分数阶小波变换

小波变换是对信号时域-频域(Fourier域)的多分辨率分析,也可看作是一种Fourier域伸缩带通滤波.分数阶Fourier变换是对传统Fourier变换的推广,对信号分析处理有更大的灵活性,为了将多分辨率分析理论推广到时域-广义频域(分数阶Fourier域),提出了一种分数阶小波变换,分析了分数阶小波变换在广义频域伸缩带通滤波特性,分析信号时的时域-广义频域平面的多分辨率分析网格划分.分数阶小波变换是传统小波变换的推广,在对原小波变换核作一定改动后增加了小波变换对信号处理的灵活性.可以看到,将分数阶小波变换的变换角度取为π/2,便得到与传统小波变换多分辨率分析理论完全一致的结果.理论分析和计算机仿真表明了所提理论的正确性和有效性.     

分数阶小波变换的性质

小波变换是对信号时域-频域(Fourier域)的多分辨率分析,是一种线性时不变伸缩带通滤波.分数阶小波变换将小波变换的多分辨率分析理论推广到时域-广义频域(分数阶Fourier域),对信号分析处理有更大的灵活性.分析了分数阶小波变换的线性时变特性、存在正交分教阶小波的条件、分数阶Fourier域传递函数,以及分数阶小波变换在分数阶Fourier域的伸缩带通滤波.     

一种新的分数阶Fourier时频图示法

分数阶Fourier（FRFT）是传统Fourier的广义形式。分数阶Fourie域（FRFD）是一个统一的时频变换域,分数阶Fourier变换是角度为口的时频面旋转。随着角度α从0逐渐增加到π/2,分数阶Fourier变换展示出信号从时域到频域的全过程。本文依据分数阶FOUrier变换的定义,随着角度α的变化给出了一种新的更为直观的分数阶Fourier的时频图示方法,以供读者参考。     

短时Fourier变换与小波变换的比较

本文首先介绍了短时Fourier变换和小波变换的基本概念，然后从离散变换与框架，正交基与多分辨分析等方面，对短时Fourier变换和小波变换作了分析比较，最后讨论了两者的适用范围和优劣评价。     

基于分数阶小波变换与QR分解 的盲数字水印算法

通过对现有的盲数字水印算法的研究,本文提出一种新颖的基于分数阶小波变换(FRWT)和QR分解的盲数字 图像水印算法,相比于现有的基于DWT域数字图像水印方法,该方法兼具分数阶小波变换(FRWT)和DWT的优点,对数字水 印处理具有更高的灵活性。仿真结果表明嵌入水印后的宿主图像的峰值信噪比均高于40dB,充分说明该算法不仅可以很好 地平衡水印的不可见性,而且对几何攻击具有较好的鲁棒性。     

多参数分数Fourier变换

分数Fourier变换具有多样性,这是分数阶算子的本质属性.文中发现了加权类分数Fourier变换多样性的一个新来源,可将加权系数推广为包含两个向量参数M,N∈Z~M的广义形式.使用推广的加权系数可以定义一种多参数分数Fourier变换,特征分析发现该变换给出了分数Fourier变换一种统一的理论框架.它不但包含已知类型的分数Fourier变换作为特例,还引入了新类型的分数Fourier变换,该方法还适用于其他线性算子的分数化.最后,利用Hermite-Gauss函数的线性组合及矩形函数作为原始信号,通过数值仿真图解多参数分数Fourier变换对信号的变换.     

分数阶Fourier变换在数字水印中的应用

基于混沌置乱以及离散分数阶Fourier变换,提出一种数字水印算法,该算法在分数阶傅里叶域嵌入水印,并用相关性检测的方法来提取水印。混沌序列的伪随机性和初值敏感性以及分数阶Fourier变换的变换阶数为数字水印的安全性提供了保证,通过对算法的仿真以及抗攻击性能测试,该数字水印有较好的不可感知性,算法对JPEG压缩、滤波、噪声等攻击具有良好的鲁棒性。     

小波变换图像编码的研究进展

基于小波变换的图像编码是当今十分流行的编码方法。文章结合小波变换的特点,介绍了小波图像压缩的基本原理及基于小波的图像多分辨率分析方法,阐述了小波图像压缩方法及其进展。     

一种基于分数阶Fourier域的数字水印

提出了一种分数阶Fourier域的水印嵌入算法。将一复伪随机序列作为水印信息嵌入到图像的分数阶Fourier域中。分数阶Fourier变换的变换角度(α,β)为水印增加了两个自由度,增强了水印的安全性。仿真结果验证了该算法的有效性。     

分数阶Fourier变换在多分量信号谱图分析中的应用

作为时频分析方法的一种,谱图对多分量信号分析时受交叉项影响,特别是当信号相隔很近时尤为严重,而且频率分辨率会受影响。给出了结合分数阶Fourier变换（FrFT）对多分量信号进行谱图分析的方法。首先利用分数阶二阶矩极值点而找到相应的最优旋转阶数,对所给多分量信号按此阶数做分数阶Fourier变换,再在此基础上做谱图分析。仿真实例表明,该方法对初始频率、调频率很接近的多分量的chirp信号能有效识别,交叉项可得到较好的抑制。     

A novel fractional wavelet transform and its applications

The wavelet transform (WT) and the fractional Fourier transform (FRFT) are powerful tools for many applications in the field of signal processing.However,the signal analysis capability of the former is limited in the time-frequency plane.Although the latter has overcome such limitation and can provide signal representations in the fractional domain,it fails in obtaining local structures of the signal.In this paper,a novel fractional wavelet transform (FRWT) is proposed in order to rectify the limitations of the WT and the FRFT.The proposed transform not only inherits the advantages of multiresolution analysis of the WT,but also has the capability of signal representations in the fractional domain which is similar to the FRFT.Compared with the existing FRWT,the novel FRWT can offer signal representations in the time-fractional-frequency plane.Besides,it has explicit physical interpretation,low computational complexity and usefulness for practical applications.The validity of the theoretical derivations is demonstrated via simulations.     

Research progress of the fractional Fourier transform in signal processing

While solving a heat conduction problem in 1807, a French scientist Jean Baptiste Jo-seph Fourier, suggested the usage of the Fourier theorem. Thereafter, the Fourier trans-form (FT) has been applied widely in many scientific disciplines, and has played i…     

改进的Morlet小波及其变换

提出了一个改进的Morlet小波,并在此基础上给出了Morlet小波变换的完全重构公式,这个重构公式不需要Morlet小波满足小波容许条件,使得Morlet小波变换在理论上趋于完善。改进后的Morlet小波其尺度参数替换为小波主频参数,参数有明确的物理意义,用它作为核函数的小波变换把时间信号映射到时间-频率域。重构公式的提出可拓宽Morlet小波的应用范围,引进了一个由Morlet小波变换及其逆变换构建的时-频滤波器并将其用于地震信号处理以提高其分辨率。从理论上分析了Morlet小波变换与S变换的区别,并用实际算例验证了分析结果。     

The multiple-parameter fractional Fourier transform

The fractional Fourier transform （FRFT） has multiplicity, which is intrinsic in fractional operator. A new source for the multiplicity of the weight-type fractional Fourier transform （WFRFT） is proposed, which can generalize the weight coefficients of WFRFT to contain two vector parameters m,n ∈ Z^M . Therefore a generalized fractional Fourier transform can be defined, which is denoted by the multiple-parameter fractional Fourier transform （MPFRFT）. It enlarges the multiplicity of the FRFT, which not only includes the conventional FRFT and general multi-fractional Fourier transform as special cases, but also introduces new fractional Fourier transforms. It provides a unified framework for the FRFT, and the method is also available for fractionalizing other linear operators. In addition, numerical simulations of the MPFRFT on the Hermite-Gaussian and rectangular functions have been performed as a simple application of MPFRFT to signal processing.     

Research progress on discretization of fractional Fourier transform

As the fractional Fourier transform has attracted a considerable amount of attention in the area of optics and signal processing, the discretization of the fractional Fourier transform becomes vital for the application of the fractional Fourier transform. Since the discretization of the fractional Fourier transform cannot be obtained by directly sampling in time domain and the fractional Fourier domain, the discretization of the fractional Fourier transform has been investigated recently. A summary of discretizations of the fractional Fourier transform developed in the last nearly two decades is presented in this paper. The discretizations include sampling in the fractional Fourier domain, discrete-time fractional Fourier transform, fractional Fourier series, discrete fractional Fourier transform （including 3 main types： linear combination-type; sampling-type; and eigen decomposition-type）, and other discrete fractional signal transform. It is hoped to offer a doorstep for the readers who are interested in the fractional Fourier transform.     

Digital computation of the weighted-type fractional Fourier transform

The paper reveals the time-frequency symmetric property of the weighted-type fractional Fourier transform (WFRFT) by investigating the original definition of the WFRFT, and proposes a discrete algorithm of the WFRFT based on the weighted discrete Fourier transform (WDFT) algorithm with constraint conditions of the definition of the WFRFT and time-domain sampling. When the WDFT is considered in digital computation of the WFRFT, the Fourier transform in the definition of the WFRFT should be defined in frequency (Hz) but not angular frequency (rad/s). The sampling period Δt and sampling duration T should satisfy Δt = T/N = 1/N(1/2) when N-point DFT is utilized. Since Hermite-Gaussian functions are the best known eigenfunctions of the fractional Fourier transform (FRFT), digital computation based on eigendecomposition is also carried out as the additional verification and validation for the WFRFT calculation.     

基于小波变换多分辨率特性的语音消噪方法的研究

小波分析是一种信号的时间——尺度分析方法,特别适合于非平稳信号的分析,具有多分辨率分析的特性,而且在时频两域都具有表征信号局部特征的能力。通过分析语音信号的特性,利用小波变换的多分辨率分析特性,提出了首先对信号进行清浊音判断,其次运用多尺度多闽值方法来抑制包含有噪声的语音信号在不同尺度上的噪声小波系数,从而实现在重构语音信号中消噪的目的,并通过计算机仿真结果验证了该方法的有效性。     

The fractional Fourier transform: theory, implementation and error analysis

The fractional Fourier transform is a time–frequency distribution and an extension of the classical Fourier transform. There are several known applications of the fractional Fourier transform in the areas of signal processing, especially in signal restoration and noise removal. This paper provides an introduction to the fractional Fourier transform and its applications. These applications demand the implementation of the discrete fractional Fourier transform on a digital signal processor (DSP). The details of the implementation of the discrete fractional Fourier transform on ADSP-2192 are provided. The effect of finite register length on implementation of discrete fractional Fourier transform matrix is discussed in some detail. This is followed by the details of the implementation and a theoretical model for the fixed-point errors involved in the implementation of this algorithm. It is hoped that this implementation and fixed-point error analysis will lead to a better understanding of the issues involved in finite register length implementation of the discrete fractional Fourier transform and will help the signal processing community make better use of the transform.