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.     

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     

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     

空间高阶时频分布及虚拟时频ESPRIT算法

基于L-Wignrer分布(LWD),定义了一种全新的空间高阶累积量L-Wigner分布,并利用虚拟时频ESPRIT算法与阵列扩展,提出了一种多个多项式相位信号(PPS)的DOA估计算法.该方法利用平滑的伪WVD分布以实现PPS信号的LWD分布极大地抑制了其中的交叉项并提高了信号时频聚集性.同时,将时频分析与高阶累积量结合,使算法具有虚拟阵列扩展的特点,高分辨高精度的估计了PPS信号的DOA.理论分析和计算机仿真表明该方法具有很好的参数估计性能.     

Unified approach to noise analysis in the Wigner distribution and spectrogram

An analysis of time-frequency representations of noisy signals is performed. Using the method for time-frequency signal analysis which was recently defined by Stankovic (the S-method), the influence of noise on the two most important distributions (spectrogram and Wigner distribution) is analyzed in unified manner. It is also shown that, for signals whose instantaneous frequency is not constant, an improvement over the spectrogram and the Wigner distribution performances in a noisy environment may be achieved using the S-method. The expressions for mean and variance are derived. Results are given for several illustrative and numerical examples.     

An analysis of instantaneous frequency representation usingtime-frequency distributions-generalized Wigner distribution

This paper presents an analysis of the representation of instantaneous frequency using time-frequency distributions of energy density domain. Similarity to the “ideal” instantaneous frequency presentation is chosen as a criterion for comparison of various distributions. Although all the commonly used distributions suffer from the artifacts along frequency axis, it is shown that the Wigner distribution is the best among them, with respect to this criterion. The generalization of Wigner distribution-LWD-is introduced to decrease the artifacts. The properties of the LWD are analyzed. It is shown that, at the expense of an insignificant increase in computation time, much better results are obtained. The theory is illustrated by a numerical example with the frequency modulated signals     

基于多尺度线性调频基信号稀疏分解的多分量LFM信号检测

该文针对传统的基于二次时频分析和原子追踪匹配方法处理多分量LFM信号时存在的时频干扰和等振幅交叉分解等问题,提出了一种基于多尺度线性调频基信号稀疏分解的多分量LFM信号检测方法,该方法采用多尺度的线调频基函数对多分量LFM信号进行投影分解,通过从不同的时间支撑区内投影系数最大的基函数中寻找出使分解信号能量最大的基元函数组合,逐次获得信号包含的能量最大的LFM信号分量,从基元函数连接形成的频率曲线即可获得LFM信号分量瞬时频率的估计,再对分量瞬时频率求起始时间点的频率值和曲线斜率便可得到该LFM分量的中心频率及调频斜率,仿真试验表明该文方法能精确地提取等振幅多分量LFM信号的瞬时频率,并具有很强的抗噪声干扰能力。     

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.     

A new time-frequency transform for non-stationary signals with any nonlinear instantaneous phase

A new generic adaptive time-frequency transform based on the Wigner distribution is proposed for amplitude estimation of transient signals with any nonlinear non-polynomial variation of the instantaneous frequency. It is for use in situations where independent synchronous measurements of the instantaneous frequency are available. It shows that the new transform tracks the instantaneous frequency and estimates signal amplitude without errors along the curve of the instantaneous frequency. The paper also applies the new transform to signals with the non-linear sinusoidal and exponential variations in the instantaneous phase and determines formulae for transform in these cases. The paper compares the new transform with the Wigner distribution in several cases and demonstrates that the new transform is more effective at amplitude estimation of signals with nonlinear variation of the instantaneous frequency. New analytic formulae are obtained for the new transform and the Wigner distribution in both the sinusoidal and the exponential cases. An analytic formula is obtained which relates the new transform to the Wigner distribution in the sinusoidal case. This formula is inverted to obtain a previously unknown formula for the Wigner distribution of any signal. It is shown that the new transform could be used for adaptive processing of transient signals with instantaneous frequency variations. The paper studies the performance of the new transform with no adaptation, partial adaptation and complete adaptation.

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     

Separation of micro-Doppler signals based on time frequency filter and Viterbi algorithm

Micro-Doppler (m-D) effect is potential useful in radar target detection, recognition, and classification. While the m-D signals are always multicomponent, it is important to separate the m-D signals for feature extraction. This paper introduces a separation algorithm based on time-frequency filter (TFF). When the m-D multicomponent signals are always overlapped in TF plane, Viterbi algorithm on time-frequency distribution is used to firstly estimate the instantaneous frequencies, then an automatic TFF is designed to filter and synthesize the interesting m-D signal. Simulation results show that the proposed algorithm can effectively extract the m-D signals even in a relatively high noise environment.     

The Wigner distribution of noisy signals with adaptivetime-frequency varying window

Time-frequency representations using the Wigner distribution (WD) may be significantly obscured by the noise in the observations. The analysis performed for the WD of discrete-time noisy signals shows that this time-frequency representation can be optimized by the appropriate choice of the window length. However, the practical value of this analysis is not significant because the optimization requires knowledge of the bias, which depends on the unknown derivatives of the WD. A simple adaptive algorithm for the efficient time-frequency representation of noisy signals is developed in this paper. The algorithm uses only the noisy estimate of the WD and the analytical formula for the variance of this estimate. The quality of this adaptive algorithm is close to the one that could be achieved by the algorithm with the optimal window length, provided that the WD derivatives were known in advance. The proposed algorithm is based on the idea that has been developed in our previous work for the instantaneous frequency (IF) estimation. Here, a direct addressing to the WD itself, rather than to the instantaneous frequency, resulted in a time and frequency varying window length and showed that the assumption of small noise and bias is no longer necessary. A simplified version of the algorithm, using only two different window lengths, is presented. It is shown that the procedure developed for the adaptive window length selection can be generalized for application on multicomponent signals with any distribution from the Cohen (1989, 1990, 1992) class. Simulations show that the developed algorithms are efficient, even for a very low value of the signal-to-noise ratio     

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.     

Highly concentrated time-frequency distributions: pseudo quantumsignal representation

Distributions that are highly concentrated in the time-frequency plane are presented. Since the idea for these distributions originated from the Wigner representation in the quantum mechanics, a review of this representation is done in the first part of the paper. Abstracting the physical sense of the quantum mechanics representation, we defined the “pseudo quantum” signal representation. On the basis of a signal, the “pseudo wave function” with the corresponding “pseudo particle” having the “pseudo momentum” _p=ℏ_fω is generated. By varying the value of ℏ_f, one is in a position to influence the concentration of the “pseudo quantum” (time-“pseudo momentum”) signal's presentation while keeping its most important time-frequency properties invariant. From this reflection, an efficient distribution for the time-frequency (time-“pseudo momentum”) signal analysis is obtained. This distribution produces as high a concentration in the time-frequency (time-“pseudo momentum”) plane as the L-Wigner distribution; however, it may satisfy the marginal properties. The theory is illustrated with examples     

Time-frequency window leakage in the short-time Fourier transform

Conventional frequency-domain window-leakage analysis accurately describes the leakage in the short-time Fourier transform only for stationary signals. Leakage in the time-frequency plane from concentrated transient or nonstationary signals can be effectively analyzed by use of a time-frequency window-leakage envelope with rectangular contours. This envelope is obtained from the Wigner distribution of the analysis window, with appropriate corrections for the sidelobe leakage. The time-frequency window-leakage envelope gives insight into the tradeoffs in time-frequency leakage between various windows and allows quick and accurate estimates of the leakage in the short-time Fourier transform.A simple technique for constructing signals with Wigner distributions that are linear transformations of the Wigner distribution of a known signal is developed. With this technique, windows with a variety of time-frequency orientations and leakage behavior can be developed.This work has been supported by NSF Grant No. ECS 83-14006 and by an NSF graduate fellowship.     

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     

Tracking of unknown nonstationary chirp signals using unsupervisedclustering in the Wigner distribution space

This paper is concerned with the problems of (1) detecting the presence of one or more FM chirp signals embedded in noise, and (2) tracking or estimating the unknown, time-varying instantaneous frequency of each chirp component. No prior knowledge is assumed about the number of chirp signals present, the parameters of each chirp, or how the parameters change with time. A detection/estimation algorithm is proposed that uses the Wigner distribution transform to find the best piecewise cubic approximation to each chirp's phase function. The first step of the WD based algorithm consists of properly thresholding the WD of the received signal to produce contours in the time-frequency plane that approximate the instantaneous frequency of each chirp component. These contours can then be approximated as generalized lines in the (ω, t, t²) space. The number of chirp signals (or equivalently, generalized lines) present is determined using maximum likelihood segmentation. Minimum mean square estimation techniques are used to estimate the unknown phase parameters of each chirp component. The authors demonstrate that for the cases of (i) nonoverlapping linear or nonlinear FM chirp signals embedded in noise or (ii) overlapping linear FM chirp signals embedded in noise, the approach is very robust, highly reliable, and can operate efficiently in low signal-to-noise environments where it is hard for even trained operators to detect the presence of chirps while looking at the WD plots of the overall signal. For multicomponent signals, the proposed technique is able to suppress noise as well as the troublesome cross WD components that arise due to the bilinear nature of the WD     

Performance of quadratic time-frequency distributions as instantaneous frequency estimators

General performance analysis of the shift covariant class of quadratic time-frequency distributions (TFDs) as instantaneous frequency (IF) estimators, for an arbitrary frequency-modulated (FM) signal, is presented. Expressions for the estimation bias and variance are derived. This class of distributions behaves as an unbiased estimator in the case of monocomponent signals with a linear IF. However, when the IF is not a linear function of time, then the estimate is biased. Cases of white stationary and white nonstationary additive noises are considered. The well-known results for the Wigner distribution (WD) and linear FM signal, and the spectrogram of signals whose IF may be considered as a constant within the lag window, are presented as special cases. In addition, we have derived the variance expression for the spectrogram of a linear FM signal that is quite simple but highly signal dependent. This signal is considered in the cases of other commonly used distributions, such as the Born-Jordan and the Choi-Williams distributions. It has been shown that the reduced interference distributions outperform the WD but only in the case when the IF is constant or its variations are small. Analysis is extended to the IF estimation of signal components in the case of multicomponent signals. All theoretical results are statistically confirmed.     

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.     

A resolution comparison of several time-frequency representations

Two signal components are considered resolved in a time-frequency representation when two distinct peaks can be observed. The time-frequency resolution limit of two Gaussian components, alike except for their time and frequency centers, is determined for the Wigner distribution, the pseudo-Wigner distribution, the smoother Wigner distribution, the squared magnitude of the short-time Fourier transform, and the Choi-Williams distribution. The relative performance of the various distributions depends on the signal. The pseudo-Wigner distribution is best for signals of this class with only one frequency component at any one time, the Choi-Williams distribution is most attractive for signals in which all components have constant frequency content, and the matched filter short-time Fourier transform is best for signal components with significant frequency modulation. A relationship between the short-time Fourier transform and the cross-Wigner distribution is used to argue that, with a properly chosen window, the short-time Fourier transform of the cross-Wigner distribution must provide better signal component separation that the Wigner distribution