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.     

周期非均匀采样信号的重建中的优化子频带划分

对于周期非均匀采样,由于每个均匀采样流的采样率通常都是小于Nyquist率的,因此,采样信号频谱中会发生频率混叠。这表明在采样信号的重建频带内将会有多于1的谱分量。因为不是所有的与重建频带相交的谱分量都会覆盖整个重建频带,所以有必要将重建频带分成若干个子频带进行分析,构造内插函数,这样有利于减小重建所必需的最小采样率。本文提出一种优化的子频带划分方法,通过该方法在重建频带上定义子频带,能在保证重建所需的采样率最低的情况下使子频带的个数最少,这对于简化内插滤波器的结构有重要的意义。     

A Versatile Iterative Framework for the Reconstruction of Bandlimited Signals from Their Nonuniform Samples

In this paper, we study a versatile iterative framework for the reconstruction of uniform samples from nonuniform samples of bandlimited signals. Assuming the input signal is slightly oversampled, we first show that its uniform and nonuniform samples in the frequency band of interest can be expressed as a system of linear equations using fractional delay digital filters. Then we develop an iterative framework, which enables the development and convergence analysis of efficient iterative reconstruction algorithms. In particular, we study the Richardson iteration in detail to illustrate how the reconstruction problem can be solved iteratively, and show that the iterative method can be efficiently implemented using Farrow-based variable digital filters with few general-purpose multipliers. Under the proposed framework, we also present a completed and systematic convergence analysis to determine the convergence conditions. Simulation results show that the iterative method converges more rapidly and closer to the true solution (i.e. the uniform samples) than conventional iterative methods using truncation of sinc series.     

基于粒子群算法的非均匀稀布阵列综合

给出了一种基于粒子群算法的非均匀稀布阵列综合方法,设计有最小阵元间距约束的稀布阵,通过加入间距约束向量改进了适应度算法,不仅减小了布阵空间,而且消除了优化过程中的不合格个体。在给定阵列孔径和阵元数的条件下,实现了最小阵元间距约束下抑制栅瓣,降低旁瓣电平的阵列综合。通过仿真实例,验证了此方法的高效可行性。     

Time-Varying and Frequency-Selective Channel Estimation with Unequally Spaced Pilot Symbols

In this paper, an accurate and computationally efficient algorithm is proposed for estimating time-varying and frequency-selective fading channel with unequally spaced pilot symbols. By employing the time-varying coefficient polynomial interpolation method, it is proved that the time-varying channel impulse response can be estimated by the product of a constant matrix and the fading information at pilot symbol positions. Furthermore, a least square off-line training algorithm is presented to optimally calculate the constant matrix, taking into consideration of the statistics of channel fading and noise. The new algorithm can also be applied for estimating flat fading channel with equally spaced pilot symbols as a special case. Simulation results indicate that our new channel estimation algorithm leads to small mean square error for fading estimation and provides bit error rate performance close to that of the perfect channel estimation.     

Interpolation of Bounded Bandlimited Signals and Applications

A method to interpolate a bounded bandlimited signal from its own samples with minimum complexity is presented that guarantees an a priori specified accuracy. The method is the result of combining a fast-converging sampling expansion with the Farrow interpolator technique. It provides important complexity reductions in various signal processing applications, from which three cases are studied. For the Farrow interpolator, it assures that the interpolation error will be smaller than a known bound, both in the time and frequency domains. For delay estimation, the method allows one to decouple a given estimation/detection algorithm in two steps. In the first one, a few finite convolutions are carried out in order to compute a set of interpolation coefficients. In the second, the algorithm is executed but with a complexity independent of the length of the temporal observation interval. Besides, it is shown how to introduce this decoupling in the matched filter, MUSIC, and conditional maximum-likelihood delay estimators. Finally, the method is employed to give a simple solution to an important efficiency problem in the simulation of communications systems: the generation of Rayleigh processes.     

带限MPSK信号码元宽度的盲估计

理想MPSK信号在码元跳变处表现为相位的不连续性。小波变换能有效检测出这些信号细节,从而为估计码元宽度提供了依据。实际系统常采用脉冲成形技术减小发射带宽,改变了信号包络,从而影响了小波变换的检测性能。本文对截获接收机输出的信号进行预处理后再利用基于Haar小波变换的算法提取码元宽度信息,理论分析和实际信号处理表明,这种算法能有效检测码元跳变,从而估计出码元宽度。     

Optical Equalization of 42.7-Gbaud Bandlimited RZ-DQPSK Signals

We demonstrate the use of an optical equalizer to allow 42.7-Gbaud (85.4-Gb/s) return-to-zero differential quadrature phase-shift keyed signals to better tolerate the narrow optical filtering required in high-spectral-efficiency systems. The equalizer passbands are repetitive, enabling the equalization of multiple channels.     

On the Design of Unequally Spaced Arrays with Nearly Equal Sidelobes

A modified iterative approach suitable for use with a modern digital computer is presented for designing unequally spaced arrays with nearly equal sidelobes. The method allows inclusion of spacing constraints in the design procedure with the result that certain practical difficulties encountered with the results of other computational schemes are avoided.     

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.     

基于迭代NUFFT的不等距阵列天线低副瓣方向图综合

针对不等间距阵列天线的低副瓣方向图综合问题,提出了一种基于非均匀快速傅立叶变换(NUFFT)的迭代算法.通过严格的数学推导,建立非均匀离散傅立叶变换(NUDFT)与天线阵因子之间的对应关系,利用过采样FFT系数的线性组合来近似表达复指数函数进而实现NUFFT,最后通过插值和反复的迭代有效地抑制了阵列天线的最大副瓣电平.结合等间距阵列和不等间距阵列的算例,对比粒子群算法的优化结果,验证了该算法的有效性和高效性.     

Unequally Spaced Channels for Upgrading WDM System from 3-Channel to 10-Channel Preserving No FWM Crosstalk

Crosstalk due to four-wave mixing (FWM) increases sharply when lightwave channels are upgraded in wavelength division multiplexing (WDM) fiber-optical communication systems. A technique is proposed for reducing crosstalk due to FWM when multichannel communication is extended from 3 channels to 10 channels in WDM system. The results of this simulation are presented. It is shown that suitable unequal channel separations can be found for which no FWM product term is superimposed on any of the transmitted channels, and the bandwidth expansion factor is not more than 2.7. Our method is better than previous ones.     

An Almost Sure Result on Sampling of Bandlimited Random Signals

The Shannon-Nyquist sampling theorem for deterministic signals is a fundamental result in the field of telecommunication and signal processing, and many re- sults on this topic have been obtained. However, very few results on random signals have been published, after Kol- mogorov mentioned the importance of Shannon-formula for stochastic signals in 1956. In this paper, following the almost sure result for bandlimited stochastic processes pro- posed by Seip in 1990, we give an almost sure result of the classical sampling theorem for bandlimited random signals with local average sampling.     

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.     

基于改进GA-FFT综合含互耦效应的不等间隔阵列赋形方向图

该文提出了一种虚拟的最小均方有源单元方向图展开方法,将不等间隔阵列的有源方向图展开为一个虚拟的均匀间隔阵列的若干单元辐射的叠加。通过该方法,对包含阵元耦合效应的不等间隔阵列方向图,可以使用快速傅里叶变换进行加速计算。并且,该文将这个方法与遗传算法(GA)相结合,得到一种改进的GA-FFT方法,可以应用于解决含阵元互耦的不均匀间隔阵列的赋形波束综合问题。最后,分别对不等间隔的偶极子阵列平顶方向图及微带阵列的余割平方方向图进行了综合,结果表明所提方法的有效性和优势。     

Concerning the Recovery of a Bandlimited Signal or Its Spectrum from a Finite Segment

The relationship between the iterative techniques of Sandberg and Papoulis is clarified. The conditions of Sandberg's theorem are not satisfied in general when the problem is that of extrapolating a bandlimited signal outside of a known segment (a problem that is "illposed"). For an interesting special case, the iteration may be applied and the signal recovered exactly.     

Reconstruction of Nonuniformly Sampled Bandlimited Signals Using a Differentiator–Multiplier Cascade

This paper considers the problem of reconstructing a bandlimited signal from its nonuniform samples. Based on a discrete-time equivalent model for nonuniform sampling, we propose the differentiator–multiplier cascade, a multistage reconstruction system that recovers the uniform samples from the nonuniform samples. Rather than using optimally designed reconstruction filters, the system improves the reconstruction performance by cascading stages of linear-phase finite impulse response (FIR) filters and time-varying multipliers. Because the FIR filters are designed as differentiators, the system works for the general nonuniform sampling case and is not limited to periodic nonuniform sampling. To evaluate the reconstruction performance for a sinusoidal input signal, we derive the signal-to-noise-ratio at the output of each stage for the two-periodic and the general nonuniform sampling case. The main advantage of the system is that once the differentiators have been designed, they are implemented with fixed multipliers, and only some general multipliers have to be adapted when the sampling pattern changes; this reduces implementation costs substantially, especially in an application like time-interleaved analog-to-digital converters (TI-ADCs) where the timing mismatches among the ADCs may change during operation.      

基于稀疏贝叶斯学习的空间紧邻信号DOA估计算法

离格(off-grid)波达方向(DOA)估计解决的是实际DOA和假设网格点的失配问题.对于空间紧邻信号的DOA,稀疏的网格点会导致精度和分辨率的下降,密集的网格点虽然可以提高估计精度却显著增加计算负担.针对此问题,该文提出基于稀疏贝叶斯学习(SBL)的空间紧邻信号DOA估计算法,主要包括3个步骤.首先,通过最大化阵列输出的边缘似然函数,推导了信号在拉普拉斯先验下的新不动点迭代方法,进行超参数的预估计,相比其他经典SBL算法提高了收敛速度;其次,利用新网格插值方法优化网格点集,并二次估计噪声方差和信号功率以分辨空间紧邻信号的DOA;最后,推导了似然函数关于角度的最大化公式以改进离格DOA搜索.仿真表明该算法比其他经典SBL类算法对空间紧邻信号的DOA具有更高的精度和分辨率,同时有计算效率的提升.     

Recovering Periodically Spaced Missing Samples [DSP Tips & Tricks]

We presented a novel method to recover periodically missing samples from appropriately bandlimited signals. We used filterbanks with fractional delay filters as building blocks, and described a windowed-sine method to approximate the fractional delays with FIR subfilters. Because the implementation of filterbanks and the approximation of fractional delay filters are well-studied problems, this makes the proposed recovery method flexible and easy to implement.     

基于稀疏贝叶斯学习的空间紧邻信号DOA估计算法

离格(off-grid)波达方向(DOA)估计解决的是实际DOA和假设网格点的失配问题。对于空间紧邻信号的DOA,稀疏的网格点会导致精度和分辨率的下降,密集的网格点虽然可以提高估计精度却显著增加计算负担。针对此问题,该文提出基于稀疏贝叶斯学习(SBL)的空间紧邻信号DOA估计算法,主要包括3个步骤。首先,通过最大化阵列输出的边缘似然函数,推导了信号在拉普拉斯先验下的新不动点迭代方法,进行超参数的预估计,相比其他经典SBL算法提高了收敛速度;其次,利用新网格插值方法优化网格点集,并二次估计噪声方差和信号功率以分辨空间紧邻信号的DOA;最后,推导了似然函数关于角度的最大化公式以改进离格DOA搜索。仿真表明该算法比其他经典SBL类算法对空间紧邻信号的DOA具有更高的精度和分辨率,同时有计算效率的提升。