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 and product theorems associated with linear canonical transform

The linear canonical transform (LCT), which is a generalized form of the classical Fourier transform (FT), the fractional Fourier transform (FRFT), and other transforms, has been shown to be a powerful tool in optics and signal processing. Many results of this transform are already known, including its convolution theorem. However, the formulation of the convolution theorem for the LCT has been developed differently and is still not having a widely accepted closed-form expression. In this paper, we first propose a generalized convolution theorem for the LCT and then derive a corresponding product theorem associated with the LCT. The ordinary convolution theorem for the FT, the fractional convolution theorem for the FRFT, and some existing convolution theorems for the LCT are shown to be special cases of the derived results. Moreover, some applications of the derived results are presented.     

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.     

Sampling and series expansion for linear canonical transform

The linear canonical transform (LCT) has been shown to be a powerful tool for optics and signal processing. This paper investigates new sampling relations in the LCT domain. Firstly, the relationship between linear canonical series (LCS) and LCT is introduced. The LCS expansion coefficients are the sampled values of LCT. Then, based on the conventional Fourier series and LCS, two new sampling relations in the LCT domain are presented, where the signal in the time domain is reconstructed from the samples of its LCT directly. The first theorem considers signals band-limited in some LCT domain, and the second deals with signals band-limited in the conventional Fourier transform domain.     

Sampling of bandlimited signals in the linear canonical transform domain

The linear canonical transform (LCT) has been shown to be a powerful tool for optics and signal processing. This paper investigates the sampling of bandlimited signals in LCT domain. First, we propose the linear canonical series (LCS) based on the LCT, which is a generalized pattern of Fourier series. Moreover, the LCS inherits all the nice properties from the LCT. Especially, the Parseval’s relation is presented for the LCS, which is used to derive the sampling theorem of LCT. Then, utilizing the generalized form of Parseval’s relation for the complex LCS, we obtain the sampling expansion for bandlimited signals in LCT domain. The advantage of this reconstruction method is that the sampling expansion can be deduced directly not based on the Shannon theorem.     

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     

On Sampling of Band-Limited Signals Associated With the Linear Canonical Transform

Sampling is one of the fundamental topics in the signal processing community. Theorems proposed under this topic form the bridge between the continuous-time signals and discrete-time signals. Several sampling theorems, which aid in the reconstruction of signals in the linear canonical transform (LCT) domain, have been proposed in the literature. However, two main practical issues associated with the sampling of the LCT still remain unresolved. The first one relates to the reconstruction of the original signal from nonuniform samples and the other issue relates to the fact that only a finite number of samples are available practically. Focusing on these issues, this paper seeks to address the above from the LCT point of view. First, we extend several previously developed theorems for signals band-limited in the Fourier domain to signals band-limited in the LCT domain, followed by the derivation of the reconstruction formulas for finite uniform or recurrent nonuniform sampling points associated with the LCT. Simulation results and the potential applications of the theorem are also proposed.      

Recovery of Bandlimited Signals in Linear Canonical Transform Domain from Noisy Samples

The linear canonical transform (LCT) has been shown to be a useful and powerful tool for signal processing and optics. Many reconstruction strategies for bandlimited signals in LCT domain have been proposed. However, these reconstruction strategies can work well only if there are no errors associated with the numerical implementation of samples. Unfortunately, this requirement is almost never satisfied in the real world. To the best of the author’s knowledge, the statistical problem of LCTed bandlimited signal recovery in the presence of random noise still remains unresolved. In this paper, the problem of recovery of bandlimited signals in LCT domain from discrete and noisy samples is studied. First, it is shown that the generalized Shannon-type reconstruction scheme for bandlimited signals in LCT domain cannot be directly applied in the presence of noise since it leads to an infinite mean integrated square error. Then an orthogonal and complete set for the class of bandlimited signals in LCT domain is proposed; and further, an oversampled version of the generalized Shannon-type sampling theorem is derived. Based on the oversampling theorem and without adding too much complexity, a reconstruction algorithm for bandlimited signals in LCT domain from discrete and noisy observations is set up. Moreover, the convergence of the proposed reconstruction scheme is also proved. Finally, numerical results and potential applications of the proposed reconstruction algorithm are given.     

Maximally concentrated sequences in both time and linear canonical transform domains

In signal analysis based on linear canonical transform (LCT), researchers often seek signal designs that have the greatest concentrations simultaneously in both time and LCT domains. This paper investigates the extent to which a sequence and its LCT can be simultaneously concentrated in their respective domains. Firstly, the most concentration of indexlimited sequences in LCT domain is derived. Then, the most concentration of (a, b, c, d)-bandlimited sequences in time domain is given. It is shown that the discrete generalized prolate spheroidal sequences (DGPSSs), which generalize the discrete prolate spheroidal sequences proposed by Slepian for Fourier transform to LCT, do the best job of simultaneous concentrations in both time and LCT domains. Associated with DGPSSs are certain related functions of frequency in LCT domain, called discrete generalized prolate spheroidal wave functions (DGPSWFs). Some interesting properties of DGPSWFs as well as two relationships between DGPSSs and DGPSWFs are also presented.     

Sampling expansion in function spaces associated with the linear canonical transform

In this paper, we investigate sampling expansion for the linear canonical transform (LCT) in function spaces. First, some properties of the function spaces related to the LCT are obtained. Then, a sampling theorem for the LCT in function spaces with a single-frame generator is derived by using the Zak Transform and its generalization to the LCT domain. Some examples are also presented.     

Active-matting-based object tracking with color cues

The linear canonical transform (LCT) has been shown to be a useful and powerful analyzing tool in optics and signal processing. Many results of this transform are already known, including its uncertainty principles (UPs). The existing UPs of the LCT for complex signals can only provide sharp bounds with LCT parameters satisfying $a_1/b_1\ne a_2/b_2$ . However, in most cases, we strive to find a lower bound, but not a sharper bound, since a lower bound often leads to optimization problems in signal processing applications. In this paper, we first present a much briefer and more transparent derivation to obtain a general uncertainty principle of the LCT for arbitrary signals via operator methods. Then, we derive lower bounds of three UPs of the LCT for complex signals, which are tighter lower bounds than the existing ones. We also prove that the derived results hold for arbitrary LCT parameters.     

线性正则域抑制频谱弥散干扰方法

频谱弥散干扰（SMSP）是一种对抗线性调频脉冲压缩雷达的新型干扰样式。自卫式干扰条件下,目标回波与干扰信号时频域重叠,传统抗干扰手段难以有效抑制。针对该问题,分析了目标回波和干扰信号线性正则域（LCT域）特征,依据特征差异提出了两种LCT域干扰抑制方法,分别为窄带滤波法和稀疏重构法,其中窄带滤波方法通过LCT域遮盖处理,滤除干扰分量,而稀疏重构方法首先利用Pei型DLCT核矩阵构造正交字典,然后基于压缩感知理论重构真实回波。仿真表明,所提两种方法均具有一定的干扰抑制效能,恢复信号与真实回波高度相似,且幅度高于真实回波,使干扰信号成为标的。      

基于分数阶Fourier变换的模糊函数及其在二次调频信号参数估计应用中的研究

作为处理非平稳信号的一种重要工具,模糊函数(ambiguity function,AF)已经被广泛应用于雷达信号处理、声纳技术等领域,并对线性调频信号信号的参数估计具有极好的处理能力。但对应用于众多领域的二次调频信号,模糊函数就显得无能为力了。作为Fourier变换的更广义形式,分数阶Fourier变换(Fractional Fourier transform)近年来受到了广泛关注。为解决二次调频信号的估计问题,本文研究了基于分数阶Fourier变换的模糊函数,给出了这种变换的一些新的重要性质,如共轭对称性、Moyal公式、时移性等,推导出了它与经典模糊函数、基于分数阶Fourier变换的Wigner分布、短时Fourier变换、小波变换等其他时频变换的关系。作为应用,最后本文用这种分数阶模糊函数来估计二次调频信号,应用实例的仿真结果表明了分数阶模糊函数在估计二次调频信号参数方面的可行性和有效性。      

Adaptive time-varying cancellation of wideband interferences inspread-spectrum communications based on time-frequencydistributions

The paper proposes an adaptive method for suppressing wideband interferences in spread-spectrum (SS) communications. The proposed method is based on the time-frequency representation of the received signal from which the parameters of an adaptive time-varying interference excision filter are estimated. The approach is based on the generalized Wigner-Hough transform as an effective way to estimate the instantaneous frequency of parametric signals embedded in noise. The performance of the proposed approach is evaluated in the presence of linear and sinusoidal FM interferences plus white Gaussian noise in terms of the SNR improvement factor and bit error rate (BER)     

Three uncertainty relations for real signals associated with linear canonical transform

Uncertainty principle plays an important role in signal processing, physics and mathematics, and it represents the relations between time spread and frequency spread (or position and velocity). Linear canonical transform (LCT) is one generalisation of Fresnel transform, fractional Fourier transform and others. The LCT has been used in physical optics and signal processing. Three novel results of uncertainty principle in the LCT domains are obtained here, in which one is connected with parameters a and b and the other one is connected with c and d; the last one is connected with the four transformation parameters a, b, c and d. Their physical meanings are given as well. These results disclose the inequalities? relations between two spreads, between two group delays and between one spread and one group delay in the LCT domains. It also shows that any one of the three cases can reduce to classical uncertainty principle in time/frequency domain. The effects of time scaling on these results? bounds are also involved.     

小波模极大值点的信号稀疏表示及重建

作为压缩感知理论的前提,稀疏表示要求信号本身是稀疏的或者在某种正交基下可以稀疏表示。本文针对信号本身及小波变换后均不够稀疏的情况,提出一种基于模极大值点的信号稀疏表示算法。该算法在小波变换的基础上,利用小波分解的结构,对各层高频小波系数通过寻找其模极大值点的方法进行稀疏化,然后通过测量矩阵得到它的测量值,对测量点数进行熵编码以实现数据压缩传输。解码时,采用正交匹配追踪算法得到模极大值点的估计值,最后通过交替投影法重构出原信号。仿真结果表明,与经典压缩感知算法相比,该算法恢复信号的质量有较大提高,且由于稀疏度增大,所以信号具有更好的可压缩性,实验表明本文算法对复杂信号效果更明显。      

A novel clipping technique for filtering FM signals embedded in intensive noise

A novel adaptive clipping technique for filtering a constant amplitude frequency modulated (FM) signal embedded in non-Gaussian noise is proposed. It is based on the analysis and processing of the estimate of probability density function of a FM signal realization. As a result, modifications of two robust estimators of FM signal amplitude are proposed. It is shown that these estimators can be used for Gaussian and non-Gaussian heavy-tail environments. The proposed clipping technique can exploit one or another obtained robust estimate of the signal amplitude for adaptive setting a threshold. Analysis of signal estimate accuracy for different noise environments is carried out. Comparative analysis of the obtained methods and known approaches based on scanning window nonlinear filtering and optimal robust L-DFT form is performed. It is demonstrated that the usage of clipping-based technique leads to the considerable improvement of the FM signal filtering efficiency in comparison to the aforementioned known approaches for different noise environments and a wide range of input SNR values.     

Convolution,correlation, and sampling theorems for the offset linear canonical transform

The offset linear canonical transform (OLCT), which is a time-shifted and frequency-modulated version of the linear canonical transform, has been shown to be a powerful tool for signal processing and optics. However, some basic results for this transform, such as convolution and correlation theorems, remain unknown. In this paper, based on a new convolution operation, we formulate convolution and correlation theorems for the OLCT. Moreover, we use the convolution theorem to investigate the sampling theorem for the band-limited signal in the OLCT domain. The formulas of uniform sampling and low-pass reconstruction related to the OLCT are obtained. We also discuss the design method of the multiplicative filter in the OLCT domain. Based on the model of the multiplicative filter in the OLCT domain, a practical method to achieve multiplicative filtering through convolution in the time domain is proposed.     

Reconstruction by weighted correlation for MRI with time-varying gradients

A general reconstruction algorithm for magnetic resonance imaging (MRI) with gradients having arbitrary time dependence is presented. This method estimates spin density by calculating the weighted correlation of the observed free induction decay signal and the phase modulation function at each point. A theorem which states that this method can be derived from the conditions of linearity and shift invariance is presented. Since these conditions are general, most of the MRI reconstruction algorithms proposed so far are equivalent to the weighted correlation method. An explicit representation of the point spread function (PSF) in the weighted correlation method is given. By using this representation, a method to control the PSF and the static field inhomogeneity effects is studied. A correction method for the inhomogeneity is proposed, and a limitation is clarified. Some simulation results are presented.     

基于稀疏优化的图像压缩感知重建算法

在信号的稀疏表示方法中,传统的基于变换基的稀疏逼近不能自适应性地提取图像的纹理特征,而基于过完备字典的稀疏逼近算法复杂度过高.针对该问题,文章提出了一种基于小波变换稀疏字典优化的图像稀疏表示方法.该算法在图像小波变换的基础上构建图像过完备字典,利用同一场景图像的小波变换在纹理上具有内部和外部相似的属性,对过完备字典进行灰色关联度的分类,有效提高了图像表示的稀疏性.将该新算法应用于图像信号进行稀疏表示,以及基于压缩感知理论的图像采样和重建实验,结果表明新算法总体上提升了重建图像的峰值信噪比与结构相似度,并能有效缩短图像重建时间.