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.     

Convolution theorems for the linear canonical transform and their applications

As generalization of the fractional Fourier transform (FRFT), the linear canonical transform (LCT) has been used in several areas, including optics and signal processing. Many properties for this transform are already known, but the convolution theorems, similar to the version of the Fourier transform, are still to be determined. In this paper, the authors derive the convolution theorems for the LCT, and explore the sampling theorem and multiplicative filter for the band limited signal in the linear canonical domain. Finally, the sampling and reconstruction formulas are deduced, together with the construction methodology for the above mentioned multiplicative filter in the time domain based on fast Fourier transform (FFT), which has much lower computational load than the construction method in the linear canonical domain.     

一种新的分数阶Fourier时频图示法

分数阶Fourier（FRFT）是传统Fourier的广义形式。分数阶Fourie域（FRFD）是一个统一的时频变换域,分数阶Fourier变换是角度为口的时频面旋转。随着角度α从0逐渐增加到π/2,分数阶Fourier变换展示出信号从时域到频域的全过程。本文依据分数阶FOUrier变换的定义,随着角度α的变化给出了一种新的更为直观的分数阶Fourier的时频图示方法,以供读者参考。     

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.     

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.     

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)     

A novel fractional wavelet transform and its applications

The wavelet transform (WT) and the fractional Fourier transform (FRFT) are powerful tools for many applications in the field of signal processing.However,the signal analysis capability of the former is limited in the time-frequency plane.Although the latter has overcome such limitation and can provide signal representations in the fractional domain,it fails in obtaining local structures of the signal.In this paper,a novel fractional wavelet transform (FRWT) is proposed in order to rectify the limitations of the WT and the FRFT.The proposed transform not only inherits the advantages of multiresolution analysis of the WT,but also has the capability of signal representations in the fractional domain which is similar to the FRFT.Compared with the existing FRWT,the novel FRWT can offer signal representations in the time-fractional-frequency plane.Besides,it has explicit physical interpretation,low computational complexity and usefulness for practical applications.The validity of the theoretical derivations is demonstrated via simulations.     

一种新型分数阶小波变换及其应用

小波变换和分数Fourier变换是应用非常广泛的信号处理工具.但是,小波变换仅局限于时频域分析信号;分数Fourier变换虽突破了时频域局限能够在分数域分析信号,却无法表征信号局部特征.为此,提出了一种新型分数阶小波变换,该变换不但继承了小波变换多分辨分析的优点,而且具有分数Fourier变换分数域表征功能.与现有分数阶小波变换相比,新型分数阶小波变换可以实现对信号在时间-分数频域的多分辨分析.此外,该变换具有物理意义明确和计算复杂度低的优点,更有利于满足实际应用需求.最后,通过仿真实验验证了所提理论的有效性.     

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.     

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.     

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。     

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。     

LFM宽带雷达信号的多通道盲压缩感知模型研究

在传统压缩感知(Compressed sensing, CS)基础上,提出了一种基于盲压缩感知(Blind compressed sensing, BCS)理论的线性调频(Linear frequency modulated, LFM)雷达信号欠采样与重构的多通道模型.这一机制在稀疏基未知的条件下,利用LFM信号在分数阶傅里叶变换(Fractional Fourier transform, FRFT)域上良好的能量聚集特性,将多个LFM信号看作是在多个未知阶次下FRFT域的稀疏表达,通过时延相关解线调和逐次消去相结合的的欠采样方法逐一估计出每个通道的LFM信号满足聚集性条件的特定分数阶傅里叶域,以此构造出该通道LFM信号对应的DFRFT正交稀疏基字典,以各DFRFT 正交基为对角块构建混合信号正交稀疏基字典,最后利用块重构算法从测量值中估计出稀疏信号,同时验证了LF M信号多通道BCS问题解的唯一性,从而实现了稀疏基未知情况下针对多路LFM宽带雷达信号的多通道盲压缩感知.     

应用分数阶傅里叶变换的形状描述方法研究

傅里叶描述子是一种经典的形状描述方法。作为傅里叶变换的推广形式,分数阶傅里叶变换在数字信号处理工程领域已有相当广泛的应用,但在形状分析领域还很少有研究工作的报道。首次研究了基于分数阶傅里叶变换的形状描述方法,比较了不同阶数下的分数阶傅里叶描述子在图像检索中的性能。通过在MPEG-7的标准图像测试集的图像检索实验,得出：阶数ρ为0.1时,分数阶傅里叶描述子的检索效果最差,随ρ=0.1的增长,检索性能总体呈上升趋势,当ρ=0.5变化到1.0时,检索性能最高。同时,与Zernike矩进行比较：当阶数为0.1时,分数阶傅里叶描述子的检索性能较差;而阶数为0.5、1.0时分数阶傅里叶描述子的检索性能均较好。     

基于TMS320C6201的离散分数阶傅立叶变换快速算法详细实现

各种离散分数阶傅立叶变换DFRFT(Discrete Fractional Fourier Transform)算法的发展促进了分数阶傅立叶变换FRFT(Fractional Fourier Transform)在数字信号处理领域的应用。本文首先介绍了FRFT的定义和特性,并给出了几种DFRFT计算方法的比较。在对Ozaktas提出的DFRFT快速算法理论分析基础上,本文给出了基于TMS320C6201定点数字信号处理器DSP(Digital Signal Processor)的快速算法详细实现。该详细充分利用FFT计算和数学处理来有效降低算法的运算量。     

基于分数阶傅里叶变换的多项式相位信号参数估计

研究了一种基于分数阶傅里叶变换(FRFT)的多项式相位信号快速估计方法,对于线性调频信号(LFM),即用信号延时相关解调的方法得到调频斜率的粗略估计,从而得到分数阶旋转角度的范围,简化为小范围的一维搜索问题。多项式相位信号的检测通过延时相关解调可转化为LFM信号的检测,再运用FRFT便可进行参数估计。理论分析与仿真结果表明该方法简单,估计性能好。     

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.     

Fractional Fourier domain equalization for single carrier broadband wireless systems

In this paper,a novel single carrier equalization approach in the fractional Fourier domain(FRFD) is proposed.It can remove all the inter-symbol interference(ISI) and avoid the considerable noise enhancement of the frequency domain-zero forcing(FD-ZF) equalizer.As the fractional Fourier transform makes a chirp spread,the impulse response of the deep fading channel may be flattened in some orders of the FRFD while it would be greatly attenuated in the FD.By searching for an optimal order under certain criterion,we take advantages of the ZF algorithm to mitigate the effects of the ISI completely.This approach can overcome the contradiction between the ISI mitigation and the noise enhancement of the FD-ZF equalizer.Theoretical analysis and simulation results show that the proposed FRFD-ZF equalizer can achieve a significant performance and have the same computation cost O(N log N) as the conventional FD linear equalizer,especially in the frequencyselective deep-fading channels.     

STLFMCW信号的周期FRFT检测与参数估计

结合对称三角线性调频连续波（STLFMCW）信号的时频特性,提出了一种采用周期分数阶傅里叶变换（FRFT）的STLFMCW信号检测与参数估计方法。分析了STLFMCW信号在周期FRFT域的能量积累特性,推导了信号周期FRFT的处理增益表达式。利用噪声在周期FRFT域的概率统计特性实现了检测门限的自适应确定。由于具有周期积累特性,在低信噪比下,该方法对调频周期、起始时间偏移、调频斜率和载频具有较好的估计性能。最后,仿真验证了该方法的有效性。     

Research progress of the fractional Fourier transform in signal processing

While solving a heat conduction problem in 1807, a French scientist Jean Baptiste Jo-seph Fourier, suggested the usage of the Fourier theorem. Thereafter, the Fourier trans-form (FT) has been applied widely in many scientific disciplines, and has played i…