Sampling of fractional bandlimited signals associated with fractional Fourier transform

Fractional Fourier transform (FRFT) plays an important role in many fields of optics and signal processing. This paper considers the problem of reconstructing a fractional bandlimited signal with FRFT. We propose a novel reconstruction method for fractional bandlimited signals using the fractional Fourier series (FRFS). The advantage is that the sampling expansion can be deduced directly not based on the Shannon theorem. By utilizing the generalized form of Parseval’s relation for complex FRFS, we obtain the sampling expansion for fractional bandlimited signals with FRFT. We show that the sampling expansion for fractional bandlimited signals with FRFT is a special case of Parseval’s relation for complex FRFS.     

High order generalized permutational fractional Fourier transforms

We generalize the definition of the fractional Fourier transform (FRFT) by extending the new definition proposed by Shih. The generalized FRFT, called the high order generalized permutational fractional Fourier transform (HGPFRFT), is a generalized permutational transform. It is shown to have arbitrary natural number M periodic eigenvalues not only with respect to the order of Hermite－Gaussian functions but also to the order of the transform. This HGPFRFT will be reduced to the original FRFT proposed by Namias and Liu's generalized FRFT and Shih's FRFT at the three limits with M=+∞, M=4k(k is a natural number), and M=4, respectively. Therefore the HGPFRFT introduces a comprehensive approach to Shih's FRFT and the original definition. Some important properties of HGPFRFT are discussed. Lastly the results of computer simulations and symbolic representations of the transform are provided.     

The concise fractional Fourier transform and its application in detection and parameter estimation of the linear frequency-modulated signal

A concise fractional Fourier transform(CFRFT) is proposed to detect the linear frequency-modulated(LFM) signal with low signal to noise ratio(SNR).The frequency axis in time-frequency plane of the CFRFT is rotated to get the spectrum of the signal in different angles using chirp multiplication and Fourier transform(FT).For LFM signal which distributes as a straight line in time-frequency plane,the CFRFT can gather the energy in the corresponding angle as a peak and improve the detection SNR,thus the LFM signal of low SNR can be detected.Meanwhile,the location of the peak value relates to the parameters of the LFM signal.Numerical simulations and experimental results show that,the proposed method can be used to efficiently detect the LFM signal masked by noise and to estimate the signal's parameters accurately.Compared with the conventional fractional Fourier transform(FRFT),the CFRFT reduces the transform complexity and improves the real-time detection performance of LFM signal.     

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.     

简明分数阶傅里叶变换及其对线性调频信号的检测和参数估计

针对低信噪比下线性调频信号的检测问题,提出了一种简明分数阶傅里叶变换方法。该变换借助chirp相乘和傅里叶变换对时频平面上的频率轴进行旋转,以获取信号在各个角度下频率轴上的频谱分布。对时频分布呈直线状的线性调频信号,简明分数阶傅里叶变换能在特定角度上将信号能量聚集成尖锐的强能量峰,从而提高信噪比,实现对线性调频信号的可靠检测和参数估计。数值仿真和实验验证结果表明,简明分数阶傅里叶变换可对较低信噪比的线性调频信号实现有效检测,并由变换域峰值的位置对信号参数进行准确估计。相比于传统的分数阶傅里叶变换方法,简明分数阶傅里叶变换的复杂度更低,离散计算效率更高,在对噪声掩盖下的线性调频信号进行检测和参数估计时能更好地满足实时处理的要求。     

The fractional Fourier transform of Airy beams using Lohmann and quadratic optical systems

Based on the generalized Huygens–Fresnel integral, the analytical expressions for Airy beams through three types of fractional Fourier transform (FRFT) optical systems (Lohmann I, Lohmann II and quadratic graded-index systems) are derived. Using the two analytical expressions, the propagation properties of Airy beams for the three types of FRFT optical systems are discussed in detail with numerical examples. Results indicate that the intensity of Airy beams periodically changes and remains form-invariant with the changing of fractional order p for the three types of FRFT optical systems.     

基于多项式调频Fourier变换的信号分量提取方法

为了从含有噪声的混合信号中有效提取各个信号分量, 提出一种基于多项式调频Fourier变换的分量提取方法. 通过研究Fourier变换和分数阶Fourier变换的信号能量积累方式及变换基函数的时频表示, 提出利用时频平面上的多项式调频曲线族代替Fourier变换和分数阶Fourier 变换的调频直线族, 将变换的适用范围扩展到非线性调频信号. 采用粒子群智能优化算法搜索调频曲线族的最优多项式参数, 使混合信号中的某一分量在多项式调频Fourier域上能量谱集中. 最后对能量谱集中的分量进行窄带滤波, 并利用多项式调频逆Fourier变换重构信号分量. 仿真实验结果表明, 该方法不仅能够提取混合信号中的线性调频分量, 还能够实现非线性调频分量的能量谱集中、信号分离和时频特征提取.     

On scaling properties of fractional Fourier transform and its relation with other transforms

Several properties of fractional Fourier transform (FRFT) have been studied recently and many are being investigated at present. In this article, scaling property of the FRFT is generalized and some of its applications are suggested. Some extensions of the sampling relations in the FRFT domain are also presented. The issues related to connections between the FRFT and other signal transforms such as scale transform, fractional Mellin transform, and chirp z-transform, are also investigated.     

基于FRFT域分形特征的压制干扰存在性检测（英）

基于分数阶傅里叶变换（FRFT）域的分形特征研究压制干扰的存在性检测问题。首先,分析了典型的三种压制干扰的分形特征,说明了典型压制干扰信号具有分形特性,并采用盒维数和信息维数定量描述它们的分形特征。然后,发现了压制干扰和高斯白噪声在FRFT域具有不同的分形特征,进而提出了一种检测压制干扰存在性的方法。最后,仿真验证了该方法的有效性和优越性。     

基于FRFT域分形特征的压制干扰存在性检测（英）

基于分数阶傅里叶变换（FRFT）域的分形特征研究压制干扰的存在性检测问题。首先,分析了典型的三种压制干扰的分形特征,说明了典型压制干扰信号具有分形特性,并采用盒维数和信息维数定量描述它们的分形特征。然后,发现了压制干扰和高斯白噪声在FRFT域具有不同的分形特征,进而提出了一种检测压制干扰存在性的方法。最后,仿真验证了该方法的有效性和优越性。     

Application of fractional Fourier transform on spatial filtering

Starting from the relationship between the Wigner transformation and the fractional Fourier transform (FRFT), we apply the FRFT to the spatial filtering. In the process of filtering, the noise can be simulated as the chirp functions or their linear combination, which can be filtered at the corresponding fractional planes, and also the actual positions can be exactly obtained. Some numerical simulations prove its possibility.     

海杂波FRFT域的分形特征分析及小目标检测方法

针对海杂波背景下海情对小目标检测的严重影响, 本文研究了实测海杂波在分数阶Fourier变换(FRFT)域的分形特征, 分别提出了单、高尺度下的分形检测方法. 由数学定义推得, FRFT 在不同阶数和尺度情况下, 不具有一致的自相似特性, 采用多重分形趋势波动分析法确定分形参数H(q), 分析了海杂波在不同海情、距离和极化条件下的分形特征. 在单尺度基础上结合FRFT的变阶优势, 提出了阶数自适应的小目标检测方法; 高尺度条件下, 比较了不同因素对海杂波FRFT域多重分形参数的影响. 结果表明:海杂波FRFT域可用变换阶数的方法检测到湮没在复杂海情中的小信号, 检测门限多数提高200%以上, 比采用时域信号提高26.3%. H(q) 在负高尺度上具有明显的多重分形特征差异, H(q)-q曲线满足反正切分布, 纯海杂波与含目标数据的拟合幅值比分别大于1.8(HH)和1.4(VV), 为海杂波背景小目标检测提供了判定依据.     

分数域滤波的分数傅里叶变换光学成像性能分析

将分数域滤波同光学成像相结合,提出了一种基于Lohmann Ⅰ型的带有滤波孔径的两级联分数傅里叶变换光学成像系统.根据分数傅里叶变换和菲涅耳衍射之间的关系以及分数傅里叶变换的分数阶可加性,结合分数域滤波分析了分数傅里叶变换光学成像的基本理论.以点扩散函数和调制传递函数作为评价成像质量的准则,详细分析了不同分数阶光学系统...     

Image encryption algorithm based on the multi-order discrete fractional Mellin transform

A multi-order discrete fractional Mellin transform (MODFrMT) is constructed and directly used to encrypt the private images. The MODFrMT is a generalization of the fractional Mellin transform (FrMT) and is derived by transforming the image with multi-order discrete fractional Fourier transform (MODFrFT) in log-polar coordinates, where the MODFrFT is generalized from the closed-form expression of the discrete fractional Fourier transform (DFrFT) and can be calculated by fast Fourier transform (FFT) to reduce the computation burden. The fractional order vectors of the MODFrMT are sensitive enough to be the keys, and consequently key space of the encryption system is enlarged. The proposed image encryption algorithm has significant ability to resist some common attacks like known-plaintext attack, chosen-plaintext attack, etc. due to the nonlinear property of the MODFrMT. Additionally, Kaplan-Yorke map is employed in coordinate transformation process of the MODFrMT to further enhance the security of the encryption system. The computer simulation results show that the proposed encryption algorithm is feasible, secure and robust to noise attack and occlusion.     

基于柱坐标系下的空心高斯光束的分数傅里叶变换

推导出分数傅里叶变换基于柱坐标系下的解析表达式,应用此解析表达式对空心高斯光束进行分析,得到其分数傅里叶变换的解析表达式。讨论其在分数傅里叶变换平面的光强分布与各种光束参数之间的变换关系,并进行数值计算,从而得出基于柱坐标系下的空心高斯光束的分数傅里叶变换的传输性质。所得结果为分析和计算这种光束的分数傅里叶变换提供了方便,同时对此类光束的传输控制提供了基础理论支持。     

Spectrum measurement in the fractional Fourier domain

Fractional Fourier transform (FRFT) plays an important role in many fields of optics and signal processing. This paper considers the problem of real-time measurement of the spectrum of a signal in the FRFT domain. In this paper, we propose two approaches for real-time measurement of the FRFT of a signal based on modulation and bandpass filtering systems. The relation is established between the linear frequency modulation (LFM or chirp) spectrum and the FRFT of its envelope. In addition, two applications for spectrum measurement are presented in the FRFT domain. The LFM signal can be bandlimited in the Fourier transform (FT) domain through spectrum measurement associated with bandpass filtering method. The results can also be useful in the problems related to swept-frequency filter for measurement in the FRFT domain.     

超洛伦兹-高斯光束SLG11模的分数傅里叶变换特性

引入了一簇互相正交的超洛伦兹-高斯光束以描述半导体激光器所产生的大角度高阶模远场分布。将分数傅里叶变换应用于超洛伦兹-高斯光束SLG11模的传输特性的研究中。利用傅里叶变换的卷积原理,导出了SLG11模经分数傅里叶变换系统后场分布的解析表达式。根据所得到的公式进行了数值计算,系统分析了分数傅里叶变换阶数和光束各参数对SLG11模在分数傅里叶变换面上光强分布的影响。结果显示：SLG11模在分数傅里叶变换面上的归一化强度分布随分数傅里叶变换的阶数呈周期性变化,周期为2;随着光束参数的增大,SLG11模在分数傅里叶变换面上的光斑尺寸增大。     

Wavelet-fractional Fourier transforms

This paper extends the definition of fractional Fourier transform （FRFT） proposed by Namias V by using other orthonormal bases for L^2（R） instead of Hermite-Gaussian functions. The new orthonormal basis is gained indirectly from multiresolution analysis and orthonormal wavelets. The so defined FRFT is called wavelets-fractional Fourier transform.     

Wavelet--fractional Fourier transforms

This paper extends the definition of fractional Fourier transform (FRFT) proposed by Namias V by using other orthonormal bases for $L^{2}\left( R \right)$ instead of Hermite--Gaussian functions. The new orthonormal basis is gained indirectly from multiresolution analysis and orthonormal wavelets. The so defined FRFT is called wavelets-fractional Fourier transform.     

Fractional Fourier transform of Lorentz beams

This paper introduces Lorentz beams to describe certain laser sources that produce highly divergent fields. The fractional Fourier transform (FRFT) is applied to treat the propagation of Lorentz beams. Based on the definition of convolution and the convolution theorem of the Fourier transform, an analytical expression for a Lorentz beam passing through a FRFT system has been derived. By using the derived formula, the properties of a Lorentz beam in the FRFT plane are illustrated numerically.