Continuous wavelet transform and exactly solvable model of nonstationary signals

An analytic expression for a continuous wavelet transform (CWT) is obtained for a nonstationary signal with spectral properties varying with time. As a model of the nonstationary signal, a superposition of elementary nonstationary signals is considered, each of which is the product of the Gaussian-shape envelope and an oscillating function. To obtain the CWT, the mother Morlet wavelet is used. The result is compared with the window Fourier transform (Gabor transform).The advantages of the CWT are illustrated by concrete examples. The applications of the exactly solvable model of nonstationary signals for simulating transient processes in physics are considered.     

Gabor transform and continuous wavelet transform for model pulsed signals

A model of a pulsed signal in the form of a superposition of elementary nonstationary signals is considered. The parameters of such a superposition are chosen so that signal amplitude A(t) strongly varies with time. For such a signal, the analytic expression for the Gabor transform (GT) and continuous wavelet transform (CWT) using the mother Morlet wavelet are analyzed. A criterion is proposed for the matched behaviors of GT and CWT with signal amplitude A(t). The advantages of the CWT adaptively selecting the window size over the GT whose explicit form depends on the window size are demonstrated. The proposed method can be used for analysis of many transient stages of time-dependent signals in various branches of physics.     

基于经典小波变换的布里渊光时域反射计光信号处理

布里渊光时域反射计结构的布里渊散射分布式光纤传感器检测的是自发散射光,信号非常微弱,而且频带在几十兆赫兹以上,难以应用普通相干解调方法。针对传感散射光信号特点,提出基于经典(Morlet)小波变换的光相干检测方法。首先采用微波电光调制产生频移参考光和自发布里渊散射光进行相干检测,再应用经典小波变换进行散射光信号的幅度解调,得到信噪比改善的布里渊散射传感光信号。给出了数值模拟和实验数据处理结果,表明经典小波变换能较好地处理布里渊光时域反射计检测信号。并和基于希尔伯特(Hilbert)变换的传感信号处理方法进行了对比,发现此方法优于基于希尔伯特变换的信号处理。     

Harmonic signal extraction from noisy chaotic interference based on synchrosqueezed wavelet transform

For the harmonic signal extraction from chaotic interference, a harmonic signal extraction method is proposed based on synchrosqueezed wavelet transform(SWT). First, the mixed signal of chaotic signal, harmonic signal, and noise is decomposed into a series of intrinsic mode-type functions by synchrosqueezed wavelet transform(SWT) then the instantaneous frequency of intrinsic mode-type functions is analyzed by using of Hilbert transform, and the harmonic extraction is realized. In experiments of harmonic signal extraction, the Duffing and Lorenz chaotic signals are selected as interference signal, and the mixed signal of chaotic signal and harmonic signal is added by Gauss white noises of different intensities.The experimental results show that when the white noise intensity is in a certain range, the extracting harmonic signals measured by the proposed SWT method have higher precision, the harmonic signal extraction effect is obviously superior to the classical empirical mode decomposition method.     

A wavelet approach to fault diagnosis of a gearbox under varying load conditions

Varying load can cause changes in a measured gearbox vibration signal. However, conventional techniques for fault diagnosis are based on the assumption that changes in vibration signal are only caused by deterioration of the gearbox. There is a need to develop a technique to provide accurate state indicator of gearbox under fluctuating load conditions. This paper presents an approach to gear fault diagnosis based on complex Morlet continuous wavelet transform under this condition. Gear motion residual signal, which represents the departure of time synchronously averaged signal from the average tooth-meshing vibration, is analyzed as source data due to its lower sensitiveness to the alternating load condition. A fault growth parameter based on the amplitude of wavelet transform is proposed to evaluate gear fault advancement quantitatively. We found that this parameter is insensitive to varying load and can correctly indicate early gear fault. For a comparison, the advantages and disadvantages of other measures such as kurtosis, mean, variance, form factor and crest factor, both of residual signal and mean amplitude of continuous wavelet transform waveform, are also discussed. The effectiveness of the proposed fault indicator is demonstrated using a full lifetime vibration data history obtained under sinusoidal varying load.     

混沌干扰中基于同步挤压小波变换的谐波信号提取方法

针对混沌干扰背景下多个谐波信号的提取问题, 提出了一种基于同步挤压小波变换(SST)的谐波信号抽取方法. 首先利用SST将混沌信号和谐波信号组成的混合信号分解为不同的内蕴模态类函数, 然后利用Hilbert变换对分离出的内蕴模态类函数进行频率识别, 从中分离出各谐波信号. 以Duffing混沌背景为例, 对混沌干扰下多谐波信号的提取进行了实验分析. 实验结果表明: 对于不同频率间隔的多个谐波分量, 本文方法的提取结果都具有较高的精度, 而且所提方法对高斯白噪声的干扰具有较好的鲁棒性, 综合提取效果优于经典的经验模态分解方法.     

小波变换应用于谐波谱线的噪声滤除与基线校正

红外光谱谐波检测系统中的噪声与基线漂移问题一直是光谱处理的热点,提出一种采用小波变换的Mallet分解算法, 解决谐波检测中各种复杂噪声以及基线漂移的问题。选取适当小波函数及分解层次将谐波曲线中含有的噪声和基线漂移与有用信号分解到不同频带;分析频带信息,设定一个检测信息频带, 应用阈值处理及系数置零的方法使频率处于此频带的信息保留下来。小波变换方法可以在一次分解与重构过程中同时去除谐波信号的噪声与基线的双重干扰,从而将谐波信号有效地测量出来。实验证明,应用小波变换进行谐波校正的方法可应用于不同的谐波检测系统,具有普遍适用性。     

基于指数尺度间隔连续小波变换的相位提取算法

只需要一幅调制图像的光栅投影测量方法主要有傅里叶变换轮廓术(FTP)、小波变换轮廓术(WTP)等。采用基于指数尺度间隔的连续小波变换与重构方法,提取调制图像的瞬时相位。针对指数尺度间隔连续小波变换,指出了足够大的噪声能够改变小波变换脊的位置,并且该脊向上移动的概率最大。因此,为了重构载频信号,选择脊及其紧邻的较大的那个尺度所对应的小波系数,采用灰度图像阈值分割中最大类间方差法(OTSU),剔除掉幅值较小的系数;针对斑点噪声的影响,对OTSU算法的结果进行了修正;使用修正后的系数集合重构载频信号,并计算该信号的瞬时相位。理论分析和实验结果表明算法有效且具有稳健性。     

Wavelet correlations of nonstationary signals

Two approaches to the analysis of nonstationary random signals are proposed and studied. The first approach is based on the adaptive Morlet wavelet that allows variations in time and frequency resolution of signals using an auxiliary control parameter. The second approach is related to the application of double correlation function that represents correlation of continuous wavelet transforms of two signals calculated in time and frequency domains. The advantages of the proposed correlation function in comparison with alternative correlation functions, in particular, analysis of both time and frequency correlations of nonstationary signals are outlined. Applications of the proposed approaches in the analysis of various transient processes in physics are discussed.     

Calculation of instantaneous radiation characteristics using the wavelet transform

To overcome the disadvantage of short time Fourier transform, the wavelet transform, which is very promising for analyzing non-stationary signals, was introduced by J. Morlet. In this paper, a new numerical calculation method for estimating the instantaneous radiation pattern by using the harmonic wavelet transform for a rectangular vibrating plate is presented.     

瞬态信号的小波变换波达方向估计

针对瞬态信号方位估计问题,提出了基于连续小波变换的多重信号分类测向算法(CWT_MUSIC)。首先由信号特征确定小波尺度参数,构造Morlet小波,对信号进行小波变换,利用获得的小波变换系数建立多分辨时频阵列信号模型,并据此模型设计基于子空间的MUSIC算法以实现瞬态信号的波达方向估计;然后对该算法的多分辨与误差性能进行分析,最后仿真实验和实际爆炸试验验证了所提出的CWT_MUSIC算法能有效地提高空间谱的分辨率和DOA估计性能。     

Adaptive harmonic wavelet transform with applications in vibration analysis

An adaptive harmonic wavelet transform is developed by taking advantage of the flexibility of the generalized harmonic wavelets. It first constructs a partition tree, which contains a great number of disjoint partitions of the frequency axis of a signal with each corresponding to an orthogonal harmonic wavelet basis. Then it searches the tree for the partition to represent the signal most sparsely. Since the corresponding basis is adapted to the composition of the signal, the transform can well reveal its characteristics. This is demonstrated with analysis examples of some simulated and vibration signals as well as comparisons with the conventional orthogonal harmonic wavelet transforms and wavelet packet transform.     

Generalized Gabor expansion associated with linear canonical transform series

The Gabor expansion (GE), which maps the time domain signal into the joint time and frequency domain, has been recognized as very useful for signal processing. However, sinusoidal analysis used in the traditional GE is not appropriate for a compact representation for chirp-type signals. In this paper, a generalized Gabor expansion (GGE) is proposed in order to rectify the limitations of the GE, the proposed expansion not only inherits the advantage of GE, but also has the capability of signal representations in the linear canonical transform (LCT) domain which is similar to the LCT. Basis functions of the proposed expansion are obtained via LCT basis. Compared with the traditional GE, the GGE can offer signal representations on a general, non-rectangular time–frequency plane tiling. Besides, the completeness and biorthogonality conditions of the GGE are derived.     

Gap detection and the auditory filter: phase effects using sinusoidal stimuli

Psychometric functions were determined for the detection of temporal gaps in sinusoidal signals at center frequencies between 0.2 and 2.0 kHz. A continuous notched-noise masker was used to restrict listening to the signal frequency region. The gap always started when the signal was at a positive-going zero crossing. There were three different conditions for the starting phase of the signal at the termination of the gap. In the standard-phase condition the signal restarted at a positive-going zero crossing, in the reversed-phase condition at a negative-going zero crossing, and in the preserved-phase condition at the phase the signal would have had if the gap had not been present. In the standard-phase and reversed-phase conditions the psychometric functions were nonmonotonic, showing oscillations with a period equal to that of the signal; maxima in the functions for the standard-phase condition coincided with minima in the functions for the reversed-phase condition, and vice versa. In the preserved-phase condition the psychometric functions were monotonic and the 75% points were roughly independent of center frequency, having a value of about 5 ms. The general form of the results can be modeled by a filter bank followed by a square-law device and a temporal integrator, but good agreement between the data and the model could not be attained across the whole range of gap durations. The deviations between data and model suggest that subjects are sensitive to the brief transitions in phase (or, equivalently, in frequency) in some conditions.     

基于连续小波变换FTIR鉴定青葙子及鸡冠子的研究

借助OMNI采样器,应用傅里叶变换红外光谱方法直接、快速、准确地测定青葙子及其伪品鸡冠子的红外光谱。利用连续小波变换对真伪品的红外光谱进行放大处理,以有效突出真伪品间的红外光谱差异程度,从而提高鉴定正确率。采用对信号奇异性具有良好探测能力的Morlet小波做小波母函数,对青葙子真伪品的红外光谱进行若干尺度的一维连续小波变换,在各个尺度下观察青葙子的真伪品的红外光谱的差异程度,从中选择一个差异程度最为明显的尺度来区分青葙子真伪品。结果表明：基于连续小波变换的傅里叶红外光谱分析方法能够有效鉴别同科同属不同种植物中药材。     

基于小波变换的堆积啁啾光脉冲时频分析

以复Morlet函数作为小波母函数推导了2M路线性啁啾高斯脉冲堆积形成的整形脉冲的时间分辨能量谱密度解析表达式。通过小波变换获得了堆积啁啾脉冲功率和瞬时波长随时间的分布; 堆积啁啾高斯脉冲功率随时间呈现振荡变化, 瞬时波长在功率接近零的极小值点发生跳变; 啁啾的增大会加剧功率的时域振荡和瞬时波长的跳变; 瞬时波长的跳变仅发生在功率接近零的极小值附近, 脉冲的能量仍然主要集中在由基元脉冲啁啾决定的波长范围内。同时, 研究表明脉冲重叠因子的微小变化会极大地影响整形脉冲峰值功率与等效平均功率之比和包络平坦度, 且两者不能同时达到最佳, 需折衷选取。     

On the Effects of Gaps and Uses of Approximation Functions on the Time-Scale Signal Analysis: A Case Study Based on Space Geophysical Events

The presence of gaps is quite common in signals related to space science phenomena. Usually, this presence prevents the direct use of standard time-scale analysis because this analysis needs equally spaced data; it is affected by the time series borders (boundaries), and gaps can cause an increase of internal borders. Numerical approximations can be used to estimate the records whose entries are gaps. However, their use has limitations. In many practical cases, these approximations cannot faithfully reproduce the original signal behaviour. Alternatively, in this work, we compare an adapted wavelet technique (gaped wavelet transform), based on the continuous wavelet transform with Morlet wavelet analysing function, with two other standard approximation methods, namely, spline and Hermite cubic polynomials. This wavelet method does not require an approximation of the data on the gap positions, but it adapts the analysing wavelet function to deal with the gaps. To perform our comparisons, we use 120 magnetic field time series from a well-known space geophysical phenomena and we select and classify their gaps. Then, we analyse the influence of these methods in two time-scale tools. As conclusions, we observe that when the gaps are small (very few points sequentially missing), all the methods work well. However, with large gaps, the adapted wavelet method presents a better performance in the time-scale representation. Nevertheless, the cubic Hermite polynomial approximation is also an option when a reconstruction of the data is also needed, with the price of having a worse time-scale representation than the adapted wavelet method.     

基于实时小波变换信号处理的相干检测布里渊光时域反射计

布里渊光时域反射计检测信号较微弱的自发布里渊散射光,其布里渊频移只有11 GHz左右.相干检测是一较好的方法,但其接收信号是一宽带的幅度调制信号,难以直接解调.采用实时小波变换对传感信号进行处理,设计了布里渊光时域反射计传感器实验系统,实现了25 km的分布式温度传感,在5 m的空间分辨率和1 min的响应时间下,达到40 με的应变分辨率.     

Emerging applications of wavelets: A review

Although most of its popular applications have been in discrete-time signal processing for over two decades, wavelet transform theory offers a methodology to generate continuous-time compact support orthogonal filter banks through the design of discrete-time finite length filter banks with multiple time and frequency resolutions. In this paper, we first highlight inherently built-in approximation errors of discrete-time signal processing techniques employing wavelet transform framework. Then, we present an overview of emerging analog signal processing applications of wavelet transform along with its still active research topics in more matured discrete-time processing applications. It is shown that analog wavelet transform is successfully implemented in biomedical signal processing for design of low-power pacemakers and also in ultra-wideband (UWB) wireless communications. The engineering details of analog circuit implementation for these continuous-time wavelet transform applications are provided for further studies. We expect a flurry of new research and technology development activities in the coming years utilizing still promising and almost untapped analog wavelet transform and multiresolution signal representation techniques.