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。     

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。     

低信噪比线性调频信号目标的方位估计

线性调频(LFM)信号目标的方位估计是水声探测研究的重要内容,在进行方位估计时,若存在强干扰信号源与强背景噪声,阵元接收信号的信噪比会显著降低,严重影响LFM信号目标方位估计结果的准确性.针对该问题,提出了一种简明分数阶滤波方法,并将其与常规波束形成方法(CBF)相结合来实现低信噪比条件下LFM信号目标的方位估计.简明分数阶傅里叶变换能在正交角度上将LFM信号的能量聚集在特定频点处并形成明显的能量峰,利用该特性,可对阵列各阵元接收的低信噪比LFM信号在简明分数阶域聚集的能量峰进行最佳滤波,以滤除干扰信息及背景噪声.对滤波输出进行逆简明分数阶傅里叶变换可得到增强信干比和信噪比的阵元域信号,进一步用于目标方位估计,就能获得更加准确的目标方位。数值仿真结果和海试实验数据处理结果验证表明,本文所提出的方法可有效抑制干扰和背景噪声,并对低信噪比LFM信号进行准确、稳健的方位估计。     

On the core of the fractional Fourier transform and its role in composing complex fractional Fourier transformations and Fresnel transformations

By a quantum mechanical analysis of the additive rule F_α[F_β[f]] = F_α+β[f], which the fractional Fourier transformation (FrFT) F_α[f] should satisfy, we reveal that the position-momentum mutualtransformation operator is the core element for constructing the integration kernel of FrFT. Based on this observation and the two mutually conjugate entangled-state representations, we then derive a core operator for enabling a complex fractional Fourier transformation (CFrFT), which also obeys the additive rule. In a similar manner, we also reveal the fractional transformation property for a type of Fresnel operator.     

Parameter estimation of optical fringes with quadratic phase using the fractional Fourier transform

Optical fringes with a quadratic phase are often encountered in optical metrology. Parameter estimation of such fringes plays an important role in interferometric measurements. A novel method is proposed for accurate and direct parameter estimation using the fractional Fourier transform (FRFT), even in the presence of noise and obstacles. We take Newton׳s rings fringe patterns and electronic speckle pattern interferometry (ESPI) interferograms as classic examples of optical fringes that have a quadratic phase and present simulation and experimental results demonstrating the performance of the proposed method.     

基于分数阶傅立叶变换的线性调频回波参数估计

本文提出了一种适用于宽带线性调频回波的分数阶傅立叶变换参数估计方法。它通过寻找合适的信号代替发射信号做模版,使得模版信号与接收信号的模糊函数匹配度高,峰值尖锐,失真率低。再利用分数阶傅立叶变换与模糊函数的关系,使用两次分数阶域的峰值搜索代替原有的一次二维搜索降低计算量。仿真实验结果表明,本文提出的方法在精确估计宽带回波的时延和频移的同时降低了计算复杂度。     

Nonlinear joint fractional Fourier transform correlation for target detection in hyperspectral image

In this paper, we present a spectral joint fractional Fourier transform correlation (SJFRTC) for detecting and identifying very small target with only a few pixels using reflectance spectral information in the hyperspectral imagery. In this technique, fractional correlation (FC) between input spectral signature and known reference spectral signature is performed digitally or optically. Binary differential technique is utilized to enhance the performance of the SJFRTC, resulting in delta-function-like correlation peaks without dc interference. Distinctive and distinguishable correlation performance is obtained using peak-to-clutter ratio (PCR) instead of absolute correlation peak value, which makes the proposed technique independent from the intensity variation of the spectral signature. Simulation results using real-life hyperspectral images show that the binary SJFRTC can detect both single or/and multiple targets successfully with excellent detection capability, as compared to the other traditional hyperspectral detection algorithms.     

Motion detection,the Wigner distribution function,and the optical fractional Fourier transform

It is shown that both surface tilting and translational motion can be independently estimated by use of the speckle photographic technique by capturing consecutive images in two different fractional Fourier domains. A geometric interpretation, based on use of the Wigner distribution function, is presented to describe this application of the optical fractional Fourier transform when little prior information is known about the motion.     

Deriving the integral representation of a fractional Hankel transform from a fractional Fourier transform

We derive the integral representation of a fractional Hankel transform (FRHT) from a fractional Fourier transform. Some basic properties of the FRHT such as Parseval's theorem and its optical implementation are discussed qualitatively.     

Spherical angular spectrum and the fractional order Fourier transform

The notion of a spherical angular spectrum leads to the decomposition of the field amplitude on a spherical emitter into a sum of spherical waves that converge onto the Fourier sphere of the emitter. Unlike the usual angular spectrum, the spherical angular spectrum is propagated as the field amplitude, in a way that can be expressed by a fractional order Fourier transform.     

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.     

Ince-Gaussian series representation of the two-dimensional fractional Fourier transform

We introduce the Ince-Gaussian series representation of the two-dimensional fractional Fourier transform in elliptical coordinates. A physical interpretation is provided in terms of field propagation in quadratic graded-index media whose eigenmodes in elliptical coordinates are derived for the first time to our knowledge. The kernel of the new series representation is expressed in terms of Ince-Gaussian functions. The equivalence among the Hermite-Gaussian, Laguerre-Gaussian, and Ince-Gaussian series representations is verified by establishing the relation among the three definitions.     

Lensless optical implementation of the coincidence fractional Fourier transform

A lensless optical system for implementing the coincidence fractional Fourier transform (FRT) is proposed. The conditions for the lensless optical system to implement the coincidence FRT with incoherent light and entangled photon pairs are discussed. The results offer a novel scheme for FRTs and thus suggest useful applications.     

Femtosecond digital in-line holography with the fractional Fourier transform: application to phase-contrast metrology

The possibility to analyze transparent phase objects using femtosecond digital in-line holography is demonstrated. We show that the diffraction patterns produced by the diffraction of 20-fs pulses by pure phase objects can be advantageously processed by using the fractional Fourier transform. The optimal fractional orders lead to the longitudinal location of the phase object while the analysis of the reconstructed patterns leads to its diameter and to the value of the phase shift. Simulations and experimental results are in good concordance. This work shows new important applications of these reconstruction techniques in the domain of digital phase-contrast metrology. We further show that the use of femtosecond pulses results in a significant reduction of coherent noise.     

Free-space Fresnel diffraction for the approximation of fractional Fourier transform

Fractional Fourier transformation of an object can be approximated by the object's free-space Fresnel diffraction pattern under some restricted conditions and plane wave illumination according to Hua's method. A better approximation is achieved under least-squared conditions developed in this paper. Simulation results verify that our theoretical development works for any fractional order a compared with the previous approach.     

二维Duffing振子的大参数随机共振及微弱信号检测研究

研究了二维Duffing振子在绝热近似条件下的随机共振特性,针对大参数条件,提出二维Duffing振子的大参数随机共振,并探讨二维Duffing振子变尺度随机共振和参数调节随机共振的关联性,揭示大参数条件下Duffing振子随机共振检测特征信号的机理,扩展其在微弱信号检测领域中的应用.     

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.     

分数傅里叶变换面上余弦-高斯光束的变换特性

 利用Wigner分布函数的方法，研究了余弦-高斯光束的分数傅里叶变换特性。导出了余弦-高斯光束在分数傅里叶变换面上光强分布和束宽的解析计算公式，并对此进行了数值模拟计算。研究表明：分数傅里叶变换阶数对余弦-高斯光束的光强分布有明显影响，余弦-高斯光束的轴上光强随分数傅里叶变换阶数呈周期性变化，束宽随分数傅里叶变换阶数也呈周期性变化，周期为2；对给定调制参数的余弦-高斯光束，通过适当选取分数傅里叶变化阶数可以获得平顶的光强分布。