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.     

基于回转椭球波函数的光学系统分析

回转椭球波函数在有限空间域和无限空间域内都是一组完备正交函数集,适合分析孔径尺寸有限的实际光学系统。线性正则变换是一种重要的时频分析工具,同时菲涅耳变换也是一种特殊的线性正则变换,因此线性正则变换可以模拟一个光学系统。研究了有限空间域和有限频域条件下的回转椭球波函数的补偿线性正则变换,以补偿线性正则变换模拟一个二维光学系统,并以回转椭球波函数作为信号函数,分析了信号通过该系统的能量损失情况,根据椭球波函数本征值性质,其本征值反映了椭球波函数的能量保存比。数值计算结果表明信号函数通过该系统的能量比与通过函数本征值的能量比的估计值一致,表明了该方法的有效性。     

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.     

基于Gabor空间的未知调频率LFM信号压缩采样与重构

针对现有压缩采样系统在宽带线性调频信号压缩采样过程中存的采样系统不适用、调制信息依赖等问题,提出了一种基于Gabor空间的线性调频信号压缩采样与重构方法,在未知调频率的条件下,实现了线性调频信号的压缩采样与有效重构。首先,结合压缩感知以及平移不变空间理论,设计了基于Gabor空间的压缩采样系统,分析了压缩采样系统组成部分以及工作原理。然后,利用信号在Gabor空间的稀疏性,建立了Gabor系数的压缩重构模型,并在充分考虑噪声、失配的条件下,分析了原始信号重构误差上界。最后,通过数值仿真实验,验证了所提方法的有效性,实验结果表明,基于Gabor空间的压缩采样系统具有采样频率低,采样点数少,以及工作稳定性高等优点。      

分数阶Fourier域上非均匀采样信号的频谱重构研究

 本文研究了分数阶Fourier变换域上非均匀采样信号的重构问题.首先得到周期非均匀采样信号经非均匀分数阶Fourier变换后的频谱表达式,研究了该分数阶频谱和信号连续分数阶频谱之间的关系,并基于该关系式提出了一种分数阶Fourier域周期非均匀采样信号的频谱重构算法;其次,讨论了分数阶Fourier变换域上更加一般情况下非均匀采样信号重构问题;最后,给出了周期非均匀采样信号频谱重构的仿真结果.     

非均匀采样信号的分数阶数字频谱研究

利用分数阶Fourier变换对实际采样过程中出现的非均匀、非理想采样信号进行了分数阶频谱分析与研究,得到了这类非均匀采样信号在分数阶Fourier变换域的数字频谱表达形式.由此进一步得到了非均匀采样Chirp信号在分数阶域的频谱表达式,并分析了非均匀采样Chirp信号在分数阶Fourier变换域的分数阶频谱性质,最后仿真结果证明了结论的正确性.     

基于高度冗余Gabor框架的欠Nyquist采样系统子空间探测

基于指数再生窗Gabor框架的欠Nyquist采样系统对窄脉冲信号完成采样与重构一般情况下效果较好,但是当框架高度冗余时,使用传统面向系数域的方法对信号进行子空间探测会面临失败或较大误差。该文采用面向信号域的思想,构建了分块的对偶Gabor字典,并对信号分块稀疏表示;根据信号的分块表示推导了采样系统的测量矩阵,提出了测量矩阵受字典相干性约束的分块-相干性;将信号合成模型引入多观测向量问题,提出基于分块-闭包的同步正交匹配追踪算法(SOMPB,F ),用于信号子空间探测。此外还证明了算法的收敛约束条件。仿真结果表明,所提子空间探测方法相比传统方法提高了信号重构成功率,降低了采样通道数,并增强了系统鲁棒性。     

基于Bayes准则自适应Gabor系数缩减的信号恢复

信号在联合时频平面内是相对稀疏的,鉴于此,本文通过假设信号Gabor展开系数先验分布和噪声Gabor展开系数的先验分布,然后应用Bayes准则从含噪信号Gabor展开系数的后验分布估计未知信号的Gabor展开系数,实现时变滤波.数值实验表明该方法是合理有效的.     

RFRFT域非均匀采样信号重构算法研究

讨论了非均匀采样信号简化分数阶傅立叶变换（RFRFT）域频谱重构的方法,推导了利用原信号的连续频谱无偏估计重构信号的算法,并得到了重构公式,验证了RFRFT域非均匀采样信号重构的可实现性.同时仿真了RFRFT域上非均匀采样信号的重构实施例,验证了该方法在RFRFT域上非均匀采样信号重构的准确性和稳定性.     

分数阶Fourier域非均匀采样LFM信号的检测概率

为了减少分数阶Fourier域线性调频(LFM)信号处理的计算量,满足实时性要求,提出了一种新的非均匀采样方法.与传统的非均匀采样方法相比,该方法既解决了整个采样时段内由于过采样造成的数据冗余性,也消除了信号时变特征造成的局部冗余性.最后,应用该方法得到了非均匀采样LFM信号在分数阶Fourier变换域的频谱表达式,并对不同信噪比(SNR)条件下信号的检测概率进行了研究.结果表明,该方法具有较好的抗噪声能力.     

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.     

Short-time Fourier transform: two fundamental properties and an optimal implementation

Shift and rotation invariance properties of linear time-frequency representations are investigated. It is shown that among all linear time-frequency representations, only the short-time Fourier transform (STFT) family with the Hermite-Gaussian kernels satisfies both the shift invariance and rotation invariance properties that are satisfied by the Wigner distribution (WD). By extending the time-bandwidth product (TBP) concept to fractional Fourier domains, a generalized time-bandwidth product (GTBP) is defined. For mono-component signals, it is shown that GTBP provides a rotation independent measure of compactness. Similar to the TBP optimal STFT, the GTBP optimal STFT that causes the least amount of increase in the GTBP of the signal is obtained. Finally, a linear canonical decomposition of the obtained GTBP optimal STFT analysis is presented to identify its relation to the rotationally invariant STFT.     

Recent developments in the core of digital signal processing

In this article members of the Digital Signal Processing (DSP) Technical Committee (TC) report recent breakthroughs in signal processing fundamentals that have happened in the last two decades. These breakthroughs include various advances and extensions from old techniques to new techniques. For example, signal processing techniques have moved from single-rate to multirate processing, from time-invariant to adaptive processing, from frequency-domain (the traditional Fourier transform, as we know it) to time-frequency analysis, and from linear to non-linear signal processing. Recent developments in these areas have not only renovated the theory of digital signal processing, they have also resulted in new tools that find applications in various domains. For instance, multirate signal processing has triggered recent advances in modem technology and speech/audio coding; adaptive filtering has made echo cancellation and noise suppression possible; time-frequency analysis has found its way into various applications in radar and medical signal processing; and non-linear processing has made engineers rethink various problems in speech recognition and image analysis. This article provides an extensive list of highlights from these recent developments.     

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.     

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     

Envelope detection using generalized analytic signal in 2D QLCT domains

The hypercomplex 2D analytic signal has been proposed by several authors with applications in color image processing. The analytic signal enables to extract local features from images. It has the fundamental property of splitting the identity, meaning that it separates qualitative and quantitative information of an image in form of the local phase and the local amplitude. The extension of analytic signal of linear canonical transform domain from 1D to 2D, covering also intrinsic 2D structures, has been proposed. We use this improved concept on envelope detector. The quaternion Fourier transform plays a vital role in the representation of multidimensional signals. The quaternion linear canonical transform (QLCT) is a well-known generalization of the quaternion Fourier transform. Some valuable properties of the two-sided QLCT are studied. Different approaches to the 2D quaternion Hilbert transforms are proposed that allow the calculation of the associated analytic signals, which can suppress the negative frequency components in the QLCT domains. As an application, examples of envelope detection demonstrate the effectiveness of our approach.     

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.