Double wavelet transform of frequency-modulated nonstationary signal

A mathematical model is proposed for a frequency-modulated signal in the form of a system of Gaussian peaks randomly distributed in time. An analytic expression is obtained for continuous wavelet transform (CWT) of the model signal. For signals with time-varying sequence of peaks, the main ridge of the skeleton characterized by frequency ν _max ^MFB (t) is analyzed. The value of ν _max ^MFB (t) is determined for any instant t from the condition of the CWT maximum in the spectral range of the main frequency band (MFB). Double CWT of function ν _max ^MFB (t) is calculated for a frequency-modulated signal with a transition regions of smooth frequency variation (trend) as well as with varying frequency oscillations relative to the trend. The duration of transition periods of the signal is determined using spectral integrals E _ν(t). The instants of emergence and decay of low-frequency spectral components of the signal are determined. The double CWT method can be used for analyzing cardiac rhythms and neural activity, as well as nonstationary processes in quantum radio physics and astronomy.     

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.     

Time-domain quantification of multiple-quantum-filtered (23)Na signal using continuous wavelet transform analysis

The application of continuous wavelet transform (CWT) analysis technique is presented to analyze multiple-quantum-filtered (MQF) (23)Na magnetic resonance spectroscopy (MRS) data. CWT acts on the free-induction-decay (FID) signal as a time-frequency variable filter. The signal-to-noise ratio (SNR) and frequency resolution of the output filter are locally increased. As a result, MQF equilibrium longitudinal magnetization and the apparent fast and slow transverse relaxation times are accurately estimated. A developed iterative algorithm based on frequency signal detection and components extraction, already proposed, was used to estimate the values of the signal parameters by analyzing simulated time-domain MQF signals and data from an agarose gel. The results obtained were compared to those obtained by measurement of signal height in frequency domain as a function of MQF preparation time and those obtained by a simple time-domain curve fitting. The comparison indicates that the CWT approach provides better results than the other tested methods that are generally used for MQF (23)Na MRS data analysis, especially when the SNR is low. The mean error on the estimated values of the amplitude signal and the apparent fast and slow transverse relaxation times for the simulated data were 2.19, 6. 63, and 16.17% for CWT, signal height in frequency domain, and time-domain curve fitting methods, respectively. Another major advantage of the proposed technique is that it allows quantification of MQF (23)Na signal from a single FID and, thus, reduces the experiment time dramatically.     

Double-beam weighting in Fourier transform spectroscopy

This paper describes a double-beam weighting system in slow-scanning Fourier transform spectroscopy by use of a photo-conductive detector with fast response. It uses the method of amplitude modulation. The intensity of reference beam (A) and that of sample beam (B) are obtained by summing and subtracting the signals (A + B) and (A ? B), which are obtained in this form from a modulated signal by means of an electronic system. Merits arising from using a detector with fast response in double-beam weighting are also discussed. The operational performance of the system has been evaluated and it has been found that the spectra thus obtained are reproducible because they are kept within 3% bound and each measured value coincides with the averaged trace of single-beam measurements within ±2%.     

Time-Domain Quantification of Multiple-Quantum-Filtered Na Signal Using Continuous Wavelet Transform Analysis

The application of continuous wavelet transform (CWT) analysis technique is presented to analyze multiple-quantum-filtered (MQF) ²³Na magnetic resonance spectroscopy (MRS) data. CWT acts on the free-induction-decay (FID) signal as a time-frequency variable filter. The signal-to-noise ratio (SNR) and frequency resolution of the output filter are locally increased. As a result, MQF equilibrium longitudinal magnetization and the apparent fast and slow transverse relaxation times are accurately estimated. A developed iterative algorithm based on frequency signal detection and components extraction, already proposed, was used to estimate the values of the signal parameters by analyzing simulated time-domain MQF signals and data from an agarose gel. The results obtained were compared to those obtained by measurement of signal height in frequency domain as a function of MQF preparation time and those obtained by a simple time-domain curve fitting. The comparison indicates that the CWT approach provides better results than the other tested methods that are generally used for MQF ²³Na MRS data analysis, especially when the SNR is low. The mean error on the estimated values of the amplitude signal and the apparent fast and slow transverse relaxation times for the simulated data were 2.19, 6.63, and 16.17% for CWT, signal height in frequency domain, and time-domain curve fitting methods, respectively. Another major advantage of the proposed technique is that it allows quantification of MQF ²³Na signal from a single FID and, thus, reduces the experiment time dramatically.     

Analysis on fringe pattern demodulation by use of 2-D CWT

We discuss the wavelet transform profilometry based on the continuous wavelet transform technique as viewed from frequency analysis. We deduce the expression of one-dimensional (1-D) and two-dimensional (2-D) wavelet transform in frequency domain and analyze their characteristics in the application of demodulating the fringe patterns. We also compare 1-D CWT and 2-D CWT in demodulating the oblique fringe patterns with dual carrier frequency components. When oblique fringe patterns are processed, the direction normal to the grating line and x axis or y axis is not identical. By 1-D CWT, in which wavelet transform is carried out row by row, we cannot obtain the most similarity between local signal and the wavelet functions with different dilation values. While a fan 2-D continuous wavelet transformation can deal with the fringe pattern as a 2-D unit as well as has multi-directions, its advantage is that it can be used to exact the information in the spatial direction. However, its spatial localization ability is not very good, which leads that it is not suitable for demodulating the fringe patterns with high phase variation. Computer simulations and experiments have verified our analysis.     

Optical pulse compression using the temporal Radon-Wigner transform

The temporal Radon-Wigner transform (RWT), which is the squared modulus of the fractional Fourier transform (FRT) for a varying fractional order p, is here employed as a tool for pulse compression applications. To synthesize the compressed pulse, a selected FRT irradiance is optically produced employing a photonic device that combines phase modulation and dispersive transmission. For analysis purposes, the complete numerical generation of the RWT with 0 < p < 1 is proposed to select the value of p required for pulse compression. To this end, the amplitude and phase of the signal to be processed should be known. In order to obtain this information we use a method based on the recording of two different FRT irradiances of the pulse. The amplitude and phase errors of the recovered signal, which are inherent to the recording process, are discussed in connection with the RWT production. Numerical simulations were performed to illustrate the implementation of the proposed method. The technique is applied to compress signals commonly found in fiber optic transmission systems, such as chirped gaussian pulses, pulses distorted by second and third-order dispersion and nonlinear self-modulated pulses.     

Moment-wavelet quantization: a first principles analysis of quantum mechanics through continuous wavelet transform theory

The space of polynomials is invariant under affine maps. This suggests that a moment based analysis can facilitate a first principles incorporation of continuous wavelet transform (CWT) theory into quantum mechanics. We show that this is indeed the case for a large class of Hamiltonians and mother wavelet functions. We establish the equivalence between moment quantization (MQ) and CWT. By so doing, we clearly demonstrate the inherent multiscale structure of MQ analysis with regards to determining the physical energies and corresponding wavefunctions.     

Fractional Gabor transform

A fractional Gabor transform (FRGT) is proposed. This new transform is a generalization of the conventional Gabor transform (GT) based on the Fourier transform to the windowed fractional Fourier transform (FRFT). The FRGT provides analyses of signals in both the real space and the FRFT frequency domain simultaneously. The space-FRFT frequency pattern can be rotated as the fractional order changes. The FRGT has an additional freedom, compared with the conventional GT, i.e., the transform order. The FRGT may offer a useful tool for guiding optimal filter design in the FRFT domain in signal processing.     

Defect detection and location in switch rails by acoustic emission and Lamb wave analysis: A feasibility study

An acoustic emission (AE) based approach is proposed in this study to identify and locate newly initiated defects or the propagation of existing defects in railroad switch rails. Defect-induced AE signals are identified through frequency analysis, as frequencies of these signals are much higher than those induced by structural vibration. Continuous wavelet transform (CWT) is employed to analyze the Lamb wave dispersion of the detected signal, so that two characteristic points can be selected on the CWT contour map to locate the defect. Using this approach, defects in a damaged switch rail can be located using a single sensor.     

Phase retrieval of speckle fringe pattern with carriers using 2D wavelet transform

A novel two-dimensional (2D) Gabor continuous wavelet transform (CWT) for phase retrieval and fringe filtering of speckle fringe patterns with spatial carriers is proposed. Theoretical analysis of 2D Gabor CWT is presented and results are compared with advanced fan 2D CWT using both the computer simulated and experimental speckle fringe patterns. It is shown that noise reduction by 2D Gabor CWT demonstrates better results than that of the advanced fan 2D CWT. Two-dimensional Gabor CWT is also compared with 2D Fourier transform and results show that 2D Gabor CWT algorithm has better noise immunity.     

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。     

Phase recovery from fringe patterns using the continuous wavelet transform

Interferometry is well established as an optical technique in which a measurand is encoded as the phase of a periodically varying intensity pattern. In view of the inherent accuracy of interferometry, many methods have been developed to retrieve the phase from images of the fringe pattern. Our focus in this paper is one such technique—the continuous wavelet transform (CWT). We begin by reviewing the CWT and the space–spatial–frequency localisation properties of wavelets. We show that a path which follows the maximum modulus of the CWT (the wavelet ridge) gives the instantaneous fringe frequency as a function of spatial displacement. The phase is automatically and trivially obtained, without discontinuities, by integration. Examples of practical wavelets are given and algorithms to isolate the wavelet ridge reviewed.     

小波变换在太赫兹三维成像探测内部缺陷中的应用

采用Syn View Head 300对内部有胶和空气孔的样件进行了太赫兹二维扫描(xy轴方向),系统通过线性调频连续波技术得到样件内部的三维信息.检测薄层时,由于太赫兹源的波长在亚毫米量级,薄层的上下表面反射峰相距太近而难以辨别.为了提高太赫兹探测的纵向分辨率,采用小波变换对探测信号进行处理,对小波系数进行三维重构,获得的三维小波系数图像比原始三维探测信号更加精确.该方法有效提高了太赫兹成像的纵向检测精度,纵向分辨率可达1 mm.     

基于小波变换的混沌信号相空间重构研究

应用小波变换和非线性动力学方法研究了混沌信号在相空间中的行为，指出混沌时间序 列的小波变换实质上是在重构的相空间中，混沌吸引子向小波滤波器向量所张的空间中的投 影，与Packard等人提出的相空间重构方法本质上是一致的.实验结果表明，混沌信号经过 小波变换后，吸引子轨迹与原有轨迹具有相似的结构，同时，系统的关联维数、Kolmogorov 熵等非线性不变量仍然得到保留.这些结果表明，利用小波变换研究混沌信号是有效的. 关键词： 小波变换 相空间重构 混沌信号 脑电信号     

AN ACOUSTIC DECAY MEASUREMENT BASED ON TIME-FREQUENCY ANALYSIS USING WAVELET TRANSFORM

In acoustic decay measurement using the third-octave band pass filter, it is known that an inevitable experimental error is produced by “ringing” at the tail part of the impulse response of the third-octave band pass filter. This ringing gives rise to distortion of the decay curve. In order to reduce this error and to obtain an acceptable acoustic decay curve, it has been recommended that the product of the 3 dB bandwidth B of the third-octave band pass filter and the reverberation time T₆₀ of the room under test be at least 16. For a listening room having short reverberation time and at the low-frequency band with narrow bandwidth, the decay curve cannot, therefore, be measured reliably by using the third-octave band pass filter. In this paper, the continuous wavelet transform (CWT) has been proposed to determine accurately the decay curve with a low value of BT₆₀. The CWT decomposes an acoustic decay signal into time-frequency domain using the third-octave band wavelet filter bank. When the CWT is applied to the measurement of an acoustic decay curve, it is found that the requirement BT₆₀>16 can be replaced by the replacementBT₆₀ >4.     

Secret shared multiple-image encryption based on row scanning compressive ghost imaging and phase retrieval in the Fresnel domain

A multiple-image encryption method is proposed that is based on row scanning compressive ghost imaging, (t, n) threshold secret sharing, and phase retrieval in the Fresnel domain. In the encryption process, after wavelet transform and Arnold transform of the target image, the ciphertext matrix can be first detected using a bucket detector. Based on a (t, n) threshold secret sharing algorithm, the measurement key used in the row scanning compressive ghost imaging can be decomposed and shared into two pairs of sub-keys, which are then reconstructed using two phase-only mask (POM) keys with fixed pixel values, placed in the input plane and transform plane 2 of the phase retrieval scheme, respectively; and the other POM key in the transform plane 1 can be generated and updated by the iterative encoding of each plaintext image. In each iteration, the target image acts as the input amplitude constraint in the input plane. During decryption, each plaintext image possessing all the correct keys can be successfully decrypted by measurement key regeneration, compression algorithm reconstruction, inverse wavelet transformation, and Fresnel transformation. Theoretical analysis and numerical simulations both verify the feasibility of the proposed method.     

Monte Carlo simulation of film growth in a phase separating binary alloy model

Using a kinetic Monte Carlo method, we simulate binary film (A_0.5B_0.5/A) growth on an L×L square lattice with the focus on the domain growth behaviour. We compute the average domain area, A(t), as a measure of domain size. For a sufficiently large system, we find that A(t) grows with a power law in time with A(t)∼t^2/3 after the initial transient time. This implies that the dynamic exponent for domain growth with non-conserved order parameter is z=3, a value which was theoretically predicted for the conserved order parameter case. Further analysis reveals that such a power-law behaviour emerges because the order parameter is approximately conserved after the early stage of growth.     

Optimal wavelet filter for suppression of coherent noise with an application to seismic data

Wavelet analysis is combined with the Karhunen-Loève (KL) transform into an innovative hybrid method for locally filtering coherent noise. In applying our method, the original time series is first decomposed with wavelet transform, the scales more contaminated with noise are reduced by an attenuation factor A_f, and the signal is reconstructed using the inverse wavelet transform. Then the KL transform is applied to the reconstructed signal and the behavior of the first energy modes is analyzed as a function of A_f. The point corresponding to a minimum in the first mode is identified with the maximum extraction of the coherent noise. Our methodology is applied with success to seismic data with the aim of locally extracting the relevant coherent noise, namely the ground roll noise. The procedure can be easily extended to other situations where an undesirable signal is associated with a specific set of energy modes.