Periodic Nonuniform Sampling Research Articles

A modification of the periodic nonuniform sampling involving derivatives with a Gaussian multiplier

A modification of the periodic nonuniform sampling involving derivatives with a Gaussian multiplier

Delay sampling theorem: A criterion for the recovery of multitone signal

Periodic nonuniform sampling (PNS) is widely used in sub-Nyquist sampling schemes of multitone signals. However, what sampling pattern of PNS enables the signal to recover from the sub-Nyquist samples? Although this problem has been discussed in different schemes, the corresponding answers vary greatly. In this paper, we introduce the recurrent delay times to describe a PNS and provide a mathematical definition of signal recoverability from the perspective of delay. On this basis, we review the well-known Shannon–Nyquist sampling theorem and suggest a delay sampling theorem (DST) that a multitone signal bandlimited to [0,fmax] can be perfectly reconstructed from its periodic nonuniform samples when its minimum combined delay time τmin is not more than the Nyquist interval 1/2fmax. To explain and demonstrate the DST, subspace theory-based algorithms and active aliasing and de-aliasing algorithm (AADA) are proposed and used to recover the spectrum or waveform of multitone signals. The high consistency between practical and theoretical results evidently verifies the correctness of DST. In terms of sampling and recovery of multitone signal, DST extends the Shannon–Nyquist sampling theorem and naturally unifies several existing sampling strategies from the perspective of delay. DST provides a new guideline for the sub-Nyquist sampling of multitone signals. According to the DST criterion, we can optimize the sampling patterns to reduce cosets and improve the supremum frequency such that the reconstructed spectrum is alias-free. By generalizing the sampling theorem from the view of delay, DST opens many avenues for the development of multitone signal processing.

Block-MUSIC in blade tip timing: Performance study of block snapshot matrix

Periodic nonuniform sampling can be viewed as a combination of uniform sampling and non-uniform sampling. Blade tip timing (BTT) technique conforms to this sampling pattern, where non-uniformity and uniformity are caused by non-uniform probe layout and rotational motion, respectively. By introducing block structure into multiple signal classification (MUSIC) for the extension of snapshot matrix, Block-MUSIC has been proved to be anti-aliasing and can filter out synchronous frequency in BTT application. However, the theoretical justification of the block structure is lacking. In this paper, a performance study of Block-MUSIC is conducted from three aspects: identifiability, stability, and resolution. Additionally, a pseudo 2-dimensional signal model is proposed to represent the periodic nonuniform sampled signal under the assumption of equally spaced probes. To support the performance study, simulations and experiments are carried out to demonstrate how the size of the snapshot matrix, the noise level, and the amplitude of sinusoidal signals affect algorithm performance. Simulation codes are available at https://github.com/superjdg/block-music4btt.

Improvement of BTT precision considering transient rotating speed fluctuations

A novel method is proposed to correct errors caused by rotational speed fluctuations in the BTT system. The analysis shows that the traditional BTT measurement using the constant speed assumption has low-frequency error due to the integration effect when the rotational speed fluctuation exists. Therefore, a filtering method based on the periodic non-uniform sampling (PNS) theorem is proposed. The details of the PNS-based method were investigated. The results show recommended order of the digital FIR filter is a length of 4 to 6 revolutions. The process of the PNS-based method for the unsteady state is given. Monitoring the condition number of the delayed phase matrix A is recommended when using the PNS-based method in the unsteady state. The effectiveness of the method was verified by a laboratory experiment and a field test. The laboratory experiment results showed that the error was reduced by 30 to 70 μm in the steady state. When the blade vibration was weak, the error was even halved. In the unsteady state, the PNS-based method controlled the error below 200 μm. In the field test, the PNS-based method also achieved the desired effect. The proposed method can significantly improve BTT measurement accuracy, showing great value and the prospect of blade health monitors in turbomachines.

Sampling and Interpolation of Periodic Nonuniform Samples Involving Derivatives

Sampling and Interpolation of Periodic Nonuniform Samples Involving Derivatives

Super-resolution wavefront reconstruction

Context.Cutting-edge, ground-based astronomical instruments are fed by adaptive optics (AO) systems that are aimed at providing high performance down to the visible wavelength domain on 10 m class telescopes and in the near infrared for the first generation instruments of Extremely Large Telescopes (ELTs). Both applications lead to a large ratio between the telescope diameter,D, and the coherence length or Fried parameter,r0, that isD/r0. As the parameter that defines the required number of degrees of freedom of the AO system,D/r0drives the requirement to reconstruct the incoming wavefront with ever-higher spatial resolution. In this context, super-resolution (SR) appears as a potential game changer. Indeed, SR promises to dramatically expand the range of spatial frequencies that can be reconstructed from a set of lower resolution measurements of the wavefront.Aims.As a technique that seeks to upscale the resolution of a set of measured signals, SR retrieves higher-frequency signal content by combining multiple lower resolution sampled data sets. It is well known both in the temporal and spatial domains and widely used in imaging to reduce aliasing and enhance the resolution of coarsely sampled images. This study applies the SR technique to the bidimensional wavefront reconstruction. In particular, we show how SR is intrinsically suited for tomographic multi-wavefront sensor (WFS) AO systems, revealing many of its advantages with minimal design effort.Methods.We provide a direct space and Fourier optics description of the wavefront sensing operation and we demonstrate how SR can be exploited through signal reconstruction, especially within the framework of periodic non-uniform sampling. We investigate both meta-uniform and non-uniform sampling schemes and we show that under some conditions, both sampling schemes enable a perfect reconstruction of band-limited signals. We also provide a SR bi-dimensional model for a Shack-Hartmann (SH) WFS, along with an analysis of the characteristics of the sensitivity function. We validated the SR concept with numerical simulations of representative multi-WFS SH AO systems. Finally, we explored the extension of the method to pyramid WFSs.Results.Our results show that combining several WFS samples in a SR framework grants access to a greater number of modes than the native one offered by a single WFS (despite the fixed sub-aperture size across samples). We show that the wavefront reconstruction achieved with four WFSs can be equivalent to a single WFS providing a sampling resolution that is twice greater (linear across the telescope aperture). We also show that the associated noise propagation is not degraded under SR. Finally, we show that the concept can be extended to the signal produced by single pyramid WFS, with its four re-imaged pupils serving as multiple non-redundant samples.Conclusions.We find that SR applied to wavefront sensing and reconstruction (WFR) offers a new parameter space to explore, as it decouples the size of the sub-aperture from the desired wavefront sampling resolution. By shifting away from outdated assumptions, new and more flexible, better-performing AO designs have now become possible.

Uniform and non-uniform sampling of bandlimited functions at minimal density with a few additional samples

A family of sampling theorems for the reconstruction of bandlimited functions from their samples is presented. Taking one or more additional samples is shown to yield more rapidly convergent series with lower truncation errors, as well as to facilitate the reconstruction of bandlimited functions with polynomial growth on the real line. The theorems apply to both uniform sampling and a large class of non-uniform sampling sets, i.e., complete interpolating sequences for the Paley–Wiener space. A number of examples and numerical illustrations accompany the general theory. These include uniform sampling, uniform sampling with finitely many points moved, periodic non-uniform sampling, and sampling at the zeros of Bessel functions J_nu (pi x) for non-integer v > -1.

Sampling with Derivatives in Periodic Shift-Invariant Spaces

We study the sampling problem involving derivatives in periodic shift-invariant spaces. We provide a necessary and sufficient condition for periodic nonuniform stable sampling with derivatives, based on block circulant matrices. Additionally, we derive an iterative algorithm to reconstruct a function from its irregular samples involving derivatives.

Dual-band signal reconstruction based on periodic nonuniform sampling at optimal sampling rate

A continuous-time multi-band signal with arbitrary spectral support can be reconstructed from its periodic nonuniform samples with the average sampling rate being arbitrarily close to the Nyquist-Landau rate. However, according to the existing literature, the theoretical uniform sampling rate in each channel of a periodic nonuniform sampling system would be infinitely small. Also, the required number of channels would be impractically large, leading to numerous filters. Consequently, the corresponding systems would be too complicated for practical implementation. It is important to consider the tradeoff between sampling rate and complexity for such systems. In this paper, we address the exact recovery of the continuous-time dual-band signal from its periodic nonuniform samples at optimal sampling rate with a simple sampling and reconstruction system. Optimal sampling rate means that the sampling rate is reduced as close to the Nyquist-Landau rate as possible. Firstly, we develop constraints on the aliasing-free sampling frequency via first-order sampling. Then, sufficient conditions are given to reconstruct a uniform sequence from its periodic nonuniform samples at optimal sampling rate. The complexity of the proposed method is calculated, which is far less than that of the conventional scheme. Our method shows potential applications in orthogonal frequency-division multiplexing (OFDM) systems.

Fast and [formula omitted]-optimal recovery for periodic nonuniformly sampled bandlimited signal

Uniformly-sampled sequence reconstruction from periodic nonuniform samples is addressed. In this problem setting, a bandlimited continuous-time signal is sampled with the same rate and different time offsets to obtain the periodic nonuniform samples, in which time offset information is unknown. We use a modified sequential forward selection algorithm to determine the sampling pattern corresponding to the time offset information, aiming to keep the condition number of the related matrix as small as possible. In our proposed algorithm, the reconstruction is formulated as a L2-norm optimization problem. Besides, proper matrix partition is included to reduce the complexity of large matrix pseudo-inverse. Numerical examples are presented to demonstrate the effectiveness of the derived reconstruction algorithm and its stability problem, even in the presence of noise.

Gaussian Regularized Periodic Nonuniform Sampling Series

The periodic nonuniform sampling plays an important role in digital signal processing and other engineering fields. In this paper, we introduce the Gaussian regularization method to accelerate the convergence rate of periodic nonuniform sampling series. We prove that the truncation error of the Gaussian regularized periodic nonuniform sampling series decays exponentially. Numerical experiments are presented to demonstrate our result.

An L2-stability estimate for periodic nonuniform sampling in higher dimensions

We consider sampling strategies for a class of multivariate bandlimited functions f that have a spectrum consisting of disjoint frequency bands. Taking advantage of the special spectral structure, we provide formulas relating f to the samples f(y),y∈X, where X is a periodic nonuniform sampling set. In this case, we show that the reconstruction can be viewed as an iterative process involving certain Vandermonde matrices, resulting in a link between the invertibility of these matrices to the existence of certain sampling sets that guarantee a unique recovery. Furthermore, estimates of inverse Vandermonde matrices are used to provide explicit L2-stability estimates for the reconstruction of this class of functions.

Fractional Spectral Analysis of Randomly Sampled Signals and Applications

Nonuniform sampling can be utilized to achieve certain desirable results. Periodic nonuniform sampling can decrease the required sampling rate for signals. Random sampling can be used as a digital alias-free signal processing method in analog-to-digital conversion. In this paper, we first present the fractional spectrum estimation of signals that are bandlimited in the fractional Fourier domain based on the general periodic random sampling approach. To show the estimation effect, the unbiasedness, the variance, and the optimal estimation condition are analyzed. The reconstruction of the fractional spectrum from the periodic random samples is also proposed. Second, the effects of sampling jitters and observation errors on the performance of the fractional spectrum estimation are analyzed, where the new defined fractional characteristic function is used to compensate the estimation bias from sampling jitters. Furthermore, we investigate the fractional spectral analysis from two widely used random sampling schemes, i.e., simple random sampling and stratified random sampling. Finally, all of the analysis results are applied and verified using a radar signal processing system.

Filterbank reconstruction of band‐limited signals from multichannel samples associated with the LCT

The linear canonical transform (LCT) has been shown to be a powerful tool for optics and signal processing. This study addresses the problem of filterbank implementation for multichannel sampling in the LCT domain. First, the interpolation and sampling identities in the LCT domain are derived by the properties of LCT. The interpolation identity is the key result of the current study, which establishes the equivalence of two signal processing operations. One of these uses continuous-time-domain filters, whereas the other uses discrete processing. Then, applying the interpolation identity, the particularly efficient filterbank implementation for derivative sampling, periodic non-uniform sampling and multichannel sampling are obtained in the LCT domain. Moreover, the authors present filterbank implementation using polyphase structures in LCT domain. Furthermore, the simulations are carried out to verify the effectiveness of the results and the potential applications of the multichannel sampling are also presented. Finally, combining the sampling and interpolation identity, they explore the relationship between the multichannel sampling and the filterbank in the LCT domain.

Periodic Nonuniform Dynamical Sampling in ℓ2(ℤ) and Shift-Invariant Spaces

ABSTRACTIn this article, we mainly study the periodic nonuniform dynamical sampling in ℓ2(ℤ) and shift-invariant spaces. We first provide a sufficient and necessary condition for c∈ℓ2(ℤ) which can be reconstructed by its spatial and temporal samples. Then we give a concrete example to show that the sufficient and necessary condition is feasible. Finally, we discuss the periodic nonuniform dynamic sampling problem in shift-invariant spaces.

Spectral reconstruction of signals from periodic nonuniform subsampling based on a Nyquist folding scheme

Periodic nonuniform sampling occurs in many applications, and the Nyquist folding receiver (NYFR) is an efficient, low complexity, and broadband spectrum sensing architecture. In this paper, we first derive that the radio frequency (RF) sample clock function of NYFR is periodic nonuniform. Then, the classical results of periodic nonuniform sampling are applied to NYFR. We extend the spectral reconstruction algorithm of time series decomposed model to the subsampling case by using the spectrum characteristics of NYFR. The subsampling case is common for broadband spectrum surveillance. Finally, we take example for a LFM signal under large bandwidth to verify the proposed algorithm and compare the spectral reconstruction algorithm with orthogonal matching pursuit (OMP) algorithm.

Generalized Sampling Expansions with Multiple Sampling Rates for Lowpass and Bandpass Signals in the Fractional Fourier Transform Domain

The objective of generalized sampling expansion (GSE) is the reconstruction of an unknown, continuously defined function $f\left(t\right)$ from samples of the responses from $M$ linear time-invariant (LTI) systems that are each sampled using the $1/M$ th Nyquist rate. In this paper, we investigate the GSE for lowpass and bandpass signals with multiple sampling rates in the fractional Fourier transform (FRFT) domain. First, we propose an improvement of Papoulis’ GSE, which has multiple sampling rates in the FRFT domain. Based on the proposed GSE, we derive the periodic nonuniform sampling scheme and the derivative interpolation method by designing different fractional filters and selecting specific sampling rates. In addition, the Papoulis GSE and the previous GSE associated with FRFT are shown to be special instances of our results. Second, we address the problem of the GSE of fractional bandpass signals. A new GSE for fractional bandpass signals with equal sampling rates is derived. We show that the restriction of an even number of channels in the GSE for fractional bandpass signals is unnecessary, and perfect signal reconstruction is possible for any arbitrary number of channels. Further, we develop the GSE for a fractional bandpass signal with multiple sampling rates. Lastly, we discuss the application of the proposed method in the context of single-image super-resolution reconstruction based on GSE. Illustrations and simulations are presented to verify the validity and effectiveness of the proposed results.

English

A traditional sampling method is that the signal should be sampled at a rate exceeding twice the highest frequency. This is based on the assumption that the signal occupies the entire bandwidth. In practice, however, many signals are sparse so that only part of the bandwidth is used. Compressed sampling has been developed for low-rate sampling of continuous time sparse signals in shift-invariant spaces generated by m kernels with period T . However, in general the reconstruction of compressed sampling signals is unstable. To reconstruct the signal, continuous reconstruction is replaced by generalized inverse. In this paper, periodic non-uniform sampling and the reconstruction of functions in shift-invariant spaces are discussed, the unique sparse expression is obtained by using the minimal L 1 norm. Also, necessary condition and error of reconstruction were analyzed. Finally, the method was validated via simulation and it was shown that the method was effective.

Wideband Photonic Compressive Sampling System

We present experimental results of a wideband photonic compressive sampling system that relies on periodic nonuniform sampling. The system unambiguously identifies signals in a 5-GHz instantaneous band using an average nonuniform sample rate an order of magnitude less than the equivalent Nyquist uniform sample rate. We demonstrate spectrum reconstruction of continuous-wave, pulsed carrier, and communications signals from dc to 50 GHz. The system demonstrates high-fidelity reconstruction of the spectral shape of signals with bandwidth in both single- and multisignal environments.

Periodic Non-uniform Sampling for Sparse Signals in Shift-invariant Spaces

Periodic Non-uniform sampling in shift-invariant spaces is an effective method for lowrate sampling of continuous time sparse signals, but, recovery of sampled time sparse signals must be unstable in general. In the paper, the possibility of stable recovery under a combination of sufficient sparsity is proposed. The sampling and reconstruction of sparse signals were transformed into matrix and vector operations by using minimum L1 normal. Necessary condition of reconstruction is analyzed. Finally, we are taking multiband sinusoidal signals as an example to prove that the method can achieve the sampling and reconstruction of sparse signals from recovery successful rate and integer of system.