Nonuniform Sampling Theorem Research Articles

Nonuniform Sampling Theorem for Non-decaying Signals in Mixed-Norm Spaces $L_{\vec{p},\frac{1}{\omega }}(\mathbb{R}^{d})$

Nonuniform Sampling Theorem for Non-decaying Signals in Mixed-Norm Spaces $L_{\vec{p},\frac{1}{\omega }}(\mathbb{R}^{d})$

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.

Monitoring the vertical changes of a tidal flat using a MEMS accelerometer array

Tidal flats are constantly affected by the tide and are alternately exposed and submerged. Therefore, it is very difficult to monitor their topographic vertical changes (subsidence or uplifting) as the geographical environment is unique. In this study, a monitoring system based on the Micro-Electro-Mechanical System (MEMS) accelerometer array is proposed for monitoring subsidence or uplift in tidal flats. The optimal arrangement of the MEMS accelerometer array was determined using the non-uniform sampling theorem (frame theory) and the amplitude distribution characteristics of the tidal flat, and was as follows: the arc length interval between the sensors should be less than double the highest frequency of the tidal flat which is a baseband signal. We theoretically proved that the MEMS accelerometer array which satisfies the requirement of the non-uniform sampling with equal arc length could completely reconstruct the tidal flat. The two-dimensional arc model for reconstructing the tidal flat was then presented. To validate and correct the non-uniform sampling conditions obtained, and the calculation model, a test was conducted on a tidal flat in Zhairuoshan Island. The wave period and the vertical change of the tidal flat are measured. The results demonstrate the feasibility of the proposed system for effective tidal flat monitoring.

Nonuniform Sampling Theorems for Bandlimited Signals in the Offset Linear Canonical Transform

Nonuniform sampling is an important kind of sampling, which arises in various real applications due to imperfect timebase or random events. As the offset linear canonical transform (OLCT) has shown to be a powerful tool for optics and signal processing, it is therefore worthwhile and interesting to explore the nonuniform sampling theorems for deterministic signals and random signals in the OLCT domain. In this paper, we address the problem of the nonuniform sampling of deterministic signals and random signals associated with the OLCT. First, some special nonuniform sampling models are briefly introduced. Then, the reconstruction theorems for deterministic signals from these nonuniform samples in the OLCT domain have been obtained. In addition, by applying the results, the nonuniform sampling theorems for random signals in the OLCT domain also have been derived. Finally, the simulation results are presented to show the advantages and effectiveness of the methods.

Nonuniform sampling theorems for random signals in the linear canonical transform domain

ABSTRACTNonuniform sampling can be encountered in various practical processes because of random events or poor timebase. The analysis and applications of the nonuniform sampling for deterministic signals related to the linear canonical transform (LCT) have been well considered and researched, but up to now no papers have been published regarding the various nonuniform sampling theorems for random signals related to the LCT. The aim of this article is to explore the nonuniform sampling and reconstruction of random signals associated with the LCT. First, some special nonuniform sampling models are briefly introduced. Second, based on these models, some reconstruction theorems for random signals from various nonuniform samples associated with the LCT have been derived. Finally, the simulation results are made to prove the accuracy of the sampling theorems. In addition, the latent real practices of the nonuniform sampling for random signals have been also discussed.

What Is Variable Bandwidth?

We propose a new notion of variable bandwidth that is based on the spectral subspaces of an elliptic operator where p > 0 is a strictly positive function. Denote by the orthogonal projection of Ap corresponding to the spectrum of Ap in ; the range of this projection is the space of functions of variable bandwidth with spectral set in Λ.We will develop the basic theory of these function spaces. First, we derive (nonuniform) sampling theorems; second, we prove necessary density conditions in the style of Landau. Roughly, for a spectrum the main results say that, in a neighborhood of , a function of variable bandwidth behaves like a band‐limited function with local bandwidth .Although the formulation of the results is deceptively similar to the corresponding results for classical band‐limited functions, the methods of proof are much more involved. On the one hand, we use the oscillation method from sampling theory and frame‐theoretic methods; on the other hand, we need the precise spectral theory of Sturm‐Liouville operators and the scattering theory of one‐dimensional Schrödinger operators. © 2017 Wiley Periodicals, Inc.

A non-uniformly under-sampled blade tip-timing signal reconstruction method for blade vibration monitoring.

High-speed blades are often prone to fatigue due to severe blade vibrations. In particular, synchronous vibrations can cause irreversible damages to the blade. Blade tip-timing methods (BTT) have become a promising way to monitor blade vibrations. However, synchronous vibrations are unsuitably monitored by uniform BTT sampling. Therefore, non-equally mounted probes have been used, which will result in the non-uniformity of the sampling signal. Since under-sampling is an intrinsic drawback of BTT methods, how to analyze non-uniformly under-sampled BTT signals is a big challenge. In this paper, a novel reconstruction method for non-uniformly under-sampled BTT data is presented. The method is based on the periodically non-uniform sampling theorem. Firstly, a mathematical model of a non-uniform BTT sampling process is built. It can be treated as the sum of certain uniform sample streams. For each stream, an interpolating function is required to prevent aliasing in the reconstructed signal. Secondly, simultaneous equations of all interpolating functions in each sub-band are built and corresponding solutions are ultimately derived to remove unwanted replicas of the original signal caused by the sampling, which may overlay the original signal. In the end, numerical simulations and experiments are carried out to validate the feasibility of the proposed method. The results demonstrate the accuracy of the reconstructed signal depends on the sampling frequency, the blade vibration frequency, the blade vibration bandwidth, the probe static offset and the number of samples. In practice, both types of blade vibration signals can be particularly reconstructed by non-uniform BTT data acquired from only two probes.

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.

On Bandlimited Signals Associated With Linear Canonical Transform

We first show that the bandlimited signals associated with linear canonical transform (LCT) form a reproducing kernel Hilbert space. An orthogonal basis for the class of bandlimited signals associated with LCT is then proposed by use of the reproducing kernel, with respect to which the coordinates of signal are actually values taken by the signal at certain instants of time. Finally, a nonuniform sampling theorem for bandlimited signals associated with LCT is presented.

Coded cooperation diversity for uncoded oversampled OFDM systems

Recently, user-cooperation diversity has been introduced as an effective scheme that can bring about antenna diversity in wireless networks. In this paper, we introduce a coded cooperation diversity technique for single antenna uncoded orthogonal frequency division multiplexing (OFDM) systems, in which implementation of coded cooperation is provided by the oversampling potential in OFDM, instead of using extra channel coding. In fact, zero-padding followed by IFFT in OFDM is similar to oversampling and could be an alternative to applying correction codes. Furthermore, we use this oversampling to implement an iterative receiver at the partner terminal. This receiver works based on the nonuniform sampling theorem for reconstructing of lost symbols. The lost symbols appear at the partner terminal of cooperative network because of dividing each OFDM block into two segments through the puncturing at the user terminal of the cooperative network. We provide simulation results for our proposed scenario, and observe significant gains over the non-cooperative oversampled OFDM systems without any need whatever for using extra channel coding.

PERTURBATION OF NONHARMONIC FOURIER SERIES AND NONUNIFORM SAMPLING THEOREM

For an entire function f whose Fourier transform has a compact support confined to [−π, π] and restriction to R belongs to L2(R), we derive a nonuniform sampling theorem of Lagrange interpolation type with sampling points λn ∈ R, n ∈ Z, under the condition that lim sup n→∞ |λn − n| < 1 4 .

On Simple Oversampled A/D Conversion in Shift-Invariant Spaces

It has been found that the quantization error e for a conventional oversampled analog-to-digital (A/D) conversion behaves like /spl par/e/spl par//sup 2/=O(/spl tau//sup 2/) with respect to the sampling rate /spl tau/. Recently, conventional A/D conversion has been extended to A/D conversion based on shift-invariant spaces. As consequences of such extension, it offers rich choices to build a nonideal A/D conversion system of high accuracy and low computational complexity, as well as reduces the noise sensitivity and computational complexity in digital-to-analog (D/A) conversion. Therefore, it is necessary to establish the estimate of quantization error for the extended A/D conversion based on shift-invariant spaces. In this paper, we introduce a constructive method to establish an estimate of the quantization error as |e|/sup 2/=O(/spl tau//sup 2/) for oversampled A/D conversion in shift-invariant spaces. Meanwhile, we demonstrate that the bit rate required to encode the converted digital signal in such A/D conversion scheme only increases as the logarithm of the sampling ratio. Therefore, the quantization error is an exponentially decaying function of the bit rate. In order to establish such an estimate, we need the nonuniform sampling theorem for shift-invariant spaces, which, as the necessary preparation, is studied prior to introducing the constructive method.

Interpolation of Non-Uniformly Sampled Sequences and Reconstruction of Sampled Sequences with Any Uniform Interval Based on Non-Uniform Sampling Theorem

In this paper we psesent a new non-uniform sampling theorem for band-limited signals, together with an associated interpolation formula of non-uniformly sampled sequences, and propose a practical algorithm for reconstructing sampled sequences of any uniform interval from a non-uniform one by using the frequency sampling technique and FFT algorithm. After a brief review of the ordinary uniform sampling theorem, the non-uniform sampling theorem is given, together with an aliasing free condition being made clear. Based on the fundamentals, an interpolation formula of non-uniformly sampled sequences is derived and a practical implementation algorithm to reconstruct uniformly sampled sequences of any desired interval from a non-uniformly sampled version is proposed by way of the frequency sampling technique and FFT algorithm. The principle as well as the effectiveness of the proposed algorithm are ascertained through numerical experiments with sinusoidal and FM signals of finite bandwidths.

Sampling and generalized almost periodic extension of functions

The author explores the connections between nonuniform sampling of a certain function and the almost periodic extension of its Fourier transform. It is shown that the Fourier transform of a bandlimited function can be extended (as a weighted sum of translates) as a Stepanoff almost periodic function, to the whole frequency axis. This result leads to a generalized nonuniform sampling theorem which, unlike previous results, does not require the continuity of the Fourier transform of the sampled function, and is valid for finite-energy band-limited functions.

Nonuniform sampling theorems for bandpass signals at or below the Nyquist density

In this paper, the problem of recovery of bandpass signals from a set of nonuniform samples is discussed. Interpolation formulas for particular sets of irregular samples are given. Issues such as the Nyquist rate are studied, and practical methods for sampling and recovery are suggested.

Nonuniform sampling and antialiasing in image representation

A unified approach to the representation and processing of a class of images which are not bandlimited but belong to the space of locally bandlimited signals is presented. A nonuniform sampling theorem (Clark et al, 1985) for functions belonging to this space is extended, and a class of nonstationary stochastic processes is considered. The space of locally bandlimited signals is shown to be a reproducing-kernel space. A generalized projection theorem can therefore be applied, yielding either a continuous or a discrete projection filter. The former can be used for image conditioning prior to nonuniform sampling, while the latter provides a general tool for image representation by nonuniform sampling schemes. The problem of finding the local bandwidth of a given signal, in order to generate an optimal sampling scheme, is addressed in the context of signal representation in the combined position-frequency space. The stochastic estimation of parameters which characterize the local bandwidth is discussed. Bounds on the error resulting from the utilization of nonexact position-varying signal parameters are derived.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Two-dimensional nonuniform sampling expansions an iterative approach. i. theory of two-dimensional bandlimited signals

The major aim is to establish a nonuniform sampling theorem for two-dimensional functions bandlimited to a parallelogram centered at the origin by an iterative application of one-dimensional expansion formulae. For this purpose results on bandlimited square integrable functions are given, and an inequality of S.M. Nikol'skii for samples taken at equally spaced nodes is generalized to the nonuniform case. A first version of the desired sampling theorem is derived here; the main ones follow in Part II

Two-dimensional nonuniform sampling expansions an iterative approach. ii. reconstruction formulae and applications

The classical sampling theorems state that bandlimited square integrable functions of one or two variables can be reconstructed from their samples taken at suitably dense equally spaced sets of nodes. In one dimension, the uniform sampling points can be replaced by certain irregularly spaced ones. By making use of the properties of bandlimited functions and a generalization of an inequality of Nikol'skii shown in [22], in this part a nonuniform sampling theorem for two-dimensional square integrable functions bandlimited to a parallelogram centered at (0,0) is established. The reconstruction formula is obtained by iterative application of sampling formulae for functions of one variable. It is shown that the resulting series converges absolutely provided the sampling points are not too wildly scattered. Several applications are worked out, including those for hexagonal and octogonal sets of sampling points. Further generalizations are discussed

Sampling theorems using nonuniform sample points and interpolation functions with band‐limited continuous spectra

AbstractThis paper presents a unified theory for the nonuniform sampling theorem for the low‐pass band‐limited waveform. In most practical cases, the mean interval of the sampling points is in the so‐called oversampling state. One can also assume that there exists a certain period in the sampling point sequence. Those properties of the sampling point sequence are assumed in this paper. Yen has derived a sampling theorem for the case where the sampling‐point sequence has a period. In his theorem, how ever, the interpolation function has a spectrum which is a stepwise discontinuous function, not suited to the realization of the interpolation function by a digital or other similar filters. One of the greatest features of the sampling theorem proposed in this paper is that the interpolation function has a continuous spectrum. the problem has been discussed by one of the authors for the case where the sampling frequency and the band‐end frequency of the waveform are in an integer ratio.In this paper, the sampling theorem is derived in a unified and integrated form without imposing such a condition. the variance of the approximation error is also derived for the case where a statistically independent additive error is superposed on the sampled value, and it is shown that the proposed sampling theorem is better in that respect. the discussion is extended to the multidimensional band‐limited waveform, and a sampling theorem is shown which has an interpolation function with a continuous spectrum.

On n-dimensional sampling theorems

Two n-dimensional nonuniform-sampling theorems are proved, which possess additional freedom for constructing recovery functions. One of these implies the n-dimensional uniform-sampling theorem for functions with band-limited Fourier transforms, which also has been proved here through distribution theory. Balth van der Pol's 1959 conjecture on nonuniform sampling for time function is a consequence of these two theorems. For either case, the recovery functions are analytic.