分数傅里叶变换的快速算法及计算全息图的研究

通过分析菲涅耳衍射积分的快速算法,依据Lohmann提出的任意阶的分数傅里叶变换的单透镜光学实验装置,详细分析了光场在此单透镜系统中的传播过程,提出了一种基于傅里叶变换的分数傅里叶变换快速算法,并对基于此快速算法的分数傅里叶变换全息图的计算机生成与数字重现进行了研究。实验结果示出了分数傅里叶变换全息图及其在重构过程中分数阶匹配与否的实验结果,验证了分数傅里叶变换分数阶的重要性质和笔者提出算法的可行性。     

分数傅里叶变换的快速算法及计算全息图的研究

通过分析菲涅耳衍射积分的快速算法，依据Lohmann提出的任意阶的分数傅里叶变换的单透镜光学实验装置，详细分析丁光场在此单透镜系统中的传播过程，提出了一种基于傅里叶变换的分数傅里叶变换快速算法，并对基于此快速算法的分数傅里叶变换全息图的计算机生成与数字重现进行了研究。实验结果示出了分数傅里叶变换全息图及其在重构过程中分数阶匹配与否的实验结果，验证了分数傅里叶变换分数阶的重要性质和笔者提出算法的可行性。     

分数傅里叶变换合成计算全息与数字重现

传统的计算全息图大多采用菲涅耳衍射积分得到,但菲涅耳衍射积分在描述整个衍射光场中,没有一个统一的采样方法对积分进行数值计算,从而给计算带来不便。为了更好地研究计算全息图问题,文章引入了分数傅里叶变换,通过利用分数傅里叶变换的一种快速数值模拟算法,提出了一种基于分数傅里叶变换的合成空间三维物体全息图的新方法,并用计算机模拟了合成的全息图及其数字重构的结果。实验结果表明:由于分数阶的引入,得到一种处理光场衍射问题的统一算法,因此用分数傅里叶变换来处理光场衍射问题是十分理想的。     

分数傅里叶全息图的算法研究及显示

为了快速获取更好的全息图显示效果,在研究了分数傅里叶变换与菲涅耳衍射光场的紧密联系的基础上,首先给出了一种分数傅里叶变换的快速数值算法,并将分数傅里叶变换应用到计算全息图中提出了一种用于计算全息图的分数傅里叶变换方法;然后分析了用这种记录全息的方法得到的全息图优越于传统的傅里叶变换获得的全息图,同时给出了计算机的模拟实验结果;最后利用拆卸的投影装置搭建出了以空间光调制器——DMD (数字微镜装置)为核心的全息显示光学系统,并在该系统下获得了用分数傅里叶变换计算得到的全息图的全息显示结果。     

基于离散分数傅里叶变换的SVG数字水印技术

提出了一种基于离散分数傅里叶变换(DFRFT)的SVG图形水印算法,用于SVG的版权保护.将水印嵌入到SVG图形控制点坐标构造的复数信号的离散分数傅里叶变换频谱中.经实验结果验证,该算法具有较强的透明性,对于通常的图形几何变换以及局部修改攻击,均有令人满意的鲁棒性.     

分数傅里叶域图像数字水印方案

根据离散分数傅里叶变换（DFRFT）,提出了一种基于分数傅里叶变换的图像数字水印方案。分数傅里叶变换具有空域和频城双城表达能力,可以对原始图像和水印信号分别进行不同阶次的分数傅里叶变换以增强水印安全性。将水印信号的分数傅里叶谱叠加在原始图像在视觉上的次重要分量上。在JPEG压缩、图像旋转、高斯低通滤波的攻击方式下,对水印图像进行了鲁棒性分析,实验表明该算法具有良好的鲁棒性。     

基于分数傅里叶变换的二维工程图水印算法

提出一种基于离散分数傅里叶变换（DFRFT）的二维工程图数字水印算法。该算法分块提取工程图中线段的相对坐标线构造复值信号量,将水印嵌入复值信号量的分数傅里叶变换频谱（FRT）中。实验表明,该算法对平移、旋转、缩放、部分实体删除或添加等攻击具有良好的鲁棒性,同时具有良好的安全性。     

基于SFFT的宽带信号互谱法测向算法

在无线电频谱监测中,随着数据采集能力和采样频率的不断提高,对算法的时效性提出了更高要求。对于宽带信号测向系统,提出基于稀疏快速傅里叶变换的互谱法相位测量算法,该算法利用信号频域的稀疏特性,通过频谱重排、滤波、降采样和估值,能快速计算出频谱中K（信号稀疏度）个拥有最大值的傅里叶系数。利用这K个大值点计算平均时延,在保证与传统快速傅里叶变换有相同精度的同时,降低算法的时间复杂度。分析表明,该算法的时间复杂度与信号稀疏度K呈亚线性关系。该方法提高了算法效率。仿真分析对比了基于稀疏快速傅里叶变换的互谱法和基于快速傅里叶变换的互谱法的误差,表明了该算法的有效性。     

分数傅立叶变换计算全息图的算法研究

传统的计算机产生全息图方法由于在标量衍射的光场描述中没有一个统一的数值计算方法,从而计算复杂度高而且重构的3D图像的体积和视场都比较小,离市场化的要求距离远.因此,在目前的计算机硬件条件下,算法和显示系统的研究仍有很大的空间.将分数傅里叶变换引入到全息图的计算中,提出一种分数傅立叶变换产生计算全息图的方法.实验结果表明,分数傅里叶变换相对于传统的傅里叶变换在记录全息数据方面的优越性,并用计算机模拟效果表明了算法的优越性.     

基于小波与分数傅里叶变换的图像水印算法

载体图像的空域隐藏Chirp信号可以通过分数傅里叶变换在变换域中进行盲检测。为了提高该算法的鲁棒性能,该文研究直接离散化方法,合理选取分数傅里叶变换的算子阶数,将Chirp 信号隐藏在图像信号的低频小波域中。仿真实验表明,改进后的水印算法提高了直接在空域进行信息隐藏的鲁棒性。     

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.     

基于FRFT的线性调频信号映射方法的快速算法

针对基于分数傅立叶变换(Fractional Fourier transform，FRFT)的线性调频信号(LFM)的参数估计方法，提出了基于Radon变换的判决准则，并建立了相应的快速算法。该映射方法将具有同样调频斜率、同样多普勒中心频率、不同初始相位的信号映射到同一个点。该快速算法的计算量等同于FRFT的计算量，计算结果的精确度取决于连续两次的FRFT运算。     

分数阶Fourier变换域极值搜索的混沌优化算法研究

正如傅里叶变换采用正弦基,单频信号能够在频域形成峰值,分数阶Fourier变换采用线性调频基,线性调频(LFM)信号能够在分数阶Fourier域上实现聚焦,利用此聚焦性通过搜索峰值可实现LFM信号检测和参数估计.通常采用步进式搜索方法,效率低下.为了克服该缺点,通过对分数阶Fourier域优化问题本质的研究,将混沌优化算法引入到分数阶Fourier域极值搜索中.仿真结果表明:本文的方法优于传统的步进式搜索法.     

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.     

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.     

MIMO-OFDM system based on fractional Fourier transform and selecting algorithm for optimal order

In the rapidly time-varying channel environment, the performance of traditional MIMO-OFDM system is deteriorated due to the intercarrier interference. In this paper, a novel MIMO-OFDM system is proposed, in which the modulation and de- modulation of the symbols are implemented by the fractional Fourier transform instead of traditional Fourier transform. Through selecting the optimal order of the fractional Fourier transform, the modulated signals can match the time-varying channel characteristics, which results in a mitigation of the intercarrier interference. Furthermore, an algorithm is presented for selecting the optimal order of fractional Fourier transform, and the impact of system parameters on the optimal order is analyzed. Simulation results show that the proposed system can concentrate the power of desired signal effectively and improve the performance over rapidly time-varying channels with respect to the traditional MIMO-OFDM system.     

短时分数阶傅里叶变换的基本性质

作为一种不会对信号时频结构在解线调时产生压缩扭曲的线性时频分析工具,短时分数阶傅里叶变换(STFrFT)相比于分数阶傅里叶变换更适于处理多项式相位信号.证明了短时分数阶傅里叶变换的一些基本性质,例如:重构条件和帕塞瓦尔定理等.以chirp信号为例给出了STFrFT的窗函数和窗口参数的选择依据.本文结论为短时分数阶傅里叶变换的应用提供参考.