Shannon Series Research Articles

An Optimal Convergence Rate for the Gaussian Regularized Shannon Sampling Series

We consider the reconstruction of a bandlimited function from its finite localized sample data. Truncating the classical Shannon sampling series results in an unsatisfactory convergence rate due to the slow decay of the sinc function. To overcome this drawback, a simple and highly effective method, called the Gaussian regularization of the Shannon series, was proposed in the engineering and has received remarkable attention. It works by multiplying the sinc function in the Shannon series with a regularized Gaussian function. Recently, it was proved that the upper error bound of this method can achieve a convergence rate of the order , where 0 < δ < π is the bandwidth and n the number of sample data. The convergence rate is by far the best convergence rate among all regularized methods for the Shannon sampling series. The main objective of this article is to present the theoretical justification and numerical verification that the convergence rate is optimal when 0 < δ < π/2 by estimating the lower error bound of the truncated Gaussian regularized Shannon sampling series.

Truncation Error Bound for the Kravchenko–Kotelnikov Series

The truncation error of the Kravchenko–Kotelnikov series, which is a generalization of the Whittaker–Kotelnikov–Shannon series, is studied. The basis functions of the Kravchenko–Kotelnikov series are the spectra Fa(t) of the atomic function ha(x), linearly transformed with respect to the argument. In this case, the function Fa(t) is defined by an infinite product. Two theorems on the truncation error bound for the Kravchenko–Kotelnikov series are proven. A practically important case in which the infinite product Fa(t) is replaced by a partial one is considered. A comparative analysis of the obtained formulae is carried out.

On the convergence of cardinal interpolation by parameterized radial basis functions

We consider cardinal interpolation on gridded data by using various radial basis functions associated with one or two parameters, one of which leads asymptotically to so-called flat-limits. Previously it had been shown that the classical Paley–Wiener functions can be recovered by such cardinal interpolations as the parameter tends to infinity. In this article, we extend the results by relaxing the requirements on the approximand functions from several points of view. The radial basis functions that we are concerned with and which are of special interest contain the celebrated multiquadrics, inverse multiquadrics and shifted thin-plate spline radial basis functions for instance. We also generalise the classes of admitted approximands as well as the radial basis functions to generalised multiquadrics in place of the well-known ordinary or for example inverse multiquadrics. An interesting analytical aspect of this work is that – unlike the classical Whittaker–Shannon theorem – functions (approximands) may be reproduced for the parameter c→∞ in the generalised multiquadrics cardinal approximands, where the usual Shannon series does not converge with theses approximands due to the slow decay of the sinc-function which does not allow e.g. polynomials as approximands. In contrast to the latter, the generalised multiquadrics cardinal functions employed here decay sufficiently fast for each fixed parameter c that even polynomials may be admitted as approximands and are reproduced when then the parameter tends to infinity.

Convergence Analysis of the Gaussian Regularized Shannon Sampling Series

ABSTRACTIn this article, we consider the reconstruction of a bandlimited function from its finite localized sample data. Truncating the classical Shannon sampling series results in an unsatisfactory convergence rate due to the slow decay of the sinc function. To overcome this drawback, a simple and highly effective method called the Gaussian regularization of the Shannon series was proposed in engineering and has received remarkable attention. It works by multiplying the sinc function in the Shannon series with a regularized Gaussian function. L. Qian established the convergence rate of for this method, where δ<π is the bandwidth and n is the number of sample data. C. Micchelli et al. proposed a different regularized method and obtained the corresponding convergence rate of . This latter rate is by far the best among all regularized methods for the Shannon series. However, their regularized method involves the solving of a linear system and is implicit and more complicated. The main objective of this article is to show that the Gaussian regularization of the Shannon series can also achieve the same best convergence rate as that by C. Micchelli et al. We also show that the Gaussian regularization method can improve the convergence rate for the useful average sampling.

Exponential approximation of bandlimited random processes from oversampling

The Shannon sampling theorem for bandlimited wide sense stationary random processes was established in 1957, which and its extensions to various random processes have been widely studied since then. However, truncation of the Shannon series suffers the drawback of slow convergence. Specifically, it is well-known that the mean-square approximation error of the truncated series at n points sampled at the exact Nyquist rate is of the order O (1/√ n ). We consider the reconstruction of bandlimited random processes from finite oversampling points, namely, the distance between consecutive points is smaller than the Nyquist sampling rate. The optimal deterministic linear reconstruction method and the associated intrinsic approximation error are studied. It is found that one can achieve exponentially-decaying (but not faster) approximation errors from oversampling. Two practical reconstruction methods with exponential approximation ability are also presented.

Error Analysis for Shannon Sampling Series Approximation with Measured Sampled Values

Errors appear when the Shannon sampling series is applied to reconstruct a signal in practice. In this paper, we study a general model that uses linear functional to cover several errors in one formula, and consider sampling series with measured sampled values for not band-limited signals but satisfying some decay condition. We obtain the uniform truncated error bound of Shannon series approximation for multivariate Besov class. The results show this kind of Shannon series can approximate a smooth signal well.

Joint Estimation of Time Delay and Doppler Shift for Band-Limited Signals

The standard approach for joint estimation of time delay and Doppler shift of a signal is to estimate the point at which the cross ambiguity function of the original and modified signals attains its maximum modulus. Since band-limited signals can be expressed exactly by their Shannon series, we here consider approximated signals gained by truncating their Shannon series to involve only the sampled signal values. We then estimate the time delay and Doppler shift by calculating a point at which the cross ambiguity function of the approximated signals attains its maximum modulus. This cross ambiguity function has an analytic expression which allows its evaluation at any point, and we may apply Newton's method to calculate accurately and efficiently a point where the maximum modulus is attained. In the numerical experiments we conducted, our method generally outperformed other methods for estimation of both time delay and Doppler shift.

Plancherel's Theorem and the Shannon Series Derived Simultaneously

Plancherel's Theorem and the Shannon Series Derived Simultaneously

Plancherel's Theorem and the Shannon Series Derived Simultaneously

Plancherel's Theorem and the Shannon Series Derived Simultaneously

On Truncation Error Bounds for Sampling Representations of Band-Limited Signals

All sampling representations of band-limited signals involve infinite sums. The truncation error associated with a given representation is defined as the difference between the signal and an approximating sum utilizing a finite number of terms. In this paper truncation error is expressed as a contour integral for Lagrange interpolation, general Hermite interpolation, the Shannon series (cardinal series), the Fogel derivative series, and multidimensional sampling expansions. Truncation error bounds are obtained under various constraints on the signal magnitude, spectral smoothness, and energy content.