A method for improved distribution concentration in thetime-frequency analysis of multicomponent signals using the L-Wignerdistribution

The energy location in the Cohen class of time-frequency distributions is analyzed. If the instantaneous frequency is linear, then only the Wigner distribution produces the ideal energy concentration. The scaled version of the Wigner distribution (L-Wigner distribution), is used to improve the time-frequency representation of signals with nonlinear instantaneous frequencies. In the case of multicomponent signals, the cross terms, appearing in the Wigner distribution and in the L-Wigner distribution, can be easily removed or reduced in a computationally very efficient way. The theory is illustrated on the numerical examples with multicomponent noisy signals     

An analysis of some time-frequency and time-scale distributions

This paper presents an analysis of the representation of instantaneous frequency and group delay using time-frequency transforms or distributions of energy density domain. The time-frequency distributions which ideally represent the instantaneous frequency or group delay (itfd) are defined. Closeness to the itfd is chosen as a criterion for comparison of various commonly used distributions. It is shown that the Wigner distribution is the best among them, with respect to this criterion. The wavelet and scaled forms of the Wigner distribution are defined and analyzed. In the second part of the paper we extended the analysis to the multicomponent signals and cross terms effects. On the basis of that analysis, an efficient method, derived from the analysis of the Wigner distribution defined in the frequency domain, is proposed. This method provides some substantial advantages over the Wigner distribution. The theory is illustrated on numerical examples.     

A comparison of the existence of `cross terms' in the Wignerdistribution and the squared magnitude of the wavelet transform and theshort-time Fourier transform

It is shown that cross terms comparable to those found in the Wigner distribution (WD) exist for the energy distributions of the wavelet transform (WT) and the short-time Fourier transform (STFT). The geometry of the cross terms is described by deriving mathematical expressions for the energy distributions of the STFT and the WT of a multicomponent signal. From those mathematical expressions it is inferred that the STFT and the WT cross terms: (1) occur at the intersection of the respective transforms of the two signals under consideration, whereas the WD cross terms occur at mid-time-frequency of the two signals; (2) are oscillatory in nature, as are the WD cross terms, and are modulated by a cosine whose argument is a function of the difference in center times and center frequencies of the signals under consideration; and (3) can have a maximum amplitude as large as twice the product of the magnitude of the transforms of the two signals in question, like WD cross terms. It is shown that the presence of these cross terms could lead to problems in analyzing a multicomponent signal. The consequences of this effect with respect to speech applications are discussed     

An analysis of the Wigner higher order spectra of multicomponent signals

The analysis of multicomponent signal representation by the Wigner higher order spectra is done. It is shown that the cross-terms can be removed only for an odd order, having equal number of conjugated and nonconjugated terms in the local multidimensional moment function. A method for higher order time-multifrequency analysis that eliminates cross-terms is proposed. This method turns out to be a dual definition of the L-Wigner distribution. The theory is illustrated by the numerical example.     

On the time-frequency analysis based filtering

Efficient processing of nonstationary signals requires time-varying approach. An interesting research area within this approach is time-varying filtering. Since there is a certain amount of freedom in the definition of time-varying spectra, several definitions and solutions for the time-varying filtering have been proposed so far. Here we will consider the Wigner distribution based time- varying filtering form defined by using the Weyl correspondence. Its slight modification will be proposed and justified in the processing of noisy frequency modulated signals based on a single signal realization. An algorithm for the efficient determination of the filters ’region of support in the time-frequency plane, in the case of noisy signals, will be presented. In the second part of the paper, the theory is applied on the filtering of multicomponent noisy signals. The S-method is used as a tool for the filters’region of support estimation in this case. This method, combined with the presented algorithm, enables very efficient time-varying filtering of the multicomponent noisy signals based on a single realization of the signal and noise. Theory is illustrated by examples.     

Bispectrum-based technique to remove cross-terms in quadratic systems and Wigner–Ville distribution

In the analysis of multicomponent signals using quadratic nonlinear transforms or systems, along with the desired auto-terms unwanted cross-terms are also generated. In this paper, a novel bispectrum-based technique is proposed to remove these cross-terms. The concept of accumulated modified bispectrum (AMB) is developed. The cross-term removal property of the AMB is demonstrated, and its use in quadratic system analysis in presence of additive noise is illustrated. The application of AMB in interference reduction in the analysis of multicomponent non-stationary signals using Wigner–Ville distribution (WVD) is illustrated. For performance comparison, we compute an information theoretic measure, the Reńyi entropy, which indicates that the performance of the proposed method is better than the smoothed pseudo-WVD in the analysis of multicomponent signals that have same support in time or frequency domain.     

On the local frequency, group shift, and cross-terms in somemultidimensional time-frequency distributions: a method formultidimensional time-frequency analysis

Presents an analysis of the representation of local frequency and group shift using multidimensional time-frequency distributions. In the second part of the correspondence, the authors extend the analysis to the multicomponent signals and cross-terms effects. On the basis of that analysis, an efficient method, derived from the analysis of the multidimensional Wigner distribution defined in the frequency domain, is proposed. This method provides some substantial advantages over the Wigner distribution: the well known cross-terms effects are reduced or completely removed; the oversampling of signals is shown to be unnecessary; and the computation time can be significantly reduced, as well. The theory is illustrated by a two-dimensional numerical example     

一种基于分数阶Fourier变换的雷达运动目标检测算法

针对雷达回波为多分量LFM信号时,时频分析存在的交叉项干扰问题,提出了一种基于分数阶Fourier变换(Fractional Fourier Transform,FRFT)的伪Wigner分布(PWD).该方法通过在参数平面按阈值进行峰值搜索确定变换域阶次,再在相应的分数阶Fourier域计算PWD,有效地抑制了交叉项的干扰,有利于更好地提取信号的时频信息.仿真实验证明了在强背景噪声下该算法的有效性.     

Signal reconstruction from two close fractional Fourier power spectra

Based on the definition of the instantaneous frequency (signal phase derivative) as a local moment of the Wigner distribution, we derive the relationship between the instantaneous frequency and the derivative of the squared modulus of the fractional Fourier transform (fractional Fourier transform power spectrum) with respect to the angle parameter. We show that the angular derivative of the fractional power spectrum can be found from the knowledge of two close fractional power spectra. It permits us to find the instantaneous frequency and to solve the phase retrieval problem up to a constant phase term, if only two close fractional power spectra are known. The proposed technique is noniterative and noninterferometric. The efficiency of the method is demonstrated on several examples including monocomponent, multicomponent, and noisy signals. It is shown that the proposed method works well for signal-to-noise ratios (SNRs) higher than about 3 dB. The appropriate angular difference of the fractional power spectra used for phase retrieval depends on the complexity of the signal and can usually reach several degrees. Other applications of the angular derivative of the fractional power spectra for signal analysis are discussed. The proposed technique can be applied for phase retrieval in optics, where only the fractional power spectra associated with intensity distributions can be easily measured.     

Parameters estimation and detection of MIMO-LFM signals using MWHT

This article proposed an improved Wigner–Hough Transform (WHT) for multicarrier LFM signals of MIMO radars (MIMO-LFM). First, the signal model of the intercepted MIMO-LFM signals and the localisation of conventional WHT for this signal model are analysed. Therefore, we present the new WHT with multiple matching components, which is called as multicomponent WHT (MWHT). Then the detection and parameters estimation performance of MWHT are deduced, and analytical results indicate that MWHT is superior to conventional WHT for MIMO-LFM. In order to reduce the computation cost, a coarse estimation method is introduced. Finally, the numerical simulations demonstrate the validity of MWHT, as well as analytical results.     

Tomographic time-frequency analysis and its application towardtime-varying filtering and adaptive kernel design for multicomponentlinear-FM signals

Since line integrals through the Wigner spectrum can be calculated by dechirping, calculation of the Wigner spectrum may be viewed as a tomographic reconstruction problem. In the paper, the authors show that all time-frequency transforms of Cohen's class may be achieved by simple changes in backprojection reconstruction filtering. The resolution/cross-term tradeoff that occurs in time-frequency kernel selection is shown to be analogous to the resolution-ringing tradeoff that occurs in computed tomography (CT). “Ideal” reconstruction using a purely differentiating backprojection filter yields the Wigner distribution, whereas low-pass differentiating filters produce cross-term suppressing distributions such as the spectrogram or the Born-Jordan distribution. It is also demonstrated how this analogy can be exploited to “tune” the reconstruction filtering (or time-frequency kernel) to improve the ringing/resolution tradeoff. Some properties of the projection domain, which is also known as the Radon-Wigner transform, are characterized, including the response to signal delays or frequency shifts and projection masking or convolution. Last, time-varying filtering by shift-varying convolution in the Radon-Wigner domain is shown to yield superior results to its analogous Cohen's class adaptive transform (shift-invariant convolution) for the multicomponent, linear-FM signals that are investigated     

二维信号Wigner分布的数值计算

Wigner分布函数作为一个同时表达空间 频率变量的函数 ,在光学领域中有着广泛的应用。提出了二维信号Wigner分布函数的数值计算方法 ,并模拟了一些二维实信号 (圆孔 ,圆环和方孔 )的Wigner分布函数     

Time-frequency distributions with complex argument

A distribution highly concentrated along the group delay or the instantaneous frequency (IF) is presented. It has been defined by introducing a signal with a complex argument in time-frequency (TF) analysis. Realization of a signal with a complex argument, using a signal with a real argument, is described. The reduced interference realization of the complex argument distribution, in the case of multicomponent signals, is presented. The proposed distribution is used for the IF estimation. It is shown that the estimation can be improved with respect to the Wigner distribution based one since the bias can be significantly reduced with only a slight increase of the variance. The theory is illustrated by examples     

Theory and applications of adaptive constant-Q distributions

A new class of signal adaptive time-frequency representations called the adaptive constant-Q distribution (AQD) is introduced. The AQD exploits a priori knowledge about a signal's instantaneous frequency and bandwidth to perform signal-dependent smoothing of the Wigner distribution. The objective is to achieve a good tradeoff between reducing the variance and preserving the resolution by means of time-frequency dependent smoothing (specifically for use in medical Doppler ultrasound). A numerical, alias-free implementation of the AQD is presented. Deterministic, multicomponent signals as well as synthetic Doppler ultrasound signals were analyzed with the AQD. The performance of the AQD was compared with the power spectrogram, the exponential distribution, and the adaptive optimum kernel representation as well as with theory. The error was consistently lowest for the AQD. In conclusion, a new signal adaptive class of time-frequency distributions has been developed, and its potential in nonstationary signal analysis has been demonstrated     

Analysis and filtering using the optimally smoothed Wignerdistribution

The authors consider the analysis and filtering of a deterministic signal with slowly time-varying spectra using the optimally smoothed Wigner distribution (OSWD). They compare this mixed time-frequency representation (MTFR) to other MTFRs such as the spectrogram, the short-time Fourier transform (STFT), and the Wigner and pseudo-Wigner distributions. The authors propose an approach to designing linear time-varying filters for slowly time-varying signals which is based on the concept of local nonstationarity cancellation and show that it is equivalent to masking the optimal STFT. The performance of the filter in suppressing white noise and in decomposing a slowly time-varying signal into its components is studied and compared to the performance of the techniques based on the STFT     

Time-frequency representation of electrocorticograms in temporallobe epilepsy

Three time-frequency distributions are evaluated in terms of their efficacy in representing nonstationary electrocorticograms (ECoG's) in human temporal lobe epilepsy. The results of a new method, the exponential distribution, are compared with those of the spectrogram and the Wigner distribution. It is shown that the exponential distribution represents a considerable improvement over the spectrogram in terms of resolution and markedly reduces cross-terms present in the Wigner distribution. Exponential distribution representations of ECoG's from different stages of an epileptic record are developed as contour maps. These high-resolution representations offer a lucid display of temporal-spectral features of the rapidly varying signals that constitute ECoG's recorded in temporal lobe epilepsy.     

Orthonormal-basis partitioning and time-frequency representation of cardiac rhythm dynamics

Although a number of time-frequency representations have been proposed for the estimation of time-dependent spectra, the time-frequency analysis of multicomponent physiological signals, such as beat-to-beat variations of cardiac rhythm or heart rate variability (HRV), is difficult. We thus propose a simple method for 1) detecting both abrupt and slow changes in the structure of the HRV signal, 2) segmenting the nonstationary signal into the less nonstationary portions, and 3) exposing characteristic patterns of the changes in the time-frequency plane. The method, referred to as orthonormal-basis partitioning and time-frequency representation (OPTR), is validated using simulated signals and actual HRV data. Here we show that OPTR can be applied to long multicomponent ambulatory signals to obtain the signal representation along with its time-varying spectrum.     

Parameter estimation for micro-Doppler signals based on cubic phase function

An estimator for evaluating the parameters from the radar returned multicomponent micro-Doppler (m-D) signals is presented in this paper. While time frequency distribution (TFD) is commonly used to analyze the time-varying m-D frequency features in TF domain, the proposed algorithm is based on cubic phase function (CPF) that can transform the signal to time frequency rate domain. In order to estimate the parameters of multicomponent m-D signal, the extended Hough transform (HT) of CPF is employed to estimate linear frequency modulation (LFM) or sinusoidal frequency modulation (SFM) components. For the m-D signal composed of both LFM and SFM components, the estimates involve two steps of HT-CPF. Firstly, LFM components are estimated by HT-CPF and removed from the time frequency rate plane, and then, HT of the modified time frequency rate distribution is applied to estimate SFM ones. Compared with HT-TFD, this algorithm needs less dimension of HT space and is thus computationally efficient. In addition, simulations show that the algorithm has almost the same performance signal-to-noise threshold as HT of Wigner–Ville distribution method.     

Analysis of polynomial-phase signals by the integrated generalizedambiguity function

The aim of this work is the performance analysis of a method for the detection and parameter estimation of mono or multicomponent polynomial-phase signals (PPS) embedded in white Gaussian noise and based on a generalized ambiguity function. The proposed method is shown to be asymptotically efficient for second-order PPS and nearly asymptotically efficient for third-order PPSs. The method presents some advantages with respect to similar techniques, like the polynomial-phase transform, for example, in terms of (i) a closer approach to the Cramer-Rao lower bounds, (ii) a lower SNR threshold, (iii) a better capability of discriminating multicomponent signals     

一种低副瓣无混叠的线性调频信号时频分析方法

作为通信与勘探中广泛使用的一类信号,线性调频信号的参数分析经常采用基于Wigner-Ville分布(WVD)的时频分析方法。该方法具有高时频分辨率,但在交叉项、高副瓣以及频谱混叠问题上存在缺陷。该文提出一种名为空间变迹重排Wigner-Ville分布(SVA-rWVD)的时频分析方法,结合空间变迹技术(SVA)的副瓣抑制能力及短时傅里叶变换(STFT)的无混叠无交叉项特性,得到一个新的时频分布。基于单分量和多分量线性调频信号的仿真实验结果表明,该方法得到的时频分布可以降低副瓣水平至–40 dB以下同时消除交叉项及频谱混叠现象。