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.     

Fractional order [proportional derivative] controller for a class of fractional order systems

Recently, fractional order systems (FOS) have attracted more and more attention in various fields. But the control design techniques available for the FOS suffer from the lack of direct systematic approaches. In this paper, we focus on a given type of simple model of FOS. A fractional order [proportional derivative] (FO-[PD]) controller is proposed for this class of FOS, and a practical and systematic tuning procedure has been developed for the proposed FO-[PD] controller synthesis. The fairness issue in comparing with other controllers such as the traditional integer order PID (IO-PID) controller and the fractional order proportional derivative (FO-PD) controller has been addressed under the same number of design parameters and the same specifications. Fair comparisons of the three controllers (i.e., IO-PID, FO-PD and FO-[PD]) via the simulation tests illustrate that, the IO-PID controller designed may not always be stabilizing to achieve flat-phase specification while both FO-PD and FO-[PD] controllers designed are always stabilizing. Furthermore, the proposed FO-[PD] controller outperforms FO-PD controller for the class of fractional order systems.     

基于M周期离散分数傅里叶变换的数字水印算法*

由于DFRFT同时包含了频域与时域的信息,在水印的嵌入与提取算法中,根据所取DFRFT的幂次α的不同结果也不一样,增强了水印的保密性。在仿真实验的结果中表明用DFRFT实现的数字水印算法比用傅里叶变换及小波变换实现的数字 算法具有更好的安全性。     

基于分数维变换的图象恢复

对时不变退化模型的图象恢复可采用估计时间为O（NlogN)的Wiener线性滤波,对某些具有时变退化模型或非平衡过程的图象进行恢复。若进行线性估计,则需用O（N^2)时间完成。这里提出的采用分数傅里叶变换域滤波的方法。在实现时间O（NlogN)下,即获得比在普通傅里叶变换域中滤波进行估计所具有的MSE小,且没有附加代价。实验中,对不同的退化过程（chirp函数等）引起的退化图象进行恢复,结果显示此 方法是有效的。     

Oversampling analysis in fractional Fourier domain

Oversampling is widely used in practical applications of digital signal processing. As the fractional Fourier transform has been developed and applied in signal processing fields, it is necessary to consider the oversampling theorem in the fractional Fourier domain. In this paper, the oversampling theorem in the fractional Fourier domain is analyzed. The fractional Fourier spectral relation between the original oversampled sequence and its subsequences is derived first, and then the expression for exact reconstruction of the missing samples in terms of the subsequences is obtained. Moreover, by taking a chirp signal as an example, it is shown that, reconstruction of the missing samples in the oversampled signal is suitable in the fractional Fourier domain for the signal whose time-frequency distribution has the minimum support in the fractional Fourier domain. Supported partially by the National Natural Science Foundation of China for Distinguished Young Scholars (Grant No. 60625104), the National Natural Science Foundation of China (Grant Nos. 60890072, 60572094), and the National Basic Research Program of China (Grant No. 2009CB724003)     

基于混沌序列和分数傅里叶变换的图像加密技术

采用混沌序列的特性和分数阶傅里叶变换,实现图像双重加密。由密钥生成实数值混沌序列,对数字图像进行空域置乱,然后进行分数傅里叶变换,实现图像的双重加密,只有同时清楚混沌序列和分数阶傅里叶变换阶数才可能对加密图像进行有效解密。计算机模拟表明该方法是有效的,具有很好的加密效果。     

一种新的分数阶Fourier域的Chirp类水印方案

提出了一种新的分数阶Fourier域Chirp类数字水印方案,该方案利用分数阶Fourier变换基函数的正交性和旋转相加性,在不同的分数阶Fourier域嵌入Chirp水印,并利用分数阶Fourier域Chirp信号的聚集性进行盲检测。接着结合该水印嵌入方案,利用分数阶Fourier变换的旋转相加性和酉性,推导出分数阶Fourier域水印容量的计算公式。仿真实验表明该算法由于可以选择嵌入在不同分数阶Fourier域,使得嵌入方法灵活安全,同时算法的不可见性好,对高斯白噪声干扰、裁剪及其它常见图像处理过程具有一定的鲁棒性。     

基于FRFT和LBP的笑脸识别

利用FRFT的时频双重特性和LBP算子能提取纹理图像微小特征的优点,提出一种将分数阶Fourier变换(FRFT)与局域二值模式(LBP)算子相结合的笑脸识别算法.对训练样本进行分数阶Fourier变换,取其变换的幅值信息作为脸部表情特征,与LBP融合进行分类判别,同时采用总体识别率和笑脸识别率统计结果,在RML表情数据库进行仿真验证.实验结果表明,该方法在笑脸识别中相比其他方法的识别性能更好.     

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。     

基于混沌系统和分数阶傅里叶变换的图像加密算法

设计一种基于混沌系统和FRFT（Fractional Fourier Transform）的图像加密算法。图像加密过程分两个步骤：首先利用混沌系统生成的置乱矩阵对图像进行置乱加密,然后利用FRFT将置乱加密的图像进行双随机相位加密得到最终的加密图像。该方案的安全性依赖于混沌系统的参数敏感性、相位掩膜和FRFT阶的随机性。理论分析和仿真实验表明,该方案具有良好的图像加密效果。     

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。     

Research on resolution between multi-component LFM signals in the fractional Fourier domain

When the initial frequencies and chirp rates of multi-component linear frequency modulation (LFM or chirp) signals are close,the signals may not be distinguished in the fractional Fourier domain (FRFD).Consequently,some signals cannot be detected.In this paper,first,the spectral distribution characteristics of a continuous LFM signal in the FRFD are analyzed,and then the spectral distribution characteristics of a LFM signal in the discrete FRFD are analyzed.Second,the critical resolution distance between the peaks of two LFM signals in the FRFD is deduced,and the relationship between the dimensional normalization parameter and the distance between two LFM signals in the FRFD is also deduced.It is discovered that selecting a proper dimensional normalization parameter can increase the distance.Finally,a method to select the parameter is proposed,which can improve the resolution ability of the fractional Fourier transform (FRFT).Its effectiveness is verified by simulation results.     

FRFT域的数字图像chirp噪声抑制方法

针对含有chirp噪声的图像,应用传统的滤波方法难以实现有效的信噪分离,提出了一种基于分数阶傅里叶变换域的数字图像chirp噪声的抑制方法。该方法是将chirp信号分数阶傅里叶域的滤波算法引入到数字图像的处理中,是一种新的改善图像质量的手段。仿真结果表明,对于含有chirp噪声干扰这一特定退化模型的图像,采取最优估计下的分数阶傅里叶变换相比普通傅里叶变换和线性平滑滤波,图像恢复的效果更佳,它能有效地去除图像中的chirp噪声。     

Sampling expansion for irregularly sampled signals in fractional Fourier transform domain

Real-world signals are often not band-limited, and in many cases of practical interest sampling points are not always measured regularly. The purpose of this paper is to propose an irregular sampling theorem for the fractional Fourier transform (FRFT), which places no restrictions on the input signal. First, we construct frames for function spaces associated with the FRFT. Then, we introduce a unified framework for sampling and reconstruction in the function spaces. Based upon the proposed framework, an FRFT-based irregular sampling theorem without band-limiting constraints is established. The theoretical derivations are validated via numerical results.     

Fractional variational iteration method for fractional initial-boundary value problems arising in the application of nonlinear science

In this paper, we suggest a fractional functional for the variational iteration method to solve the linear and nonlinear fractional order partial differential equations with fractional order initial and boundary conditions by using the modified Riemann-Liouville fractional derivative proposed by G. Jumarie. Fractional order Lagrange multiplier has been considered. Solution has been plotted for different values of α.     

分数阶Fourier域多分量LFM信号间的分辨研究

在分数阶Fourier域内,当多分量线性调频(LFM)信号的初始频率和调频率相近时,信号的尖峰会出现无法分辨的现象,导致目标信号漏检.文中分析了LFM信号在分数阶Fourier域的频谱分布特征,以及离散分数阶Fourier变换计算条件下LFM信号的频谱分布特征.推导了两个LFM信号在分数阶Fourier域的临界分辨距离,以及两个LFM信号尖峰之间的距离与量纲归一化因子的变化关系,发现选择合理的量纲归一化因子可以增大两个信号尖峰之间的距离.文中提出一种量纲归一化因子优化选择的方法,该方法可以提高分数阶Fourier变换对多分量LFM信号的分辨能力.最后,仿真结果验证了该方法的有效性.     

利用短时分数阶PWVD实现多分量Chirp信号分离

针对加性高斯白噪声背景下多分量Chirp信号的分离问题,采用一种基于短时分数阶傅里叶的伪魏格纳变换来实现对多分量Chirp信号的分离。该方法利用分数阶傅里叶变换四阶中心矩寻找极值点来确定最佳变换域,在最佳变换域对信号进行旋转的短时傅里叶变换,并进行伪魏格纳变换,最后把在时频面得到的冲激信号变换到时域再进行分数阶傅里叶逆变换,实现了多分量Chirp信号的分离。仿真实验证明该方法可有效地实现多分量Chirp信号分离,有助于后续对各分量的参数估计。     

An algorithm for solving multi-term diffusion-wave equations of fractional order

In this paper an algorithm, based on a new modified homotopy perturbation method (MHPM), is presented to obtain approximate solutions of multi-term diffusion-wave equations of fractional order. To illustrate the method some examples are provided. The results show the simplicity and the efficiency of the algorithm.     

Simultaneous state estimation and parameter identification in linear fractional order systems using coloured measurement noise

In the present paper, the identification and estimation problem of a single-input–single-output (SISO) fractional order state-space system will be addressed. A SISO state-space model is considered in which parameters and also state variables should be estimated. The canonical fractional order state-space system will be transformed into a regression equation by using a linear transformation and a shift operator that are appropriate for identification. The identification method provided in this paper is based on a recursive identification algorithm that has the capability of identifying the parameters of fractional order state-space system recursively. Another subject that will be addressed in this paper is a novel fractional order Kalman filter suitable for the systems with coloured measurement noise. The promising performance of the proposed methods is verified using two stable fractional order systems.