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Specification of the observation‐error covariance matrix for data assimilation systems affects the observation information content retained by the analysis, particularly for observations known to have correlated observation errors (e.g., geostationary satellite and Doppler radar data). A widely adopted approach for estimating observation‐error covariance matrices uses observation‐minus‐background and observation‐minus‐analysis residuals, which are routinely produced by most data assimilation systems. Although this approach is known to produce biased and noisy estimates, due to sampling and misspecification errors, there has been no systematic study of sampling errors with this approach to date. Furthermore, the eigenspectrum of the estimated observation‐error covariance matrix is known to influence the analysis information content and numerical convergence of variational assimilation schemes. In this work, we provide new theorems for the sampling error and eigenvalues of the estimated observation‐error covariance matrices with this approach. We also conduct numerical experiments to illustrate our theoretical results. We find that this method produces large sampling errors if the true observation‐error standard deviation is large, while the other error characteristics, including the true background‐error standard deviation and observation‐ and background‐error correlation length‐scales, have a relatively small effect. We also find that the smallest eigenvalues of the estimated matrices may be smaller or larger than the true eigenvalues, depending on the assumed and true observation‐ and background‐error statistics. These results may provide insights for practical applications: for example, in deciding on appropriate sample sizes and choosing parameters for matrix reconditioning techniques. [ABSTRACT FROM AUTHOR]

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This paper suggests a class of modified exponential estimators for estimating the population mean of the study variable by using the information on the auxiliary variable under two situations: i) when information on the study variable and the auxiliary variable is available; ii) when there is non-response on the study as well as on auxiliary variable. Various estimators are obtained from the proposed class of estimators. The expressions for the bias and mean square errors (MSE) of the proposed estimator are derived up to the first degree of approximation. Theoretical comparisons are made with existing estimators and conditions are developed, under which proposed estimators are efficient. Also, these theoretical findings are supported by the simulation and empirical study by considering three real data sets. [ABSTRACT FROM AUTHOR]

This study proposes exponential ratio type estimators of population mean in case of simple random sampling without replacement (SRSWOR) when the auxiliary variable is an attribute. The double sampling version of these estimators is also proposed. Expressions for Bias and mean square error (MSE) have been evaluated. The proposed estimators were found to be more precise and also work as an alternative to different types of estimators as already existing. In order to support the theoretical findings, efficiency of the proposed estimators has also been analyzed numerically by using different population datasets. [ABSTRACT FROM AUTHOR]

In this paper, a lower bounded of sequences on classical sampling theorem for random signals is given and is used to extend the applicable range of frame. The convergence property of sampling series, the estimate of truncation error both in mean square sense and for almost sure results on sampling theorem for multidimensional random signals from asymmetrical average sampling are analyzed. Using the new results on frames recently, the famous Shannon sampling theorem on multidimensional random signals has been extended with a new idea. [ABSTRACT FROM AUTHOR]

In this paper, a regularized sampling algorithm for band-limited signals is presented in the twodimensional case. The convergence of the regularized sampling algorithm is studied and compared with Shannon's sampling theorem and the Tikhonov regularization method by some examples. [ABSTRACT FROM AUTHOR]

The article starts with generalizations of some classical results and new truncation error upper bounds in the sampling theorem for bandlimited stochastic processes. Then, it investigates Lp ( [0, T]) and uniform approximations of φ -sub-Gaussian random processes by finite time sampling sums. Explicit truncation error upper bounds are established. Some specifications of the general results for which the assumptions can be easily verified are given. Direct analytical methods are employed to obtain the results. [ABSTRACT FROM AUTHOR]

This paper starts with new aliasing-truncation error upper bounds in the sampling theorem for non-bandlimited stochastic signals. Then, it investigates Lp([0,T]) approximations of the sub-Gaussian random signals. Explicit truncation error upper bounds are established. The obtained rate of convergence provides a constructive algorithm for determining the sampling rate and the sample size in the truncated Whittaker-Kotel’nikov-Shannon expansions to ensure the approximation of the sub-Gaussian signals with given accuracy and reliability. Some numerical examples are presented. [ABSTRACT FROM AUTHOR]

Consider a set of points $$(x_1,y_1),\ldots ,(x_n,y_n)$$ with distinct $$0 \le x_i \le 1$$ and with $$-1 < y_i < 1$$ . The question of whether the function $$y = \sin (\omega x)$$ can approximate these points arbitrarily closely for a suitable choice of $$\omega $$ is considered. It is shown that such approximation is possible if and only if the set $$\{x_1,\ldots ,x_n\}$$ is linearly independent over the rationals. Furthermore, a constructive sufficient condition for such approximation is provided. The results provide a sort of counterpoint to the classical sampling theorem for bandlimited signals. They also provide a stronger statement than the well-known result that the collection of functions $$\{\sin (\omega x) : \omega < \infty \}$$ has infinite pseudo-dimension. [ABSTRACT FROM AUTHOR]

The linear canonical transform (LCT) plays an important role in optical and digital signal processing. Over the past few decades, there has been a vast amount of research on sampling theorems for a deterministic signal bandlimited in the LCT domain. However, signals are usually random in practical situations. Hence in this paper, we study sampling theorems for a random signal bandlimited in the LCT domain. We first construct a random signal theoretic framework in the LCT domain, such as the LCT power spectral density and the LCT auto-correction function. Then, we formulate uniform sampling theorem and multi-channel sampling theorem for a random signal bandlimited in the LCT domain. Finally, we analyze two kinds of reconstruction error estimates for uniformly sampling a random signal in the LCT domain: aliasing error and truncation error. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of fault estimation and accommodation for a class of networked control systems with nonuniform uncertain sampling periods. Firstly, the reason why the adaptive fault diagnosis observer cannot be applied to networked control systems is analyzed. Based on this analysis, a novel robust fault estimation observer is constructed to estimate both continuous-time fault and system states by using nonuniformly discrete-time sampled outputs. Furthermore, using the obtained states and fault information, a nonuniformly sampled-data fault tolerant control law is designed to preserve the stability of the closed-loop system. The proposed scheme can not only guarantee the impact of continuous-time uncertainties and discrete-time sampled estimation errors on the faulty system to satisfy a H ∞ performance index but also repress the negative effect of the unknown intersample behavior of continuous-time fault by use of an inequality technique. Finally, simulation results are included to demonstrate the feasibility of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We survey some aspects of the classical prediction theory for stationary processes, in discrete time in Section 1, turning in Section 2 to continuous time, with particular reference to reproducing-kernel Hilbert spaces and the sampling theorem. We discuss the discrete-continuous theories ofARMA-CARMA, GARCH-COGARCH, and OPUCCOPUC in Section 3. We compare the various models treated in Section 4 by how well they model volatility, in particular volatility clustering. We discuss the infinite-dimensional case in Section 5, and turn briefly to applications in Section 6. [ABSTRACT FROM AUTHOR]

ABSTRACT According to the theory of compressed sensing, magnetic resonance (MR) images can be well reconstructed from randomly sub-Nyquist sampling of the k-space data using a nonlinear recovery technique if certain conditions are satisfied. The sparse coefficients of the partial Fourier data can be estimated by minimizing the reconstruction cost function. Separable surrogate functional (SSF) method is one of the effective numerical techniques for minimizing mixed [ABSTRACT FROM AUTHOR]

This paper investigates the effect of sub-Nyquist sampling upon the capacity of an analog channel. The channel is assumed to be a linear time-invariant Gaussian channel, where perfect channel knowledge is available at both the transmitter and the receiver. We consider a general class of right-invertible time-preserving sampling methods which includes irregular nonuniform sampling, and characterize in closed form the channel capacity achievable by this class of sampling methods, under a sampling rate and power constraint. Our results indicate that the optimal sampling structures extract out the set of frequencies that exhibits the highest signal-to-noise ratio among all spectral sets of measure equal to the sampling rate. This can be attained through filterbank sampling with uniform sampling grid employed at each branch with possibly different rates, or through a single branch of modulation and filtering followed by uniform sampling. These results reveal that for a large class of channels, employing irregular nonuniform sampling sets, while are typically complicated to realize in practice, does not provide capacity gain over uniform sampling sets with appropriate preprocessing. Our findings demonstrate that aliasing or scrambling of spectral components does not provide capacity gain in this scenario, which is in contrast to the benefits obtained from random mixing in spectrum-blind compressive sampling schemes. [ABSTRACT FROM PUBLISHER]

Abstract: It is shown that three versions of the sampling theorem of signal analysis are equivalent in the sense that each can be proved as a corollary of one of the others. The theorems in question are the sampling theorem for functions belonging to the Bernstein space , the sampling theorem for functions in , , and the approximate sampling theorem for non-bandlimited function. One essential difference to an earlier paper of two of the authors is the avoidance of the deep Paley–Wiener theorem of Fourier analysis. [Copyright &y& Elsevier]

One of the limiting factors in the determination of gravity field solutions is the spatial sampling. Especially during phases, when the satellite repeats its own track after a short time, the spatial resolution will be limited. The Nyquist rule-of-thumb for mapping geopotential functions of a planet, also referred to as the Colombo-Nyquist rule-of-thumb, provides a limit for the maximum achievable degree of a spherical harmonic development for repeat orbits. We show in this paper that this rule is too conservative, and solutions with better spatial resolutions are possible. A new rule is introduced which limits the maximum achievable order (not degree!) to be smaller than the number of revolutions if the difference between the number of revolutions and the number of nodal days is of odd parity and to be smaller than half the number of revolutions if the difference is of even parity. The dependence on the parity is reflected in the eigenvalue spectrum of the normal matrix and becomes especially important in the presence of noise. The rule is based on applying the Nyquist sampling theorem separately in North-South and East-West direction. This is only possible for satellites in highly inclined orbits like champ and grace. Tables for these two satellite missions are also provided which indicate the passed and (in case of grace) expected repeat cycles and possible degradations in the quality of the gravity field solutions. [ABSTRACT FROM AUTHOR]

In this paper, the authors have provided a brief review of the recent advances in the Shannon sampling theory. In particular, they discuss the work related to 1-D signal reconstruction involving the samples taken below the Nyquist rate using nonlinear/time-variant systems. The extensions of the sampling theorems to the fractional Fourier and Linear canonical transform domains are also discussed. [ABSTRACT FROM AUTHOR]

Functions on the sphere appear in several applications, including geodesics, imaging and acoustics. Sampling of these functions may result in aliasing if the sampling condition is not met. The generalized sampling expansion introduced by Papoulis enables the reconstruction of a band-limited function sampled at a frequency lower than the Nyquist frequency using the outputs of several linear time-invariant systems. This paper formulates the generalized sampling expansion for functions on the sphere using spherical harmonics decomposition, facilitating sub-Nyquit sampling without aliasing error. An analysis of linear systems on the sphere and the aliasing phenomenon in the spherical harmonics domain is presented. Examples demonstrating the performance of the method and its limitations are studied. [ABSTRACT FROM PUBLISHER]

Meaningful upper bounds for the Fourier transform of polynomial exponential functions are often hard to come by. Regarding Fourier transforms of rational exponential functions, which are of importance, for example in Campbell's sampling theorem, the purpose of finding significant upper bounds is an even more demanding exercise. In this article, we propose a new approach in order to obtain significant upper bounds for Fourier transforms of general exponential functions. The technique is shown to allow further generalization in order to deal with Fourier-like integrals and rational exponential integrals. [ABSTRACT FROM AUTHOR]

We review the essential characteristics of the three different approaches to specifying a sampling strategy; the design-based approach, the model-assisted approach and the model-based approach. We then describe a unified framework for survey design and estimation that incorporates all three approaches, allowing us to contrast them in terms of their concepts of efficiency as well as their robustness to assumptions about the characteristics of the finite population. Our conclusion is that although no one approach delivers both efficiency and robustness, the model-based approach seems to achieve the best compromise between these typically conflicting objectives. [ABSTRACT FROM AUTHOR]

Abstract: We introduce a sampling theory for cyclic lattice invariant spaces in the context of locally compact Abelian groups. The key element of the theory is a new periodization condition which, for a lattice in a group G, characterizes the cyclic invariant subspaces which allow sampling. When is contained in a larger lattice , we characterize the collection of -invariant subspaces of which are also -invariant. We use this to show that, if V is cyclic with respect to both and , then V admits sampling with respect to if and only if V admits sampling with respect to . [Copyright &y& Elsevier]

The WKS sampling theorem is a fundamental result in the field of telecommunication and signal processing. It is a perfect example of the synthesis of traditionally distinct disciplines in mathematics, engineering analysis, and the sciences. The theorem recovers a bandlimited signal from its samples at uniformly points, which purpose is to analyze deterministic functions, or deterministic signals. However, there are a few counterparts of this theorem for random signals so far. This article gives a new WKS sampling principle on random signals from local averages with probability 1. [ABSTRACT FROM AUTHOR]

This article discusses the interplay between multiplex signal transmission in telegraphy and telephony, and sampling methods. It emphasizes the works of Herbert Raabe (1909-2004) and Claude Shannon (1916-2001) and the context in which they occurred. Attention is given to the role that the exceptional research atmosphere in Berlin during the 1920s and early 1930s played in the development of some of the ideas underlying these works, first in Germany and then in the USA, as some of the protagonists moved there. Raabe's thesis, published in 1939, describes and analyses a time-division multiplex system for telephony. In order to build his working prototype, Raabe had to develop the theoretical tools he needed and achieved a thorough understanding of sampling, including sampling with pulses of finite duration and sampling of low-pass and band-pass signals. His condition for reconstruction was known as 'Raabe's condition' in the German literature of the time. On the other hand, Shannon's works of 1948 and 1949 contain the classical sampling theorem, but go much further and lay down the abstract theoretical framework that underlies much of the modern digital communications. It is interesting to compare Raabe's very practical approach with Shannon's abstract work: Raabe independently developed his methods to the degree he needed, but his main purpose was to build a working prototype. Shannon, on the other hand, approached sampling independently of practical constraints, as part of information theory - which became tremendously influential. [ABSTRACT FROM AUTHOR]

It is shown that the Whittaker-Kotel'nikov-Shannon sampling theorem of signal analysis, which plays the central role in this article, as well as (a particular case) of Poisson's summation formula, the general Parseval formula and the reproducing kernel formula, are all equivalent to one another in the case of bandlimited functions. Here equivalent is meant in the sense that each is a corollary of the other. Further, the sampling theorem is equivalent to the Valiron-Tschakaloff sampling formula as well as to the Paley-Wiener theorem of Fourier analysis. An independent proof of the Valiron formula is provided. Many of the equivalences mentioned are new results. Although the above theorems are equivalent amongst themselves, it turns out that not only the sampling theorem but also Poisson's formula are in a certain sense the 'strongest' assertions of the six well-known, basic theorems under discussion. [ABSTRACT FROM AUTHOR]

The discretization size is limited by the sampling theorem, and the limit is one half of the wavelength of the highest frequency of the problem. However, one half of the wavelength is an ideal value. In general, the discretization size that can ensure the accuracy of the simulation is much smaller than this value in the traditional finite element method. The possible reason of this phenomenon is analyzed in this paper, and an efficient method is given to improve the simulation accuracy. [ABSTRACT FROM AUTHOR]

n algorithm for the synthesis of complex technical objects with greater eficiency and precision than that attainable with well-known algorithms is developed. An analysis of the systematic error of the algorithm is presented. [ABSTRACT FROM AUTHOR]

Abstract: The sampling theory is basic and crucial in engineering sciences. On the other hand, the linear canonical transform (LCT) is also of great power in optics, filter design, radar system analysis and pattern recognition, etc. The Fourier transform (FT), the fractional Fourier transform (FRFT), Fresnel transform (FRT) and scaling operations are considered as special cases of the LCT. In this paper, we structure certain types of non-bandlimited signals based on two ladder-shape filters designed in the LCT domain. Subsequently, these non-bandlimited signals are reconstructed from their samples together with the generalized sinc function, their parameter M-Hilbert transforms or their first derivatives and other information provided by the phase function of the nonlinear Fourier atom which is the boundary value of the Möbius transform, respectively. Simultaneously, mathematical characterizations for these non-bandlimited signals are given. Experimental results presented also offer a foundation for the sampling theorems established. [Copyright &y& Elsevier]

Abstract: The M-band symmetric cardinal orthogonal scaling function with compact support is of interest in several applications such as sampling theory, signal processing, computer graphics, and numerical algorithms. In this paper, we provide a complete mathematical analysis for the M-band symmetric cardinal orthogonal scaling function. Firstly, we generalize some results of the cardinal orthogonal scaling function from the special case to the most general case . Also, we find some new results. Secondly, we obtain the characterizations of the M-band symmetric cardinal orthogonal scaling function and revisit some known examples to prove our theory. [Copyright &y& Elsevier]

The functional equation for the Riemann zeta function is shown to be equivalent to the classical sampling theorem. [ABSTRACT FROM AUTHOR]

For signals in the Paley-Wiener space PW1π a reconstruction in the form of a sampling series that is uniformly convergent on compact subsets of the real axis and uniformly bounded on the whole real axis is not possible in general if the signals are sampled equidistantly at Nyquist rate. We prove that, even if the signal is non-uniformly sampled with an average sampling rate equal to the Nyquist rate, a uniformly convergent reconstruction is not possible. Additionally, we provide a detailed convergence analysis and give a sufficient condition for the uniform convergence of the Shannon sampling series without oversampling. However, if oversampling is applied, a uniformly convergent reconstruction is always possible and as far as convergence is concerned no elaborate kernel design is necessary. Moreover, we show that a projection of the reconstruction process onto the range of signal frequencies is not possible without losing the good convergence behavior. [ABSTRACT FROM AUTHOR]

We extend Unser's theory of sampling in shift-invariant spaces to the case of a Gaussian generator previously excluded because of violating the partition of unity condition. An unconditional sampling (or interpolation) theorem for these spaces is stated and proved. As an application the interpolation expansion of the dual function is given. Then, for a new class of function spaces that we call exponential Sobolev spaces, a sampling theorem is stated and proved. Exponential Sobolev spaces are more general than shift-invariant spaces but reconstruction of sampled functions takes place in a shift-invariant space with Gaussian generator. Finally it is noted that, despite a striking similarity between two arising interpolating functions, these are not the image of each other under a connected Gauss transform. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the multiwavelet sampling theorem by reproducing a kernel that is easy to use. Multiscaling functions that have orthogonality, regularity, short compact sup- port, symmetry, and high approximation order satisfy the conditions in the sampling theorem, which is not possible in the case of the scalar wavelet. We consider the general cases of the uniform noninteger and irregular sampling. A reconstruction from more general sets of points is necessary since the measurements may not be made at uniform points. Finally, we establish a general irregular sampling theorem for multiwavelet subspaces and derive an estimate for the perturbations of uniform noninteger sampling in shift invariant spaces. [ABSTRACT FROM AUTHOR]

We present the sampling theorem with sampling functions of general form for entire functions satisfying one of the growth conditions 1+&z.sfnc;y&z.sfnc;f(z)&les;A1+&z.sfnc;z&z.sfnc;N1expτ&z.sfnc;x&z.sfnc;+σ&z.sfnc;y&z.sfnc;,1+&z.sfnc;x&z.sfnc;f(z)&les;A1+&z.sfnc;z&z.sfnc;N1expτ&z.sfnc;x&z.sfnc;+σ&z.sfnc;y&z.sfnc; for some A>0, τ,σ&ges;0, N1∈N∪{0} and any z=x+iy∈C. It will be shown that many well-known sampling theorems included in SIAM J. Math. Anal. 19 (1988) 1198–1203 and Inform. Control 8 (1965) 143–158 can be interpreted as special cases of this sampling theorem. As examples, we provide sampling representations for entire functions which are bounded, of polynomial growth, or of exponential growth on R. We also provide sampling representations involving derivatives of entire functions and nonuniform sampling representations. Taking the set of sampling points in which a finite number of points are arbitrarily distributed, we obtain a sampling representation. [Copyright &y& Elsevier]

In this paper, we study the aliasing error when a nonbandlimited function is reconstructed by means of prefiltering and sampling. We give the optimal upper bound for the mean aliasing error and show that the error bound can be minimized by a suitable filter. [Copyright &y& Elsevier]

An orthonormal wavelet basis in L 2 used for microlocal filters, which decompose signals into microlocal contents, is shown to be a "stepwise" unconditional basis in L p (1 < p < ∞). Other related spaces are also treated. As part of the proof, an elementary proof of the L p version of the sampling theorem with unconditional convergence is given. Finally, an application is given to the expression of some distributions as sums of boundary values of holomorphic functions. [ABSTRACT FROM AUTHOR]

We show that every function in a spline subspace is uniquely determined and can be reconstructed by its local averages near certain points. Regular and irregular average sampling theorems for spline subspaces are obtained. [Copyright &y& Elsevier]

This paper derives a Fourier-like expansion formula in the signal space composed of spline functions, with the sampled value sequence of the signal as the expansion coefficients. Here, the domain is a finite closed interval and the boundary condition is periodic. The signal space composed of spline functions is one of the generalizations of the band-limited signal space with the continuous differentiability as the parameter. Consequently, the Fourier-like expansion formula derived in this paper is a generalization of Cahn's sampling theorem in the band limited signal space. [ABSTRACT FROM AUTHOR]

The Nyquist folding receiver (NYFR) can achieve a high-probability interception of an ultra-wideband (UWB) signal with fewer devices, while the output of the NYFR is converted into a hybrid modulated signal of the local oscillator (LO) and the received signal, which requires the matching parameter estimation methods. The linear frequency modulation (LFM) signal is a typical low probability of intercept (LPI) radar signal. In this paper, an estimation method of both the Nyquist Zone (NZ) index and the chirp rate for the LFM signal intercepted by NYFR was proposed. First, according to the time-frequency characteristics of the LFM signal, the accurate NZ and the rough chirp rate was estimated based on least squares (LS) and random sample consensus (RANSAC). Then, the information of the LO was removed from the hybrid modulated signal by the known NZ, and the precise chirp rate was obtained by using the fractional Fourier transform (FrFT). Moreover, a fast search method of FrFT optimal order was presented, which could obviously reduce the computational complexity. The simulation demonstrated that the proposed method could precisely estimate the parameters of the hybrid modulated output signal of the NYFR. [ABSTRACT FROM AUTHOR]

This is an erratum to [Random Oper. Stoch. Equ. 8 (2000), no. 2, 175-188], . [ABSTRACT FROM AUTHOR]

This is an erratum to [Random Oper. Stoch. Equ. 8 (2000), no. 2, 175-188], . [ABSTRACT FROM AUTHOR]