Optimal sub-Nyquist nonuniform sampling and reconstruction formultiband signals

We study the problem of optimal sub-Nyquist sampling for perfect reconstruction of multiband signals. The signals are assumed to have a known spectral support ℱ that does not tile under translation. Such signals admit perfect reconstruction from periodic nonuniform sampling at rates approaching Landau's (1967) lower bound equal to the measure of ℱ. For signals with sparse ℱ, this rate can be much smaller than the Nyquist rate. Unfortunately the reduced sampling rates afforded by this scheme can be accompanied by increased error sensitivity. In a previous study, we derived bounds on the error due to mismodeling and sample additive noise. Adopting these bounds as performance measures, we consider the problems of optimizing the reconstruction sections of the system, choosing the optimal base sampling rate, and designing the nonuniform sampling pattern. We find that optimizing these parameters can improve system performance significantly. Furthermore, uniform sampling is optimal for signals with ℱ that tiles under translation. For signals with nontiling ℱ, which are not amenable to efficient uniform sampling, the results reveal increased error sensitivities with sub-Nyquist sampling. However, these can be controlled by optimal design, demonstrating the potential for practical multifold reductions in sampling rate     

Perfect sampling for the wavelet reconstruction of signals

The coupling from the past (CFTP) procedure is a protocol for finite-state Markov chain Monte Carlo (MCMC) methods whereby the algorithm itself can determine the necessary runtime to convergence. In this paper, we demonstrate how this protocol can be applied to the problem of signal reconstruction using Bayesian wavelet analysis where the dimensionality of the wavelet basis set is unknown, and the observations are distorted by Gaussian white noise of unknown variance. MCMC simulation is used to account for model uncertainty by drawing samples of wavelet bases for approximating integrals (or summations) on the model space that are either too complex or too computationally demanding to perform analytically. We extend the CFTP protocol by making use of the central limit theorem to show how the algorithm can also monitor its own approximation error induced by MCMC. In this way, we can assess the number of MCMC samples needed to approximate the integral to within a user specified tolerance level. Hence, the method automatically ensures convergence and determines the necessary number of iterations needed to meet the error criteria     

改进的宽带信号有理数倍采样率变换结构

针对现有的数字器件无法满足对宽带信号进行有理数倍采样率变换的问题,提出了一种改进的宽带信号有理数倍采样率转换结构.在分析抽取器与内插器顺序的基础上,得出了等效变换的前提条件,并结合多相滤波结构使改进的结构更具有实际应用价值.仿真实验证明了改进后结构的有效性.通过与现有有理数倍采样率变换结构运算量对比分析表明,改进后的结构的运算效率为传统结构运算效率的1/(LM)倍,其中L和M为变换过程中的内插因子和抽取因子.     

Optimal sampling and reconstruction of MRI signals resulting fromsinusoidal gradients

The optimum continuous time filter and its various discrete time implementations for free induction delay signals resulting from sinusoidal gradients of magnetic resonance imaging (MRI) are formulated. It is found that the estimation error associated with implementation based on linear temporal sampling, or, equivalently, nonlinear spatial frequency sampling, is smaller than that of nonlinear temporal sampling. It is shown that the optimal maximum likelihood estimator for sinusoidal gradients has higher error variance than that of constant gradients. Experimental results verifying the theoretical predictions are presented     

Adaptive repetitive control to track variable periodic signals with fixed sampling rate

Recent research has shown that the repetitive control is very efficient in tracking periodic signals, where it is required that an integer number of samples in each period. However, in some industrial applications where the signal period varies but other requirements force a fixed sample rate, the number of samples per period may be a non-integer. To address this problem, this paper presents a new adaptive repetitive control, which deals with the non-integer samples per period due to the fixed sampling rate. The proposed adaptive repetitive control consists of two portions, the repetitive controller and nominal controller, where the former uses a fictitious sampler operating at a variable sample rate maintained at multiple times of the signal frequency, while the latter uses a fixed sampling rate. Interpolations are utilized to generate the fictitious samples required for the repetitive learning. The nearly perfect tracking was achieved for non-integer samples per period, when a simple linear interpolation is used. The error due to the interpolation is quantified, which is negligible to the residual tracking error. The comparison of the proposed and the existing schemes shows the significant improvement on the tracking performance. The experimental results on the control of a servomotor demonstrate the effectiveness of the proposed schemes.     

On the reconstruction of band-limited multidimensional signals from algebraic sampling contours

In this letter we pose and answer the following question: Is a band-limited multidimensional signal uniquely determined by a continuous set of samples whose location is given by the zeroes of a polynomial equation? Applications to the theory of reconstruction of finite-extent objects from a set of generalized projections are discussed including the classical problems of parallel X-ray and diffraction tomography.     

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.     

Perfect reconstruction formulas and bounds on aliasing error insub-Nyquist nonuniform sampling of multiband signals

We examine the problem of periodic nonuniform sampling of a multiband signal and its reconstruction from the samples. This sampling scheme, which has been studied previously, has an interesting optimality property that uniform sampling lacks: one can sample and reconstruct the class ℬ(ℱ) of multiband signals with spectral support ℱ, at rates arbitrarily close to the Landau (1969) minimum rate equal to the Lebesgue measure of ℱ, even when ℱ does not tile R under translation. Using the conditions for exact reconstruction, we derive an explicit reconstruction formula. We compute bounds on the peak value and the energy of the aliasing error in the event that the input signal is band-limited to the “span of ℱ” (the smallest interval containing ℱ) which is a bigger class than the valid signals ℬ(ℱ), band-limited to ℱ. We also examine the performance of the reconstruction system when the input contains additive sample noise     

Subband decomposition of signals with generalized sampling

The authors generalize down and up-sampling operations by proposing block sampling. Perfect reconstruction conditions for two-band subband coding with block sampling are derived. By generalizing the sampling operation, new degrees of freedom are introduced and as a result, filter banks which were not previously possible become possible. Generalized down-sampler introduces different aliasing components than that of the traditional down-sampler. This can be used to ease some the requirements of the filter bank design problem. A constructive sampling method is proposed so that coprimeness of the transfer functions of the analysis filter banks is not only a necessary but also sufficient condition for perfect reconstruction. The results are extended to the case where the filter banks are linear periodically time-varying. The multichannel case is analyzed and the relation between unimodular matrices and perfect reconstruction filter banks is discussed     

Spectrum blind reconstruction and direction of arrival estimation of multi-band signals at sub-Nyquist sampling rates

In this paper we consider the problem of spectrum blind reconstruction (SBR) and direction of arrival (DOA) estimation of constituent sources of a disjoint multi-band signal (MBS) at sub-Nyquist sampling rates. Transformation of the problem into frequency domain indicates that the steering vector is a function of both the carrier frequency and its corresponding DOA. Employing the existing two dimensional frequency-DOA search algorithms suffers from the drawbacks of increased computational complexity and ambiguity issues. To overcome these drawbacks, in this paper we propose a simple modification to the receiver architecture by introducing an additional delay channel at every sensor. Estimation algorithms based on ESPRIT is then employed to estimate the carrier frequencies, while MUSIC algorithm is employed to estimate their corresponding DOAs. Using the knowledge of both these parameters, the MBS spectrum is then reconstructed. A two-dimensional iterative grid refinement algorithm is also described to further improve the estimation accuracy in the presence of noise. Identifiability issues are addressed and the conditions for unique identifiability are discussed. Furthermore, by assuming a two dimensional uniform array the advantages of the proposed approach in terms of identifiability is also provided. We further show that an \(M \ge N+1\) sensors and an overall sampling rate of at least \(2(N+1)B\) would be sufficient to achieve SBR and DOA estimation of an MBS comprising of N disjoint bands each of maximal bandwidth B. Numerical simulations are finally presented which verifies the validity of the proposed approach and compares the performance against appropriate bounds.     

Generalised filters and optimum sampling frequencies for the reconstruction of bandlimited signals from past samples

Generalised first- and second-order filters are presented for the reconstruction of bandlimited signals. In each case the minimum sampling frequency is derived and the corresponding filter coefficients obtained. A tradeoff exists between the rate of convergence of the reconstruction and the sampling frequency.<>     

Minimum bandwidth signals with some specified waveform parameters

For each of four separate conditions placed on a waveform (amplitudes at certain epochs, and slopes at those epochs) constraints on the corresponding spectra are obtained for a system of any bandwidth. Waveforms which satisfy certain combinations of amplitude and slope constraints, within a minimum bandwidth, are shown to have unique spectra. The method confirms that Nyquist's first and second conditions are satisfied by the raised-cosine frequency spectrum, and spectra are obtained for other waveform conditions. The new results are of interest in practical systems where constraints are placed on intersymbol interference and telegraph distortion or in theoretical systems as design targets.     

Motion-aided sampling and reconstruction

Motivated by motion compensated filtering in image processing, this paper considers the problem of sampling and reconstruction of signals with sampling rates below the Nyquist rate. It is assumed that temporal dependence can be induced via motion. This way, the data consists of both spatial and temporal sampling, and here the conditions for reconstruction are analysed for a number of typical motions. Extensive simulation experiments are also provided which further support the analysis.     

Data-dependent sampling of two-dimensional signals

Samples taken at scattered points of a finite-support two-dimensional signal can be interpolated to recover an approximation of the original signal. Given a bound on the number of samples, where should they be placed to enable the most accurate reconstruction? Or, given an error bound for the reconstruction, what is the minimum number of samples required, and where should they be placed? In this paper we introduce search schemes that provide good candidate solutions to these problems, for digital signals. Natural Neighbour Interpolation is used in iterative sample removal and movement processes to obtain sparse sample patterns. For pictures and Digital Elevation Models, fewer samples are required if the interpolant is onlyC ⁰ continuous at the data sites, than if it isC ¹. Retained samples lie on the ridges and valleys of the laplacian.     

Sampling signals with finite rate of innovation

The authors consider classes of signals that have a finite number of degrees of freedom per unit of time and call this number the rate of innovation. Examples of signals with a finite rate of innovation include streams of Diracs (e.g., the Poisson process), nonuniform splines, and piecewise polynomials. Even though these signals are not bandlimited, we show that they can be sampled uniformly at (or above) the rate of innovation using an appropriate kernel and then be perfectly reconstructed. Thus, we prove sampling theorems for classes of signals and kernels that generalize the classic "bandlimited and sinc kernel" case. In particular, we show how to sample and reconstruct periodic and finite-length streams of Diracs, nonuniform splines, and piecewise polynomials using sinc and Gaussian kernels. For infinite-length signals with finite local rate of innovation, we show local sampling and reconstruction based on spline kernels. The key in all constructions is to identify the innovative part of a signal (e.g., time instants and weights of Diracs) using an annihilating or locator filter: a device well known in spectral analysis and error-correction coding. This leads to standard computational procedures for solving the sampling problem, which we show through experimental results. Applications of these new sampling results can be found in signal processing, communications systems, and biological systems     

Nonuniform sampling of bandlimited modulated signals

In this paper it is shown that an unequally spaced sampling can be implemented on the base of sinc-composition function usually used in the uniform sampling expansion. The nonuniform sampler is based on a level crossing detector (LCD) which produces a sample of carrier whenever the integral of modulating function crosses a threshold level. The information about the modulated signal is contained in the carrier samples taken at the instants which are defined by modulating function. The Riemann-Stieltjes uniform integral sum representation of the Fourier transform of signal is used as a mathematical base for the derivation of the principal results. In the particular case of the representation of single-valued real positive function, the suggested method leads to the signal representation by the equally weighted sampling at the nonuniformly distributed instants. The error estimation and numerical example illustrating the methods are given.     

Effects of reconstruction filters and sampling rate for a digitalsignal component separator on LINC transmitter performance

The authors investigate how reconstruction filters and sampling rate for a digital signal component separator affect the performance of an LINC (linear amplification with nonlinear components) transmitter. A method is described that, for a specified adjacent channel interference level, finds a nearly optimal cutoff frequency for a given filter approximation and the minimum sampling rate     

连续时间信号的采样与重构及其实现

介绍了连续时间信号的采样和重构的基本原理,并通过MATLAB实现了对指数信号的采样及重构。     

Finite sampling approximations for non-band-limited signals

Finite sampling approximations, along with bounds on the approximation error, are derived for certain deterministic and random signals which are not band-limited.     

Reconstruction of irregularly sampled discrete-time bandlimited signals with unknown sampling locations

The purpose of this paper is to develop methods that can reconstruct a bandlimited discrete-time signal from an irregular set of samples at unknown locations. We define a solution to the problem using first a geometric and then an algebraic point of view. We find the locations of the irregular set of samples by treating the problem as a combinatorial optimization problem. We employ an exhaustive method and two descent methods: the random search and cyclic coordinate methods. The numerical simulations were made on three types of irregular sets of locations: random sets; sets with jitter around a uniform set; and periodic nonuniform sets. Furthermore, for the periodic nonuniform set of locations, we develop a fast scheme that reduces the computational complexity of the problem by exploiting the periodic nonuniform structure of the sample locations in the DFT.