Joint time-frequency analysis

It has been well understood that a given signal can be represented in an infinite number of different ways. Different signal representations can be used for different applications. For example, signals obtained from most engineering applications are usually functions of time. But when studying or designing the system, we often like to study signals and systems in the frequency domain. Although the frequency content of the majority of signals in the real world evolves over time, the classical power spectrum does not reveal such important information. In order to overcome this problem, many alternatives, such as the Gabor (1946) expansion, wavelets, and time-dependent spectra, have been developed and widely studied. In contrast to the classical time and frequency analysis, we name these new techniques joint time-frequency analysis. We introduce the basic concepts and well-tested algorithms for joint time-frequency analysis. Analogous to the classical Fourier analysis, we roughly partition this article into two parts: the linear (e.g., short-time Fourier transform, Gabor expansion) and the quadratic transforms (e.g., Wigner-Ville (1932, 1948) distribution). Finally, we introduce the so-called model-based (or parametric) time-frequency analysis method     

Adaptive cone-kernel time-frequency analysis

Presents a technique to adaptively optimize the performance of the cone-kernel distribution (CKD) by varying the cone length in response to changing signal properties. The result is an adaptive CKD that preserves the outer hull of signal time support, yields excellent results on real signals, outperforms fixed-length CKDs, and requires only slightly more computation than a fixed-kernel distribution     

Wavelets and time-frequency analysis

We present a selective overview of time-frequency analysis and some of its key problems. In particular we motivate the introduction of wavelet and wavelet packet analysis. Different types of decompositions of an idealized time-frequency plane provide the basis for understanding the performance of the numerical algorithms and their corresponding interpretations within the continuous models. As examples we show how to control the frequency spreading of wavelet packets at high frequencies using nonstationary filtering and study some properties of periodic wavelet packets. Furthermore we derive a formula to compute the time localization of a wavelet packet from its indexes which is exact for linear phase filters, and show how this estimate deteriorates with deviation from linear phase     

A nonlinear time-frequency analysis method

An alternative method of time-frequency signal analysis is introduced and is compared with the conventional discrete Fourier transfer (DFT)-based methods. The structure of the proposed algorithm and its mathematical properties are presented. The proposed algorithm has a nonlinear structure that provides a frequency-adaptivity feature. It can be implemented both in analog and in digital form and is particularly suitable for real-time applications where computational efficiency is important. A comparison is made with the DFT in terms of algorithm efficiency, sensitivity, and complexity. It is shown that compared with the DFT, the algorithm is more efficient for real-time applications and is less sensitive to noise and variations in the frequency and sampling rate while maintaining simplicity of the structure and computational efficiency. Slower convergence rate of the algorithm, compared with the DFT, constitutes its main shortcoming.     

Time-frequency-based detection using discrete-time discrete-frequency Wigner distributions

During the last decade, a comprehensive theory for optimum time-frequency (TF)-based detection has been developed. This was originally proposed in the continuous-time continuous-frequency case. This paper deals with detectors operating on discrete-time discrete-frequency Wigner distributions (WDs). The purpose is to discuss some existing definitions of this distribution within the context of TF-based detection and selecting those that do not affect the performance of the decision device with which they are associated. This question is of interest since there exist several approaches for discretizing the WD, sometimes resulting in a loss of fundamental properties. First, the discrete-time discrete-frequency formulations of optimum detection are investigated. Next, the problem of the design of TF-based detectors from training data, keeping in mind severe effects of the curse of dimensionality, is considered.     

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     

The Weyl correspondence and time-frequency analysis

Describes the Weyl correspondence and its properties, showing how it gives a “window-independent” definition of time-frequency concentration for use in models in signal detection. The definition of concentration is justified by showing that it gives reasonable answers in certain intuitive cases. The Weyl correspondence expresses a linear transformation as a weighted superposition of time-frequency shifts of the signal, and then authors explain why this is not the same as “transforming” a signal into the time-frequency domain, multiplying by a weight in the transform domain and taking the inverse. The investigation into time-frequency concentration and the Weyl correspondence is justified by a new result. The authors show that convolving the Wigner distribution with a general smoothing function is equivalent to evaluating a weighted sum of spectrograms. This is a new interpretation of the process of smoothing the Wigner distribution to reduce cross-terms. It relates smoothing of the Wigner distribution to the “multiple window” technique pioneered by Thomson (1982)     

一种新的联合时频分析方法

根据信号对当前时刻的联合时频分布的贡献主要源于最近出现的信号的思想，提出了一种新的联合时频分布一渐忘分布。分布表明，渐忘分布具有非负、交叉项不明显及可用FFT实现快速计算等优点，且该分布的时频局域性可方便地通过选取不同的衰减因子进行调整。通过对典型信号的应用，进一步验证了渐忘分布的良好特性。     

Chirp信号的时频分析特征比较

Chirp信号是一个典型的非平稳信号,在通信、声纳、雷达等领域具有广泛的应用,为了更好的显示其特性,文中首先介绍了各个算法的定义和公式,然后用各种时频分析方法对该信号以及该信号添加单频正弦噪声信号进行分析,比较各类方法的特点,通过分析和比较可知希尔伯特黄变换在处理该信号具有很好的聚集性以及对单频噪声也有很好的辨别能力.最后用MATLAB软件进行仿真得到结果.     

A new transform for time-frequency analysis

The psi-decomposition of a signal, in which the signal is written as a weighted sum of certain elementary synthesizing functions, is described. The set S of synthesizing functions consists of dilated and translated copies of two parent functions, which are concentrated in both the time and the frequency domains. The weighting constants in the psi-decomposition define a transform called the phi-transform. The phi-transform of a signal captures both the frequency content and the temporal evolution of a nonstationary signal. The phi-transform is linear, continuous, and continuously invertible. The set S of synthesizing functions used in the psi-decomposition is nonorthogonal, hence considerable flexibility is permitted in its construction. It is shown with the help of two examples that the set S is easy to construct     

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.     

Super-resolved time-frequency analysis of wideband backscattereddata

A time-frequency super-resolution procedure is presented for processing wideband backscattered data containing both scattering center and natural resonance information. In this procedure, Prony's method is first applied in the frequency domain to locate scattering centers. The data is processed one slice at a time through the use of a sliding window function. Parameterized models for the scattering centers are obtained by a weighted average of the results from each segment. A similar procedure is used in the time domain to fully parameterize the natural resonances. In contrast to other time-frequency techniques, the time-frequencg display from the present super-resolution procedure is not constrained in resolution by the Fourier limit     

Discrete-time analysis of cell spacing in ATM systems

In this paper, an analysis of a general spacer mechanism as used in ATM systems and being part of the usage parameter control functions is developed. The cell process which is subject to spacing can be an arbitrarily chosen renewal process. The algorithm aims at the calculation of the spacer output process in terms of the cell inter-departure time distribution which gives insights to understand the traffic stream forming properties of the spacing mechanism. Two spacer variants are taken into account, where cell rejection and non-rejection versions of the spacing scheme are considered. It is shown that the state process of theGI/D/1 queue andGI/D/1 queue with bounded delay can be used to analyze the spacer process. Numerical results are presented to show the system performance for different traffic conditions and system parameters. Beyond the consideration of the pure spacing mechanism, we also take into account that the cell streams are changed between the ATM connection endpoints and the spacer. We model this by considering the output process of a discrete-timeGI/G/1 queue as the spacer input process. We also take the spacer output process as input for a discrete-timeG//G/1 queue to investigate how the spaced cell stream is again changed until reaching the private/public UNI.The work for this paper was performed while the author was with the Institute of Computer Science, University of Würzburg, Germany.     

基于时频分析的ISAR成像

传统的逆合成孔径雷达(ISAR)成像算法用傅里叶变换进行频谱分析,对机动目标的成像会导致成像模糊。本文应用短时傅里叶变换(STFT)、魏格纳分布(WVD)、平滑伪魏格纳分布(SPWVD)几种时频分析方法对机动目标进行瞬时成像。仿真和实验结果表明,通过该方法能得到清晰的目标的距离-瞬时多普勒像。     

A time-frequency analysis method for radar scattering

A time-frequency analysis method to study electromagnetic scattering is presented and demonstrated using canonical objects. The time-frequency analysis method utilizes the Bargmann transform to formulate the signal representation in phase space. The use of the Bargmann transform leads to an attractive parametric signal representation in terms of complex polynomials, and elliptical filters can be constructed to crop or extract selected areas of the phase plane. The signal representation and filtering operations are demonstrated using scattering responses from spheres and thin wires, and the prominent scattering features are identified and extracted     

Subspace analysis of spatial time-frequency distribution matrices

Spatial time-frequency distributions (STFDs) have been previously introduced as the natural means to deal with source signals that are localizable in the time-frequency domain. Previous work in the area has not provided the eigenanalysis of STFD matrices, which is key to understanding their role in solving direction finding and blind source separation problems in multisensor array receivers. The aim of this paper is to examine the eigenstructure of the STFD matrices. We develop the analysis and statistical properties of the subspace estimates based on STFDs for frequency modulated (FM) sources. It is shown that improved estimates are achieved by constructing the subspaces from the time-frequency signatures of the signal arrivals rather than from the data covariance matrices, which are commonly used in conventional subspace estimation methods. This improvement is evident in a low signal-to-noise ratio (SNR) environment and in the cases of closely spaced sources. The paper considers the MUSIC technique to demonstrate the advantages of STFDs and uses it as grounds for comparison between time-frequency and conventional subspace estimates     

Adaptive time-frequency analysis based on autoregressive modeling

A new adaptive method for discrete time-frequency analysis based on autoregressive (AR) modeling is introduced. The performance of AR modeling often depends upon a good selection of the model order. The predictive least squares (PLS) principle of Rissanen was found to be a good criterion for model order estimation in stationary processes. This paper presents a modified formulation of the PLS criterion suitable for non-stationary processes. Efficient lattice filters based on the covariance assumption are used to estimate the model parameters of all model orders less than some maximum order M. It is shown that the resulting complexity is no larger than M. The modified PLS criterion allows the model order to adapt to non-stationary processes and, in turn, compute adaptive AR based time-frequency representations (TFRs). Examples of time-frequency analyses for synthetic and bio-acoustical signals are provided as well as comparisons to classical time-frequency representations.     

时频联合分析在闪电定位中的应用

为了揭示闪电甚高频（very high frequency,VHF）辐射源的时频特性,探索时频分析方法在闪电VHF辐射源定位中的应用,文中利用Wigner-Ville分布（Wigner-Ville distribution,WVD）对闪电宽带VHF辐射信号进行了时频分析,提出一种半二维相关（semi-two-dimensional correlation,STDC）时延算法来实现闪电VHF辐射源的定位,并通过数值模拟和人工引雷观测数据对定位算法进行了验证.时频分析结果揭示了在纳秒尺度上,先导宽带VHF辐射"倾斜状"的时频分布特征中高频辐射先于低频成分到达,典型的倾斜斜率约为-3 MHz/ns.提出的时频定位算法可以计算信号不同频率成分的时延,便于去除干扰点影响.利用本算法对一次人工引雷直窜先导进行了定位分析,通过舍去调频（frequency modulation,FM）干扰,能够清晰描绘先导VHF辐射源的发展过程,定位结果与高速摄像图像吻合,进一步丰富了短基线VHF的定位方式.     

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.     

基于二次型时频分析方法处理非平稳信号

使用时频分析方法处理非平稳信号,提取非平稳信号的时频分布。分别采用线性时频分析和二次型时频分析处理模拟仿真信号,仿真结果显示用二次型时频分析处理所得的时频分布的时频分辨率较高。仿真结果证明,二次型时频分析方法是处理非平稳信号的有效方法。