The role of functional data analysis for instantaneous frequency estimation

This paper proposes a method for estimating the instantaneous frequency of a nonstationary signal; this method is based on a combination of empirical mode decomposition and functional data analysis. The proposed method incorporates a basis expansion technique for a functional data into time-varying phase derived by empirical mode decomposition and Hilbert transform, which provides a stable instantaneous frequency function. The superiority of the proposed method for instantaneous frequency estimation is demonstrated by various simulation studies. The analysis of multicomponent signals by the proposed method is also discussed. Furthermore, it is shown that the proposed method is highly effective for identifying groups (clusters) of nonstationary signals on the basis of the instantaneous frequency information.     

基于分数阶Fourier变换的结构瞬时频率识别北大核心CSCD

为识别时变信号的瞬时频率,由分数阶Fourier变换定义推导出了一般信号的频率与单一变量旋转角度α的关系式,从理论上解释了分数阶Fourier变换本质上是一种普通Fourier变换结合伸缩平移窗的算法,进而在分数阶Fourier域建立了非平稳信号瞬时频率的一般表达式,实现了结构瞬时频率的识别.采用任意非线性调频信号仿真算例和三自由度有阻尼时变结构系统的数值算例对提出的方法进行了比较分析.结果表明,该文提出的方法与理论值吻合良好,并具有一定的抗噪性,验证了方法的可靠性和实用性,可以应用于时变结构瞬时频率的识别.     

The structure of instantaneous frequencies of periodic analytic signals

In this paper, the structure of analytic signals is investigated by means of the relation between analytic signals and functions in the Hardy space. It is shown that an analytic signal is made up of two parts, one depending on the amplitude of the signal and another on the boundary value of an inner function. Based on this result, properties of the instantaneous frequencies of these two parts are studied, and it is found that negative instantaneous frequencies are caused by the amplitude of a signal. Finall...     

Diagnostics-based Approach to Vibration Surveillance during High Speed Milling of Flexible Details

The paper concerns vibration surveillance during ball end milling of curved flexible details and considering assurance of parameters of a modal subsystem. Here is explained dynamic analysis of non-stationary vibrating system, from which are separated subsystems: modal, structural and connective. Minimising vibration level by matching the spindle speed to optimal phase shift has been employed. Two alternatives of the approach have been developed. The first one is based on generalised condition of optimum spindle speed and relates to diagnosis of chatter vibration with only one detected frequency. The second approach concerns a few harmonic components having different frequencies. Obtained results of production experiments proved a meaning of the approaches. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Signal moments for the short‐time Fourier transform associated with Hardy–Sobolev derivatives

The short‐time Fourier transform has been shown to be a powerful tool for non‐stationary signals and time‐varying systems. This paper investigates the signal moments in the Hardy–Sobolev space that do not usually have classical derivatives. That is, signal moments become valid for non‐smooth signals if we replace the classical derivatives by the Hardy–Sobolev derivatives. Our work is based on the extension of Cohen's contributions to the local and global behaviors of the signal. The relationship of the moments and spreads of the signal in the time, frequency and short‐time Fourier domain are established in the Hardy–Sobolev space. Copyright © 2014 John Wiley & Sons, Ltd.     

Analysis of the Intrinsic Mode Functions

The Empirical Mode Decomposition is a process for signals which produces Intrinsic Mode Functions from which instantaneous frequencies may be extracted by simple application of the Hilbert transform. The beauty of this method to generate redundant representations is in its simplicity and its effectiveness. Our study has two objectives: first, to provide an alternate characterization of the Intrinsic Mode components into which the signal is decomposed and, second, to better understand the resulting polar representations, specifically the ones which are produced by the Hilbert transform of these intrinsic modes.     

A discrete Fourier transform based on Simpson's rule

Fourier analysis plays a vital role in the analysis of continuous‐time signals. In many cases, we are forced to approximate the Fourier coefficients based on a sampling of the time signal. Hence, the need for a discrete transformation into the frequency domain giving rise to the classical discrete Fourier transform. In this paper, we present a transformation that arises naturally if one approximates the Fourier coefficients of a continuous‐time signal numerically using the Simpson quadrature rule. This results in a decomposition of the discrete signal into two sequences of equal length. We show that the periodic discrete time signal can be reconstructed completely from its discrete spectrum using an inverse transform. We also present many properties satisfied by this transform. Copyright © 2012 John Wiley & Sons, Ltd.     

Main frequency band of blast vibration signal based on wavelet packet transform

As a key parameter in blasting safety criteria, accurately describing the frequency's characteristics is of practical significance. Due to the deficiency of Fourier transform in the analysis of non-periodic and non-stationary signals, this study defined a wavelet frequency domain parameter, referred to as a main frequency band. A computational method associated with the wavelet packet transform is also proposed. To verify the feasibility of main frequency band and the proposed computational method in describing blasting frequency characteristics, an application is exemplified with field blasting vibration signals monitored in a mine. The effects of explosive charge and distance on main frequency band distribution characteristics are also studied. Results show that the main frequency band based on the computational method is a sensitive, accurate and efficient frequency parameter; it can accurately describe the frequency characteristics of blasting signals and effectively overcome the drawbacks in Fourier transform. When the explosive charge is constant, the span of main frequency reduces as a whole as the distance increases, and the frequency domain energy of blast vibration signals are concentrated mainly in the low-frequency range. When the distance is constant, the peak energy of blast vibration signals increase with the increase of explosive charge, without obvious change in main frequency band. To avoid the effects of interferences on frequency characteristics, the least square method is employed to eliminate signal trend components, and the wavelet threshold method with a hard thresholding function and the Birge–Massart strategy is applied in denoising.     

Modeling of non-stationary vibration signals based on the modified Kronecker sequences

The paper establishes a representation model for non-stationary random vibration signals based on the modified Kronecker sequences. The modified Kronecker sequence constructed via generalizing golden ratio is one of the special types of low discrepancy sequences which have better dimensional projections [1]. The actual modeling and simulation of non-stationary random data is more suitable for seismological signals and not for the vehicle vibrations [2, 3]. Under these circumstances, this paper presents a new algorithm for finding the modified Kronecker sequences in order to generate non-stationary vehicle vibration signals which mostly withhold the amplitude-frequency-time distribution of the sample signal. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phaseless inverse discrete Hilbert transform and determination of signals in shift‐invariant space

In this paper, we first address the uniqueness of phaseless inverse discrete Hilbert transform (phaseless IDHT for short) from the magnitude of discrete Hilbert transform (DHT for short). The measurement vectors of phaseless IDHT do not satisfy the complement property, a traditional requirement for ensuring the uniqueness of phase retrieval. Consequently, the uniqueness problem of phaseless IDHT is essentially different from that of the traditional phase retrieval. For the phaseless IDHT related to compactly supported functions, conditions are given in our first main theorem to ensure the uniqueness. A condition on the step size of DHT is crucial for the insurance. The second main theorem concerns the uniqueness related to noncompactly supported functions. It is not the trivial generalization of the first main theorem. Our third main result is on the determination of signals in shift‐invariant spaces by phaseless IDHT. Note that the measurements used for the determination are the DHT magnitudes. They are the approximations to the Hilbert transform magnitudes. Recall that for the existing methods of phaseless sampling (a special phase‐retrieval problem), to determine a signal depends on the exact measurements but not the approximative ones. Therefore, our determination method is essentially different from the traditional phaseless sampling. Numerical simulations are conducted to examine the efficiency of phaseless IDHT and its application in determining signals in spline Hilbert shift‐invariant space.     

Convex Optimization approach to signals with fast varying instantaneous frequency

Motivated by the limitation of analyzing oscillatory signals composed of multiple components with fast-varying instantaneous frequency, we approach the time-frequency analysis problem by optimization. Based on the proposed adaptive harmonic model, the time-frequency representation of a signal is obtained by directly minimizing a functional, which involves few properties an “ideal time-frequency representation” should satisfy, for example, the signal reconstruction and concentrative time-frequency representation. FISTA (Fast Iterative Shrinkage-Thresholding Algorithm) is applied to achieve an efficient numerical approximation of the functional. We coin the algorithm as Time-frequency bY COnvex OptimizatioN (Tycoon). The numerical results confirm the potential of the Tycoon algorithm.     

Employing applied mathematics to expand the bandwidth of heterodyne carrier signals with a small phase modulation index

Frequency modulation (fm) theory states that heterodyne carrier signals cannot be frequency modulated with modulation frequencies f_m higher than the carrier frequency f_c subtracted by the maximum frequency deviation f_d,max. Thus, f_m ? f_c − f_d,max is considered as maximum limit to obtain the correct modulation signal from the carrier signal by fm demodulation. This paper proves mathematically that this limit can be breached for small phase modulation indices. The result is advanced to a new algorithm to demodulate heterodyne carrier signals with a modulation frequency of up to twice the carrier frequency if a small phase modulation index M can be assumed. This assumption is valid in heterodyne laser interferometers for ultrasonic testing where the phase modulation of a detector signal is originated by vibration amplitudes much smaller than the wavelength of the laser. Simulations of the demodulation demonstrate the functioning of the algorithm. The employment of the presented results in an interferometric system demonstrates the impact in metrology instruments for vibration measurements at ultra-high frequencies. Therefore, the influence of the presented algorithm to the measurement uncertainty of interferometric systems is also derived in this paper.     

基于S变换的地震信号降噪方法研究

S变换在分析非平稳信号时能有效地反映出频率随时间的变化,但由于其窗函数是固定不变的,在实际中应用受到了限制.从基本理论出发,推导出一种改进的S变换形式,并对合成信号分别进行傅立叶变换、s变换和改进的S变换,通过对比发现:改进的s变换方法能够更好地分辨非平稳信号的频率特性,比S变换具有更高的分辨率.最后应用改进的s变换方法对地震背景噪声数据进行了去噪处理,取得了较好的结果.     

Fractional time-varying energy distributions of ECG signals

We introduce here fractional Cohen class of time-frequency distributions (FCCTFDs) containing fractional modulations which is kernel of fractional Fourier transform (FFT). The fractional modulation depends on angular parameter α and can be interpreted as a rotation by an angle α in time-frequency plane. This distribution promotes to track time-variant energy of a biological signals and represents it in time-frequency domain. It uses the fractional ambiguity function (FAF) of signal multiplied by a suitable kernel which is designed for the biological signals generally having multi-non-stationary components. This result improves and generalizes some of the previous time-frequency distributions derived in the literature.     

Improvement of separability of time series in singular spectrum analysis using the method of independent component analysis

The separation of signal components is an important problem of time-series analysis. For example, the solution of this problem allows one to extract a trend and to separate harmonic signals with different frequencies. In the paper, the modification of the singular spectrum analysis (SSA) method is considered for improving the separability of time-series components. The new method is called SSA-AMUSE, since it is based on the AMUSE method, which is used to apply independent component analysis to signal separation. The suggested modification weakens the conditions of the so-called strong separability and, thus, improves the quality of the separation of time-series components by comparing similar methods. The paper contains proof of the algorithm, as well as the conditions of separability for the considered modification. Besides the exact separability, the asymptotic separability is also considered. The separability conditions are applied to the case of two harmonic time series. It appears that separability by SSA-AMUSE does not depend on the amplitudes of the separated harmonics, while the Basic SSA method requires different amplitudes. A numerical example demonstrates the advantage of the SSA-AMUSE method compared with a similar modification.     

Analysis of brain electrical topography by spatio-temporal wavelet decomposition

Currently, the Functional Magnetic Resonance Imaging (fMRI), Positron Emission Tomography (PET), and Electroencephalography (EEG) recordings are the major techniques of neuroimaging. The EEG with its highest temporal resolution is still a crucial measurement for localization of activities arising from the electrical behaviour of the brain. A scalp topographic map for an EEG may be a superposition of several simpler subtopographic maps, each resulting from an individual electrical source located at a certain depth. Furthermore, this source may have a temporal characteristic as an oscillation or a rhythm that extends in a certain time window which has been a basis of assumption for the time-frequency analysis methods. A method for the spatio-temporal wavelet decomposition of multichannel EEG data is proposed which facilitates the localization of electrical sources separate and/or overlapping on a continuum of time, frequency and space domains. The subtopographic maps asociated with each of these individual components are then used in the MUSIC source localization algorithm. The validations are performed on simulated EEG data. Spatio-temporal wavelet decomposition as a preprocessing method improves the source localization by simplifying the topographic data formed by the superposition of EEG generators, having possible combinations of temporal, frequency and/or spatial overlappings. Spatio-temporal analysis of EEG will help enhance the accuracy of dipole source reconstruction in neuroimaging.     

Orthonormal bases with nonlinear phases

For adaptive representation of nonlinear signals, the bank M{\cal M} of real square integrable functions that have nonlinear phases and nonnegative instantaneous frequencies under the analytic signal method is investigated. A particular class of functions with explicit expressions in M{\cal M} is obtained using recent results on the Bedrosian identity. We then construct orthonormal bases for the Hilbert space of real square integrable functions with the basis functions from M{\cal M}.     

基于Hilbert变换的LFM检测与估计

LFM(线性调频)信号是一类重要的非平稳信号,其完全被初始频率和调频斜率两个参量表征,而LFM信号的检测与估计问题是信号处理中最为重要的研究热点之一.由于调频信号在时频平面内有较好的聚集性,通常使用时频分析的方法对其进行检测和估计.线性正则变换是经典时频分布的广义形式,对LFM信号具有很好的能量聚集特性,在现有的线性正则域Hilbert变换的基础上,提出了一种不需要谱峰搜索而快速检测LFM信号和估计其参数的方法,并且通过仿真实例验证了所提出方法的优越性.     

功率放大器非线性特性及预失真建模

信号的功率放大器是电子通信系统的关键器件之一,功放的输出信号相对于输入信号可能产生非线性变形,这将带来无益的干扰信号,研究其机理并采取措施改善,具有重要意义.通过利用无记忆非线性功放和记忆非线性功放的实测数据用数学方法对其分别进行建模,而后使用前置预失真器的方法改善功放的非线性特性,并对其中预失真器的建模做了研究,采用间接学习型结构构建预失真器模型.仿真结果显示功放非线性模型结合预失真器模型能够很好地逼近实际情况,并且能很好地抑制带外频谱扩展.