线性正则变换域的带限信号采样理论研究

线性正则变换是傅里叶变换、分数阶傅里叶变换的更广义形式,是一种潜在而重要的信号变换工具,但是与之相应的采样理论目前还不十分完备,所以有必要在线性正则变换域重新研究采样定理.本文从线性正则变换的定义和性质出发,首先得到时域均匀采样信号的线性正则变换;然后在此基础上导出了线性正则变换域带限信号的采样定理和重构公式;最后以chirp信号为例仿真说明了采样定理的应用.文中得出的结论是对经典采样理论的推广,将进一步丰富线性正则变换的理论体系.     

基于分数阶傅里叶变换的线性调频信号分辨率分析

分数阶傅里叶变换(FRFT)是分析线性调频信号(LFM)最有效的工具之一.本文研究了LFM信号在分数阶傅里叶域(FRFD)上的分辨性能以及变换阶次上的分辨性能,并研究了变换阶次误差对输出信噪比的影响,分别得到了 FRFD分辨率、变换阶次分辨率、输出信噪比损失与信号持续时间及调频率的关系,为利用FRFT进行LFM信号检测和参数估计时的分辨率分析、变换阶次搜索步长的选择等提供一定的理论参考.     

基于分数阶傅里叶变换的MIMO-TDCS抗干扰性能研究

针对建立在快速傅里叶变换(FFT)的基础上只在频域之中作变换的传统MIMO-TDCS,抗线性调频干扰(LFM)信号能力弱,无法对其有效地应对和剔除等问题,引入分数阶傅里叶变换(FRFT).首先对电磁环境采样并估计出LFM干扰信号的各个参数,然后作相应阶次的分数阶傅里叶变换,在FRFT域内剔除干扰并生成相应的MIMO-TDCS基函数,从而达到收发两端联合主动躲避LFM干扰的目的.仿真结果表明,利用FRFT对信号良好的能量聚焦特性将线性调频干扰平稳化处理,能够有效地躲避LFM干扰的影响,为MIMO-TDCS抗非平稳干扰开辟了一条新思路.     

DSSS系统LFM干扰抑制技术研究

针对传统的傅里叶变换只能对平稳干扰信号具有较好的抑制能力而对非平稳的LFM干扰信号不能有效抑制,该文引入了分数阶傅里叶变换（FRFT）,在分数阶傅里叶域给出了一种LFM信号最佳旋转角度确定的算法,并利用该算法对LFM在分数阶傅里叶域进行干扰抑制。仿真结果表明,该算法能够准确确定LFM信号在分数阶傅里叶域的最佳旋转角度,...     

基于FRFT的中压配电线载波通信方法

针对中压配电线信道特性,提出了一种基于分数阶Fourier变换(FRFT)的通信方法.以线性调频信号(chirp)作为调制信号,利用线性调频信号在分数阶傅里叶变换域的能量聚焦特性,提高系统的抗噪声干扰和频率选择性衰减的能力.建立了系统的仿真模型,仿真结果证明了方法的可行性.     

一种分数阶傅里叶变换快速算法的研究

介绍了分数阶傅里叶变换的定义,接着提出了一种分数阶傅里叶变换的快速算法,其中分数阶傅里叶变换快速算法分三步进行:线性调频信号乘法,线性调频信号卷积,另一个线性调频信号乘法,从而利用FFT来计算FRFT。这种算法思想直观,结果与连续FRFT的输出接近。最后用具体的信号作了计算机仿真,并给出Matlab仿真结果图。     

基于分数阶傅里叶变换的多分量Chirp信号的检测与参数估计

分数阶傅里叶变换是傅里叶变换的广义形式。利用Chirp信号在分数阶傅里叶变换域的特点,提出了对含有多个非平稳Chirp成分的信号在噪声中的检测与参数估计方法。理论分析和仿真结果表明,与现有的基于时间域、频率域和时频域的方法相比,该方法物理意义清楚,计算简便,无交叉项干扰,抗噪声性能强。     

分数阶傅里叶变换时滞模型及其应用

实际采集到的信号中常常含有与信号频谱相同的延时噪声分量,难以用常规的滤波方法剔除。针对延时噪声干扰的特点,依据分数阶傅里叶变换（FRFT）的时移特性和乘积变换延时特性,提出了一个信号与同频噪声分离的时滞模型,通过对含噪信号进行相应的分数阶傅里叶变换,在变换域上可不断加大信号与同频噪声的距离,距离的增加与迭代次数成正比,从而能较好地分离同频干扰。实验中,对分数阶傅里叶变换的分离效果进行了仿真演示。     

短时分数阶傅里叶变换对调频信号的时频分辨能力

调频信号的检测和参数估计一直是信号处理领域的研究热点之一。为深入挖掘短时分数阶傅里叶变换对调频信号的时频分析优势,从短时分数阶傅里叶变换的定义出发,推导了其时频分辨能力与信号参数的关系,并与短时傅里叶变换进行了对比分析。结论表明,短时傅里叶变换时频分辨能力与信号频率变化率有关,而短时分数阶傅里叶变换几乎不受调频率变化率影响。最后,通过对比仿真实验证明,对于频率变化率较小的信号,两者时频分辨效果差别不明显,对于频率变化率较大的信号,短时分数阶傅里叶变换的时频分辨效果更好。     

工程采样系统的分数阶傅里叶域分析与改进

采样和重构是信号处理的基本问题。已有的分数阶傅里叶域采样理论是对理想的采样系统进行分析的,工程中的A/D和D/A转换是基于采样保持电路实现的,该文在分数阶傅里叶域对采样保持系统进行分析,提出一个可行的工程采样和重构模型,该模型仅需在传统采样保持系统上增加两级乘法器就可实现。所得结果进一步完善了分数阶傅里叶域采样定理,为分数阶傅里叶域采样定理的实用化提供了理论基础。     

On Convolution and Product Theorems for FRFT

The fractional Fourier transform (FRFT), which is considered as a generalization of the Fourier transform (FT), has emerged as a very efficient mathematical tool in signal processing for signals which are having time-dependent frequency component. Many properties of this transform are already known, but the generalization of convolution theorem of Fourier transform for FRFT is still not having a widely accepted closed form expression. In the recent past, different authors have tried to formulate convolution theorem for FRFT, but none have received acclamation because their definition do not generalize very appropriately the classical result for the FT. A modified convolution theorem for FRFT is proposed in this article which is compared with the existing ones and found to be a better and befitting proposition.     

Sampling and Sampling Rate Conversion of Band Limited Signals in the Fractional Fourier Transform Domain

The fractional Fourier transform (FRFT) has become a very active area in signal processing community in recent years, with many applications in radar, communication, information security, etc., This study carefully investigates the sampling of a continuous-time band limited signal to obtain its discrete-time version, as well as sampling rate conversion, for the FRFT. Firstly, based on product theorem for the FRFT, the sampling theorems and reconstruction formulas are derived, which explain how to sample a continuous-time signal to obtain its discrete-time version for band limited signals in the fractional Fourier domain. Secondly, the formulas and significance of decimation and interpolation are studied in the fractional Fourier domain. Using the results, the sampling rate conversion theory for the FRFT with a rational fraction as conversion factor is deduced, which illustrates how to sample the discrete-time version without aliasing. The theorems proposed in this study are the generalizations of the conventional versions for the Fourier transform. Finally, the theory introduced in this paper is validated by simulations.     

Sampling of compact signals in offset linear canonical transform domains

The offset linear canonical transform (OLCT) is the name of a parameterized continuum of transforms which include, as particular cases, the most widely used linear transforms in engineering such as the Fourier transform (FT), fractional Fourier transform (FRFT), Fresnel transform (FRST), frequency modulation, time shifting, time scaling, chirping and others. Therefore the OLCT provides a unified framework for studying the behavior of many practical transforms and system responses. In this paper the sampling theorem for OLCT is considered. The sampling theorem for OLCT signals presented here serves as a unification and generalization of previously developed sampling theorems.     

Sampling of Bandlimited Signals in Fractional Fourier Transform Domain

Fractional Fourier transformed bandlimited signals are shown to form a reproducing kernel Hilbert space. Basic properties of the kernel function are applied to the study of a sampling problem in the fractional Fourier transform (FRFT) domain. An orthogonal sampling basis for the class of bandlimited signals in the FRFT domain is then given. A nonuniform sampling theorem for bandlimited signals in the FRFT domain is also presented. Numerical experiments are given to demonstrate the effectiveness of the proposed nonuniform sampling theorem.     

A Convolution and Correlation Theorem for the Linear Canonical Transform and Its Application

As a generalization of the fractional Fourier transform (FRFT), the linear canonical transform (LCT) plays an important role in many fields of optics and signal processing. Many properties for this transform are already known, but the correlation theorem, similar to the version of the Fourier transform (FT), is still to be determined. In this paper, firstly, we introduce a new convolution structure for the LCT, which is expressed by a one dimensional integral and easy to implement in filter design. The convolution theorem in FT domain is shown to be a special case of our achieved results. Then, based on the new convolution structure, the correlation theorem is derived, which is also a one dimensional integral expression. Last, as an application, utilizing the new convolution theorem, we investigate the sampling theorem for the band limited signal in the LCT domain. In particular, the formulas of uniform sampling and low pass reconstruction are obtained.     

Generalized convolution theorem associated with fractional Fourier transform

The fractional Fourier transform (FRFT)—a generalization of the well‐known Fourier transform (FT)—is a comparatively new and powerful mathematical tool for signal processing. Many results in Fourier analysis have currently been extended to the FRFT, including the ordinary convolution theorem. However, the extension of the ordinary convolution theorem associated with the FRFT has been developed differently and is still not having a widely accepted closed‐form expression. In this paper, a generalized convolution theorem for the FRFT is proposed, and the dual of it is also presented. The ordinary convolution theorem and some of its existing extensions related to the FRFT are shown to be special cases of the derived results. Moreover, some applications of the derived results are presented. Copyright © 2012 John Wiley & Sons, Ltd.     

分数阶傅里叶域滤波器组的一般化设计方法

 分数阶傅里叶变换相对于传统的傅里叶变换具有灵活的时频分析特性,在最优分数阶傅里叶域进行滤波可以实现对某些非平稳信号的最优检测和参数估计以及对某些干扰和噪声的滤除.分数阶傅里叶域滤波器组理论的提出弥补了分数阶傅里叶域滤波不具备多尺度分析以及运算量过大的缺点,但现有的分数阶傅里叶域准确重建滤波器组设计方法不具备形式一般化的特点,很难满足很多实际工程的需要.本文从分数阶傅里叶域多抽样率信号处理基本理论和分数阶卷积定理出发,推导出了分数阶傅里叶域准确重建滤波器组的一般化设计方法,为分数阶傅里叶域滤波器组理论在实际工程中的推广应用奠定了理论基础.最后,仿真实验验证了本文所提分数阶傅里叶域滤波器组一般化设计方法的有效性.     

Relations between fractional operations and time-frequencydistributions, and their applications

The fractional Fourier transform (FRFT) is a useful tool for signal processing. It is the generalization of the Fourier transform. Many fractional operations, such as fractional convolution, fractional correlation, and the fractional Hilbert transform, are defined from it. In fact, the FRFT can be further generalized into the linear canonical transform (LCT), and we can also use the LCT to define several canonical operations. In this paper, we discuss the relations between the operations described above and some important time-frequency distributions (TFDs), such as the Wigner distribution function (WDF), the ambiguity function (AF), the signal correlation function, and the spectrum correlation function. First, we systematically review the previous works in brief. Then, some new relations are derived and listed in tables. Then, we use these relations to analyze the applications of the FRPT/LCT to fractional/canonical filter design, fractional/canonical Hilbert transform, beam shaping, and then we analyze the phase-amplitude problems of the FRFT/LCT. For phase-amplitude problems, we find, as with the original Fourier transform, that in most cases, the phase is more important than the amplitude for the FRFT/LCT. We also use the WDF to explain why fractional/canonical convolution can be used for space-variant pattern recognition     

Uncertainty principles for discrete signals associated with the fractional Fourier and linear canonical transforms

The fractional Fourier transform (FRFT), which generalizes the classical Fourier transform, has gained much popularity in recent years because of its applications in many areas, including optics, radar, and signal processing. There are relations between duration in time and bandwidth in fractional frequency for analog signals, which are called the uncertainty principles of the FRFT. However, these relations are only suitable for analog signals and have not been investigated in discrete signals. In practice, an analog signal is usually represented by its discrete samples. The purpose of this paper is to propose an equivalent uncertainty principle for the FRFT in discrete signals. First, we define the time spread and the fractional frequency spread for discrete signals. Then, we derive an uncertainty relation between these two spreads. The derived results are also extended to the linear canonical transform, which is a generalized form of the FRFT.