Adaptive instantaneous frequency estimation of multicomponent FMsignals using quadratic time-frequency distributions

An adaptive approach to the estimation of the instantaneous frequency (IF) of nonstationary mono- and multicomponent FM signals with additive Gaussian noise is presented. The IF estimation is based on the fact that quadratic time-frequency distributions (TFDs) have maxima around the IF law of the signal. It is shown that the bias and variance of the IF estimate are functions of the lag window length. If there is a bias-variance tradeoff, then the optimal window length for this tradeoff depends on the unknown IF law. Hence, an adaptive algorithm with a time-varying and data-driven window length is needed. The adaptive algorithm can utilize any quadratic TFD that satisfies the following three conditions: First, the IF estimation variance given by the chosen distribution should be a continuously decreasing function of the window length, whereas the bias should be continuously increasing so that the algorithm will converge at the optimal window length for the bias-variance tradeoff, second, the time-lag kernel filter of the chosen distribution should not perform narrowband filtering in the lag direction in order to not interfere with the adaptive window in that direction; third, the distribution should perform effective cross-terms reduction while keeping high resolution in order to be efficient for multicomponent signals. A quadratic distribution with high resolution, effective cross-terms reduction and no lag filtering is proposed. The algorithm estimates multiple IF laws by using a tracking algorithm for the signal components and utilizing the property that the proposed distribution enables nonparametric component amplitude estimation. An extension of the proposed TFD consisting of the use of time-only kernels for adaptive IF estimation is also proposed     

Adaptive time-frequency analysis of knee joint vibroarthrographic signals for noninvasive screening of articular cartilage pathology

Vibroarthrographic (VAG) signals emitted by human knee joints are nonstationary and multicomponent in nature; time-frequency distributions (TFD's) provide powerful means to analyze such signals. The objective of this paper is to construct adaptive TFD's of VAG signals suitable for feature extraction. An adaptive TFD was constructed by minimum cross-entropy optimization of the TFD obtained by the matching pursuit decomposition algorithm. Parameters of VAG signals such as energy, energy spread, frequency, and frequency spread were extracted from their adaptive TFD's. The parameters carry information about the combined TF dynamics of the signals. The mean and standard deviation of the parameters were computed, and each VAG signal was represented by a set of just six features. Statistical pattern classification experiments based on logistic regression analysis of the parameters showed an overall normal/abnormal screening accuracy of 68.9% with 90 VAG signals (51 normals and 39 abnormals), and a higher accuracy of 77.5% with a database of 71 signals with 51 normals and 20 abnormals of a specific type of patellofemoral disorder. The proposed method of VAG signal analysis is independent of joint angle and clinical information, and shows good potential for noninvasive diagnosis and monitoring of patellofemoral disorders such as chondromalacia patella.     

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.     

High Resolution Time-Frequency Distribution Based on Short-Time Sparse Representation

This paper focuses on the high resolution time-frequency distribution (TFD) of multicomponent signals with amplitude and frequency modulations, and a concise method named short-time sparse representation (STSR) is proposed. In STSR, both analysis and synthesis of the discrete signal can be achieved by exploiting the signal’s sparsity in frequency domain at each time instant. In order to fasten the STSR procedure, an efficient sparse recovery algorithm named SL0 is applied, and the signal model for each sliding window is modified to form the same dictionary, which guarantees that the whole recovery procedure adapts to the matrix form. The performance of STSR is compared with other TFD techniques and assessed in various configurations. It is shown that both preferable representation and acceptable computational cost can be obtained.     

基于条件对抗生成时频分布的多分量信号瞬时频率估计

瞬时频率（Instantaneous Frequency,IF）估计在多分量信号处理中具有重要意义,而现有方法在信号分量的IF曲线相近或相交时估计准确度不佳。针对这一问题,本文提出一种基于条件对抗生成时频分布的多分量信号IF估计方法。该方法首先采用时频分析产生信号的时频图像（例如掩膜维格纳分布）作为条件生成对抗网络（Conditional Generative Adversarial Networks, CGAN）的原始数据集,通过训练CGAN进行学习之后生成接近理想时频分布的时频图像。根据这些图像,本文利用一种改进的维特比算法提取出不同分量的IF曲线。其改进点在于增加了一个线段梯度的惩罚项,使维特比算法在分量相交的时频区域仍有准确的IF估计。实验结果表明,该方法能够有效且准确地估计分量相近或相交情况下信号的IF信息。      

Principal decomposition of time-frequency distributions

The authors present a novel time-frequency analysis technique which uses principal components analysis to map any given time-frequency distribution (TFD) of a signal into a set of three 1-D principal decomposition functions. These three functions may then be considered to be `separable' components of a time-frequency function which they refer to as the principal approximation function for the original TFD. They show how principal decomposition analysis is useful for the enhancement and frequency-tracking of nonstationary harmonic signals     

Signal enhancement by time-frequency peak filtering

Time-frequency peak filtering (TFPF) allows the reconstruction of signals from observations corrupted by additive noise by encoding the noisy signal as the instantaneous frequency (IF) of a frequency modulated (FM) analytic signal. IF estimation is then performed on the analytic signal using the peak of a time-frequency distribution (TFD) to recover the filtered signal. This method is biased when the peak of the Wigner-Ville distribution (WVD) is used to estimate the encoded signal's instantaneous frequency. We characterize a class of signals for which the method implemented using the pseudo WVD is approximately unbiased. This class contains deterministic bandlimited nonstationary multicomponent signals in additive white Gaussian noise (WGN). We then derive the pseudo WVD window length that gives a reduced bias when TFPF is used for signals from this class. Testing of the method on both synthetic and real life newborn electroencephalogram (EEG) signals shows clean recovery of the signals in noise level down to a signal-to-noise ratio (SNR) of -9 dB.     

A signal-dependent time-frequency representation: optimal kerneldesign

A new time-frequency distribution (TFD) that adapts to each signal and so offers a good performance for a large class of signals is introduced. The design of the signal-dependent TFD is formulated in Cohen's class as an optimization problem and results in a special linear program. Given a signal to be analyzed, the solution to the linear program yields the optimal kernel and, hence, the optimal time-frequency mapping for that signal. A fast algorithm has been developed for solving the linear program, allowing the computation of the signal-dependent TFD with a time complexity on the same order as a fixed-kernel distribution. Besides this computational efficiency, an attractive feature of the optimization-based approach is the ease with which the formulation can be customized to incorporate application-specific knowledge into the design process     

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.     

New Time-Frequency Distribution

Cross terms are an inherent consequence of the second order nature of Cohen's class TFDs (Time-Frequency Distributions) [5], [6]. They are manifest in a TFD of multicomponent signals as spurious artifacts arising from interactions between the various signal components, and they can often appear at times and/or frequencies inconsistent with the underlying physical nature of the signal, causing misinterpretation [2], [3], [4]. There are many time frequency distributions that avoid the cross term effect; the best are the Choi-Williams ED (Exponential Distribution) [1] and Levin's IPS (Instantaneous Power Spectrum) [9]. In this paper we combine the cross term reducing philosophy of the ED and IPS to obtain a new TFD that most effectively reduces the cross term effect. Surprisingly, the new TFD also satisfies most desired TFD properties.     

分数阶Fourier域强弱LFM信号检测与参数估计

分数阶Fourier变换（FRFT）由于其特有的性质,非常适合处理线性调频（LFM）信号,尤其是,作为一种线性变换,可以克服多分量LFM信号之间的交叉项干扰。但是采用逐次消去法检测多分量LFM信号时,每检测一个LFM信号,都要对信号分别求旋转角 的FRFT,再进行二维搜索,计算量较大。为了提高FRFT对多分量LFM信号的检测效率,本文给出一种在分数阶Fourier域检测强、弱LFM信号的新方法。首先,分析了逐次消去法和聚类分析法检测多分量LFM信号的原理,以及它们的优缺点。提出一种聚类分析和逐次消去相结合的信号检测方法,利用平面截取信号在平面(u,α)上的尖峰,并引入基于广度优先搜索邻居（BFSN）的聚类算法,对截取的信号尖峰进行聚类分析,获得每个LFM信号对应的信号尖峰,实现多个较强信号的检测与参数估计,再利用逐次消去法实现弱信号的检测。该方法可以同时检测多个能量相近的LFM信号,提高了检测效率,以及次强信号的参数估计精度,并有效地抑制了强信号对弱信号的遮蔽影响。通过对信号进行平面切割处理,减少了BFSN聚类算法中输入集样本数量,大大降低了算法的计算量。最后,仿真验证了该方法的有效性。      

Weighted least squares implementation of Cohen-Posch time-frequencydistributions

We present an improvement of the least-squares method of Sang et al. (see Proc. IEEE-SP Int. Symp. Time-Freq./Time-Scale Anal., p.165-8, 1996) for constructing nonnegative joint time-frequency distributions (TFDs) satisfying the time and frequency marginals (i.e., Cohen-Posch (1985) distributions). The proposed technique is a positivity constrained iterative weighted least-squares (WLS) algorithm used to modify an initial TFD (e.g., any bilinear TFD) to obtain a Cohen-Posch TFD. The new algorithm solves the “leakage” problem of the least-squares approach and is computationaly faster. Examples illustrating the performance of the new algorithm are presented. The results for the WLS method compare favorably with the minimum cross-entropy method previously developed by Loughlin et al. (1992)     

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.     

LMS-like AR modeling in the case of missing observations

This paper presents a new recursive algorithm for the time domain reconstruction and spectral estimation of uniformly sampled signals with missing observations. An autoregressive (AR) modeling approach is adopted. The AR parameters are estimated by optimizing a mean-square error criterion. The optimum is reached by means of a gradient method adapted to the nonperiodic sampling. The time-domain reconstruction is based on the signal prediction using the estimated model. The power spectral density is obtained using the estimated AR parameters. The development of the different steps of the algorithm is discussed in detail, and several examples are presented to demonstrate the practical results that can be obtained. The spectral estimates are compared with those obtained by known AR estimators applied to the same signals sampled periodically. We note that this algorithm can also be used in the case of nonstationary signals     

Householder transform based joint diagonal zero diagonalization for source separation using time-frequency distributions

We present a new combined joint diagonalization and zero diagonalization algorithm for separating the source signals by using time-frequency distributions (TFD). The proposed algorithm is based on the Householder transform, which exactly guarantees the orthonormality of the diagonalizer and/or zero diagonalizer. As an application, we show that blind separation of correlated sources can be achieved by applying the proposed algorithm to spatial quadratic TFD matrices corresponding to auto-source terms and/or cross-source terms. Computer simulations are provided to demonstrate the performances of the proposed algorithm and compare it with the classical ones to show the performance improvement.     

S-transform-based intelligent system for classification of power quality disturbance signals

In this paper, a new approach is presented for the detection and classification of nonstationary signals in power networks by combining the S-transform and neural networks. The S-transform provides frequency-dependent resolution that simultaneously localizes the real and imaginary spectra. The S-transform is similar to the wavelet transform but with a phase correction. This property is used to obtain useful features of the nonstationary signals that make the pattern recognition much simpler in comparison to the wavelet multiresolution analysis. Two neural network configurations are trained with features from the S-transform for recognizing the waveform class. The classification accuracy for a variety of power network disturbance signals for both types of neural networks is shown and is found to be a significant improvement over multiresolution wavelet analysis with multiple neural networks.     

基于多谱图叠加阈值的抑制WVD交叉项的新方法

本文提出了一种抑制Wigner-Ville分布(WVD)交叉项的新方法。 首先对多幅具有不同时-频分辨率的谱图进行叠加, 然后对叠加结果进行阈值处理,确定WVD自项在时频平面的支撑区域。 最后,用该区域的示性函数乘以WVD得到一个新的时频分布。 不同于传统的抑制交叉项的核函数方法, 该方法抑制交叉项的同时, 保持WVD了高时频聚集性。 实验结果表明, 该方法对由多个LFM信号构成的多分量信号和非线性调频信号都非常有效。     

Spatial averaging of time-frequency distributions for signalrecovery in uniform linear arrays

This paper presents a new approach based on spatial time-frequency averaging for separating signals received by a uniform linear antenna array. In this approach, spatial averaging of the time-frequency distributions (TFDs) of the sensor data is performed at multiple time-frequency points. This averaging restores the diagonal structure of the source TFD matrix necessary for source separation. With spatial averaging, cross-terms move from their off-diagonal positions in the source TFD matrix to become part of the matrix diagonal entries. It is shown that the proposed approach yields improved performance over the case when no spatial averaging is performed. Further, we demonstrate that in the context of source separation, the spatially averaged Wigner-Ville distribution outperforms the combined spatial-time-frequency averaged distributions, such as the one obtained by using the Choi-Williams (1989) distribution. Simulation examples involving the separation of two sources with close AM and FM modulations are presented     

Difference equation representation of chirp signals andinstantaneous frequency/amplitude estimation

We present a time-varying coefficient difference equation representation for sinusoidal signals with time-varying amplitudes and frequencies. We first obtain a recursive equation for a single chirp signal. Then, using this result, we obtain time-varying coefficient difference equation representations for signals composed of multiple chirp signals. We analyze these equations using the skew-shift operators. We show that the phases of the poles of the difference equations produce instantaneous frequencies (IF), and the magnitudes are proportional to the ratio of successive values of the instantaneous amplitudes (IA). Then algorithms are presented for the estimation of instantaneous frequencies and instantaneous amplitudes for multicomponent signals composed of chirps using the difference equation representation. The first algorithm we propose is based on the skew-shift operators. Next we derive the conditions under which we can use the so-called frozen-time approach. We propose an algorithm for IF and IA estimation based on the frozen-time approach. Then we propose an automatic signal separation method. Finally, we apply the proposed algorithms to single and multicomponent signals and compare the results with some existing methods     

Investigation and Performance Enhancement of the Empirical Mode Decomposition Method Based on a Heuristic Search Optimization Approach

Empirical mode decomposition (EMD) is a relatively new, data-driven adaptive technique for analyzing multicomponent signals. Although it has many interesting features and often exhibits an ability to decompose nonlinear and nonstationary signals, it lacks a strong theoretical basis which would allow a performance analysis and hence the enhancement and optimization of the method in a systematic way. In this paper, the optimization of EMD is attempted in an alternative manner. Using specially defined multicomponent signals, the optimum outputs can be known in advance and used in the optimization of the EMD-free parameters within a genetic algorithm framework. The contributions of this paper are two-fold. First, the optimization of both the interpolation points and the piecewise interpolating polynomials for the formation of the upper and lower envelopes of the signal reveal important characteristics of the method which where previously hidden. Second, basic directions for the estimates of the optimized parameters are developed, leading to significant performance improvements.