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.     

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     

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     

基于欠采样的宽频段信号频率估计技术

对基于欠采样的单阵元输出宽频段多信号频率估计提出了一种算法。为正确解由于欠采样所导致的频率混迭,需要对单阵元输出的信号延时延成构成通道。求解频率所用的子空间类算法PRO－ESPRIT,保证了算法只需要较小的运算时,计算机模拟结果,频率估计离差接近CRB。     

Sampling theorems for uniform and periodic nonuniform MIMO sampling of multiband signals

We examine a multiple-input multiple-output (MIMO) sampling scheme for a linear time-invariant continuous-time MIMO channel. The input signals are modeled as multiband signals with different spectral supports, and the channel outputs are sampled on either uniform or periodic nonuniform sampling sets, with possibly different but commensurate intervals on the different outputs. This scheme encompasses Papoulis' generalized sampling and several nonuniform sampling schemes as special cases. We derive necessary and sufficient conditions on the channel and the sampling rate that allow stable perfect reconstruction of the inputs or, equivalently, perfect inversion of the channel. From an implementation viewpoint, we note that it is desirable that the reconstruction filters have continuous frequency responses. We derive necessary and sufficient conditions that guarantee this continuity property. The frequency responses of the reconstruction filters are specified as solutions to a system of linear equations. Finally, we demonstrate that perfect reconstruction may be possible, even when the channel outputs are sampled at an average rate that does not allow the reconstruction of any output from its samples alone. In certain instances, this average rate can achieve the recently presented fundamental bounds on MIMO sampling density.     

Optimal reconstruction of bandlimited bounded signals

A formula is presented for restoring bounded, bandlimited signals from a finite number of samples. The restoration is linear in the samples used, and the error bound of the method is the theoretical minimum.     

Compressed sensing based multi-rate sub-Nyquist sampling system

Signal sampling is a vital component in modern information technology. As the signal bandwidth becomes wider, the sampling rate of analog-to-digital conversion(ADC) based on Shannon-Nyquist theorem is more and more high and may be beyond its capacity. However the analog to information converter(AIC) based on compressed sensing(CS) is designed to sample the analog signals at a sub-Nyquist sampling rate. A new multi-rate sub-Nyquist sampling(MSS) system was proposed in this article, it has one mixer, one integrator and several parallel ADCs with different sampling rates. Simulation shows the signals can be reconstructed in high probability even though the sampling rate is much lower than the Nyquist 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     

Frequency estimation of sinusiode from wideband using sub-Nyquist sampling

A novel frequency estimation algorithm for wideband signal with sub-Nyquist sampling is proposed in this paper. With the aid of information provided by the auxiliary delayed sampling channel and the aliased frequency estimation for wideband signal with sub-Nyquist sampling, the frequency aliasing due to sub-Nyquist sampling can be solved. This method can reduce the complexity of the overall hardware at the cost of an auxiliary sampling channel. Furthermore, in order to alleviate the computation burden for its practicability, a more simplified algorithm is put forward and its validity is proved by our numerical simulation results. The Cramer-Rao Lower Bound （CRLB） of the frequency estimation is also derived at the end of this paper.     

基于非均匀采样重构的图像插值算法

在大规模集成电路实现图像缩放时,通常使用的算法是插值。基于信号的非均匀采样重构算法,提出了一种新的图像插值算法[1]。该插值算法所使用的基函数是基于余弦和函数。本算法同其他算法相比,在插值基函数长度相同的情况下,本算法的性能最好。相较其他算法,本算法保持了图像中更多的高频分量,同时减少了插值过程中的图像混淆效应。同时并没有增加插值所需的运算资源。     

Minimum rate sampling and reconstruction of signals with arbitraryfrequency support

We examine the question of reconstruction of signals from periodic nonuniform samples. This involves discarding samples from a uniformly sampled signal in some periodic fashion. We give a characterization of the signals that can be reconstructed at exactly the minimum rate once a nonuniform sampling pattern has been fixed. We give an implicit characterization of the reconstruction system, and a design method by which the ideal reconstruction filters may be approximated. We demonstrate that for certain spectral supports the minimum rate can be approached or achieved using reconstruction schemes of much lower complexity than those arrived at by using spectral slicing, as in earlier work. Previous work on multiband signals have typically been those for which restrictive assumptions on the sizes and positions of the bands have been made, or where the minimum rate was approached asymptotically. We show that the class of multiband signals which can be reconstructed exactly is shown to be far larger than previously considered. When approaching the minimum rate, this freedom allows us, in certain cases to have a far less complex reconstruction system     

频带有限信号非一致采样重建的谱方法

本文研究带限信号的非一致采样重建问题。我们利用信号频带分割方法,导出带限信号周期非一致采样情 况下的具体重建公式,并由此给出带限信号有限采样在区间上的插值公式并对它进行了数值实验。最后利用频谱分割方法, 讨论了多带限信号一些采样策略。     

Augmented Lagrangian based reconstruction of non-uniformly sub-Nyquist sampled MRI data

MRI has recently been identified as a promising application for compressed-sensing-like regularization because of its potential to speed up the acquisition while maintaining the image quality. Thereby non-uniform k-space trajectories, such as random or spiral trajectories, are becoming more and more important, because they are well suited to be used within the compressed-sensing (CS) acquisition framework. In this paper, we propose a new reconstruction technique for non-uniformly sub-Nyquist sampled k-space data. Several parts make up this technique, such as the non-uniform Fourier transform (NUFT), the discrete shearlet transform and a augmented Lagrangian based optimization algorithm. Because MRI images are real-valued, we introduce a new imaginary value suppressing prior, which attenuates imaginary components of MRI images during reconstruction, resulting in a better overall image quality. Further, a preconditioning based on the Voronoi cell size of each NUFT data point speeds up the conjugate gradient optimization used as part of the optimization algorithm. The resulting algorithm converges in a relatively small number of iterations and guarantees solutions that fully comply to the imposed constraints. The results show that the algorithm is applicable not only to sub-Nyquist sampled k-space reconstruction, but also to MR image fusion and/or resolution enhancement.     

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.     

Method of data reduction of sampled speech signals by using nonuniform sampling and a time-variable digital filter

A method of data reduction applicable to sampled continuous signals, for example, sampled speech signals, is described and tested by computer simulation, Nonuniform sampling obtained through skipping of samples is used. The reconstruction network is a time-variable digital filter. A minimisation procedure for coefficient derivation is described.     

Noise amplification analysis and comparison of two periodic nonuniform sampling reconstruction methods used in DPCA SAR

General Sampling Expansion Reconstruction Method （GSERM） and Digital Spectrum Reconstruction Method （DSRM）, which prove effective to reconstruct azimuth signal of Displaced Phase Center Apertures （DPCA） Synthetic Aperture Radar （SAR） system from its Periodic Non-Uniform Sampling （PNUS） data sequences, would amplify the noise and sidelobe clutter simultaneously in the reconstruction. This paper formulates the relation of the system transfer matrixes of the above two methods, gives the properties, such as periodicity, symmetry, and time-shift property, of their Noise and Sidelobe Clutter Amplification Factor （NSCAF）, and discovers that DSRM is more sensitive than GSERM in the white noise environment. In addition, criteria based on initial sampling point analysis for the robust PRF selection are suggested. Computer simulation results support these conclusions.     

A generalization of nonuniform bandpass sampling

Nth-order nonuniform sampling is described for generalized bandpass signal frequency position, bandwidth, sampling rate, frequency-shift and phase-shift. A bandpass extension to the Nyquist criterion is derived, showing that restrictions on bandpass frequency position for odd orders of nonuniform sampling tend to zero as N tends to infinity. Bandpass interpolants based on the sinc function are derived for the generalized Nth-order sampled bandpass signals. It is shown that, for minimum (Nyquist) rate sampling, these interpolants are comprised of N bandpass filters, each with independent phase. The number of bandpass filters comprising the interpolant is found to decrease as the sample rate increases. The advantage of describing Nth-order sampling as the Nth replication and uniform sampling of a signal is demonstrated. Finally, digital implementation of the Nth-order bandpass sampling interpolants is discussed. It is established that it is not practicable to attempt to perform nonuniform bandpass sampling at the theoretical minimum rate, where the interpolation is to be performed digitally     

A stable nonuniform sampling expansion involving derivatives

The work of D.A. Linden and N.M. Abramson (Inf. Control, vol.3, p.26-31, 1960, and vol.4, p.95-6, 1961) is completed in the sense that a full investigation of the convergence properties of the expansion is made. By adapting the work of N. Levinson (Am. Math. Soc. Colloq. Pub. vol.26, 1940) on nonharmonic Fourier series and general results from the theory of B-splines, the author extends the result to the case of sampling with uniformly spaced samples. A stability property is proved     

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.     

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.<>