线性正则变换域的框架理论研究

框架理论是研究信号采样特别是非均匀采样问题的一个重要工具,基于框架理论的Gabor展开是在时频混合空间描述信号的非正交展开,而Weyl-Heisenberg框架理论是Gabor展开的理论基础,它描述了信号在时频平面的局部性质与特点,在信号的短时Fourier变换(STFT)、Gabor展开中起到非常重要的作用.本文探讨了W-H框架在线性正则变换下的特点,得到了在时域和线性正则变换域联合空间中信号的展开形式,为在线性正则变换域中讨论信号的展开和非均匀采样问题打下了良好的理论基础.     

一种基于有限更新率的非带限信号采样方法

针对Shannon采样定理只能处理带限信号的缺陷,推导论证了一种基于有限更新率的非带限信号采样方法.该方法处理的是更新率有限的信号,先将信号经过一个采样核滤波,然后以不低于信号更新率的采样率对其采样,就可以借助抵消滤波器和范德蒙方程从采样点中重建出原信号.计算机仿真结果表明,该方法重建原信号的精确度较高,并且具有较好的抗噪声性能,信噪比为20dB时信号估计值的均方差只有10-5的数量级,完全能满足实际工程的需求.     

基于奇异值分解的非带限信号采样算法

本文借助奇异值分解技术,研究了对狄拉克流这类特殊非带限信号的采样与重建算法。先对信号的采样点进行离散傅里叶变换,并用生成的系数形成Hankel矩阵,然后对Hankel矩阵进行奇异值分解,求取狄拉克流的位置信息,再解范德蒙方程组求得狄拉克流的权信息,重建出原信号。该算法具有计算量小、信号恢复精确率高和抗噪声能力强的特点。     

利用带限信号的部分时域和频域信息恢复离散信号

本文提出了离散情况下利用带限信号的部分时域信息和部分频域信息而不知道频带进行信号恢复的问题,分析了进行信号重构时解的可能性,给出了最小二乘解法,并且在有多个解的情况下,给出了最小能量准则,以此为基础确定了最小能量解,并且给出了最小能量解的解析形式.仿真结果表明本文提出的方法是行之有效的。     

带通信号的直接采样理论

用基本的数学方法和信号处理理论研究了带通信号的直接采样问题,得到了一般性的结论,进一步丰富和深化了信号的直接采样理论。     

小波空间中采样理论研究

本文研究了小波理论在非带限信号采样中的应用，分析了Walter子波空间采样定理及其突破Nyquist率限制的理论依据；介绍了在信号恢复的过程中可以避免无限冲激响应滤波器出现的周期性非均匀采样及导数采样理论，并以紧支信号为例分别进行了仿真验证。     

利用带限信号的部分时域信息和频域信息恢复信号

本文在介绍带限信号恢复问题及有关算法的基础上，从理论上详细讨论了一类的带限信号的恢复问题，即在只知道时域的部分值和频域的部分值而不知道频带的情况下，给出了相应的算法，并论证了算法的收敛性及算法收敛的只一性，而且对外推的数值计算误差及外推和稳定性也作了说明。本文还给出了模拟例子以及数字仿真的结果。本文提出了的算法是行之有效的。     

利用小波技术检测淹没在噪声中的带限信号

导出了小波分解层数、信号采样频率和信号频带的关系式，说明了当信号的宽带为小波分解的某一个频带时的输出信噪比为最优，仿真结果表明，利用小波技术不仅可以检测到淹没在噪声中的带限信号，而且可以估计其到达时间和脉宽。     

带限信号外推问题新的解析解

本文给出能量有限带限信号外推问题新的解析解，它比长球波函数展开法更为在于新的解析解公式，给出了相应的算法和稳定性判据。     

分数阶傅里叶变换域上带通信号的采样定理

傅里叶变换和采样定理是信号处理领域的两大基本问题,采样定理研究了傅里叶变换域上带限信号的采样和重构理论.分数阶傅里叶变换(FRFT)是傅里叶变换的一种推广,与之相应的采样理论目前还不十分完备,所以有必要从FRFT域上重新研究采样定理.本文首先得到了均匀冲激串采样信号的FRFT,然后在此基础上导出了FRFT域上带通信号和低通信号的采样定理和重构公式.这些结果是经典理论的推广,将丰富分数阶傅里叶变换的理论体系.     

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.     

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.     

采用线性正则域模糊函数的QFM信号参数估计算法输出信噪比分析

采用线性正则域模糊函数的二次调频 (quadratic frequency modulated,QFM) 信号参数估计算法简单易解,估计精度较高,误差传递小,在实际应用中有较好的前景。本文对该算法的输出信噪比进行了较为深入而详细的分析,推导了输出信噪比与输入信噪比及信号采样点数之间的关系表达式;通过仿真实验比较了在同等条件下本算法和积分广义模糊函数算法(integrated generalized ambiguity function, IGAF))以及多项式相位变换(polynomial-phase transform, PPT)算法的输出信噪比大小,以及达到相同大小输出信噪比所需采样点数。发现本算法的输出信噪比要大于IGAF算法和PPT算法,且达到相同大小的输出信噪比所需采样点数分别是IGAF算法和PPT算法的1/4和1/9,即得到相同大小的输出信噪比时本算法所需的采样点数更少.      

基于线性正则变换与短时傅里叶变换联合的时频分析方法

线性正则变换作为傅里叶变换、分数阶傅里叶变换更为广义的形式,已经在光学和信号处理等领域得到了应用.短时傅里叶变换是一种线性时频分布,避免了其他双线性时频分布中出现的交叉项干扰,是分析时频信号的有力工具.本文从线性正则变换的定义和性质出发,研究了线性正则变换与短时傅里叶变换的时频关系,提出了基于线性正则变换与短时傅里叶变换联合的时频分析方法,避免了交叉项问题能够实现chirp信号干扰抑制和多分量时频信号分离.最后用仿真实例表明,该方法是分析时频信号的有效手段.     

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Wigner-Hough变换的LFM信号检测性能分析

根据Wigner-Hough变换思想和广义似然比检验理论,分析了双门限情况下线性调频信号的检测性能,给出了虚警概率和检测概率的数学表达式,该表达式是Hough变换的积累单元数和检测门限的函数,然后对概率密度函数进行了合理近似,并给出了Hough变换的积累单元数计算公式,最后的计算机仿真比较了傅立叶变换和Wigner-Hough变换的LFM(Linear Frequency Modulated)信号检测性能和基于Wigner-Hough变换的不同目标起伏类型的LFM信号检测性能.     

Multichannel Sampling of Signals Band-Limited in Offset Linear Canonical Transform Domains

The aim of the multichannel sampling is the reconstruction of a band-limited signal f(t), from the samples of the responses of M linear time invariant systems, each sampled by the 1/Mth Nyquist rate. As the offset linear canonical transform (OLCT) has been found wide applications in signal processing and optics fields, it is necessary to consider the multichannel sampling based on offset linear canonical transform. In this paper, we develop a multichannel sampling theorem for signals band-limited in offset linear canonical transform domains. Moreover, by designing different OLCT filters, reconstruction formulas for uniform sampling from the signal, from the signal and its first derivative or its generalized Hilbert transform are obtained based on the derived multichannel sampling theorem. Since recurrent nonuniform sampling for the signal has valuable applications, reconstruction expression for recurrent nonuniform samples of the signal band-limited in the offset linear canonical transform domain is also obtained by using the derived multichannel sampling theorem and the properties of the offset linear canonical transform.