD which considerably reduces the artefacts due to the complexity of the analyzed signal. This leads to an accurate estimation process, illustrated and justified by numerical and real examples.

Moments of the second order polynomial Fourier transform are considered. Relations between moments for various parameters in the polynomial FT are established. Estimation of multidimensional FM signals parameters based on the moments is proposed and compared with the phase derivative based estimator. Implementation issues of the proposed estimator are discussed.

Local frequency (LF) estimation of multidimensional (md) signals is considered. The md-Wigner distribution (WD) is used as the LF estimator. The LF is estimated based on the positions of the WD maxima. A nonparametric algorithm for the LF estimation is developed. It is based on the intersection of confidence intervals rule. This algorithm produces an adaptive window size in the WD which gives almost minimal mean squared error of the estimate. A simplified version of this algorithm is developed, with the starting estimate being produced with the WD of one-dimensional signals. Theory is illustrated in examples.

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.

The principal axes of the time-frequency representation of a signal are defined as those mutually orthogonal directions in the time-frequency plane for which the width of the signal's fractional power spectrum is minimum or maximum. The time-frequency kernels used in the Cohen class of time-frequency representations are then rotated in the time-frequency plane, in order to align the kernels' preferred axes to the signal's principal axes. It is shown that the resulting time-frequency representations show a better reduction of cross-terms without too severely degrading the autoterms than the corresponding original time-frequency representations.

An efficient and simple procedure for filtering of signals in an impulse noise environment is proposed. It can be used for realization of all filter forms: lowpass, highpass, stopband, and bandpass. Accuracy of the proposed procedure is of the same order of magnitude as in the case of the weighted median/myriad filters admitting negative weights, recently proposed by Arce et al.

Exact expressions for the quadratic distributions' variance of signals corrupted with white stationary, white nonstationary, and colored stationary noise are derived. It has been shown that the signal-dependent part of variance is closely related to the nonnoisy distribution values.

The robust M-periodogram is defined for the analysis of signals with heavy-tailed distribution noise. In the form of a robust spectrogram (RSPEC) it can be used for the analysis of nonstationary signals. A RSPEC-based instantaneous frequency (IF) estimator, with a time-varying window length, is presented. The optimal choice of the window length can resolve the bias-variance tradeoff in the RSPEC-based IF estimation. However, it depends on the unknown nonlinearity of the IF. The algorithm used is able to provide accuracy close to the one that could be achieved if the IF to be estimated were known in advance. Simulations show good accuracy ability of the adaptive algorithm and good robustness property with respect to rare high-magnitude noise values.

Estimation of the instantaneous frequency by using quadratic distributions from the general Cohen class is analyzed. Frequency modulated signals corrupted with a white stationary noise are considered. An expression for the variance is derived. It is shown that the variance is closely related to the non-noisy distribution of a predefined signal.

The asymptotic performance of the local frequency (LF) estimator, based on the Wigner distribution (WD) of multidimensional signals, is considered. The optimal estimation window size, producing minimal mean squared error (MSE), is derived. The results are illustrated and confirmed by numerical examples.

The standard short-time Fourier transform and the Wigner distribution can be obtained as solutions of the minimization problem, with the absolute square error as a loss function. It has been shown that some other loss functions, like for example the absolute error, can produce more robust results in the case of signals corrupted with impulse, heavy-tailed, noise. This paper presents robust time frequency-signal analysis of nonstationary signals, corrupted with heavy-tailed noise. For this purpose the robust Wigner distribution is introduced, as an extension of the robust M-periodogram concept. The theory is illustrated on several examples, including application of the proposed distribution on the instantaneous frequency estimation.

Analysis of time frequency (TF) distributions, as the instantaneous frequency (IF) estimators for low noise, has been previously carried out. In this letter, we extend the analysis to high noise. This noise causes a specific error, which can dominate over all other studied errors. The crucial parameter is the ratio of auto-term (AT) magnitude and distribution standard deviation.

The L-Wigner distributions are defined in order to improve the concentration of signal's time-frequency representation. For a finite distribution order and nonlinear instantaneous frequency (IF) it gives biased IF estimates. In the case of noisy signals the optimal window length and distribution order depend on the noise variance and unknown IF. In this article an adaptive IF estimator with the time-varying and data-driven window length and distribution order is developed. Based on the analysis that has been done here, lower order time-frequency distributions are introduced.

The peak of the polynomial Wigner-Ville distribution is known to be a consistent estimator of the instantaneous frequency for polynomial FM signals. We present an algorithm for the design of an optimal time-varying window length for this estimator when noisy non-linear, not necessarily polynomial, FM signals are considered. The results obtained show that the estimator is accurate and outperforms any fixed window time-frequency distribution based estimator.

A method for the polynomial Wigner-Ville distributions realization, in the case of multicomponent signals, is presented. Using this method, one may theoretically get a sum of the polynomial Wigner-Ville distributions of each component separately. The method is illustrated by a numerical example.

A method for minimization of the mean square error (MSE) of the instantaneous frequency estimation using time-frequency distributions, in the case of a discrete optimization parameter, is presented. It does not require a knowledge of the estimation bias. The method is illustrated on adaptive window width determination in the Wigner distribution.

A method for the polynomial Wigner-Ville distributions realization, in the case of multicomponent signals, is presented. It is based on the author's previously proposed S-method. Using this method one may, theoretically, get the sum of the polynomial Wigner-Ville distributions of each component separately. An architecture for the polynomial Wigner-Ville distributions realization, starting from the short time Fourier transform, is given. The method is illustrated on a numerical example.

The L-class of distributions for time-frequency signal analysis is derived and presented, generalizing the L-Wigner distribution. Some particular distributions belonging to this class are introduced.

A time-frequency distribution that produces high concentration at the instantaneous frequency for an arbitrary signal is proposed. This distribution may be treated as a variant of the L-Wigner distribution, but it also satisfies unbiased energy condition, time marginal, as well as the frequency marginal in the case of asymptotic signals. The theory presented is illustrated by examples.

A dual form of the Wigner higher order spectra is introduced. Its analysis in the case of multicomponent signals is performed. An efficient distribution for time-frequency signal analysis (L-Wigner distribution) is derived from that analysis. The theory is illustrated on a numerical example.>