Approximation of Wide-Sense Stationary Stochastic Processes by Shannon Sampling Series

Abstract

In this paper, the convergence behavior of the symmetric and the nonsymmetric Shannon sampling series is analyzed for bandlimited continuous-time wide-sense stationary stochastic processes that have absolutely continuous spectral measure. It is shown that the nonsymmetric sampling series converges in the mean-square sense uniformly on compact subsets of the real axis if and only if the power spectral density of the process fulfills a certain integrability condition. Moreover, if this condition is not fulfilled, then the pointwise mean-square approximation error of the nonsymmetric sampling series and the supremum of the mean-square approximation error over the real axis of the symmetric sampling series both diverge. This shows that there is a significant difference between the convergence behavior of the symmetric and the nonsymmetric sampling series.

Sampling theorems and error estimates for random signals in the linear canonical transform domain

“…For simplicity, we only consider two kinds of errors in this section, i.e., aliasing error and truncation error for uniformly sampling a random signal in the LCT domain. Aliasing error and truncation error estimates have been extensively studied in the classical Shannon sampling theorem for a deterministic or random signal in the Fourier domain [4][5][6][7]13,21]. As far as we know, there does not exist any result about error estimates for sampling a random signal in the LCT domain.…”

System Approximations and Generalized Measurements in Modern Sampling Theory

“…For such applications, B ∞ π is the appropriated signal space. Moreover, in sampling and reconstruction of stochastic processes, the space PW 1 π plays a fundamental role [8,16] because the spectral densities of such processes are L 1 functions, in general. Consequently, one has to investigate the behavior of the reconstruction series for functions in PW 1 .…”

“…Theorem 10. Let Λ = {λ n } n∈Z be the zeros set of a sine-type function ϕ, let {ϕ n } n∈Z be the corresponding interpolation kernels, defined in (7), and let A N : PW 1 π → B ∞ π be defined as in (8). Then for every T > 0, one has…”

Aliasing-Truncation Errors in Sampling Approximations of Sub-Gaussian Signals

“…WKS theorems are extensively used in communications and information theory. Various new refine results are published regularly by engineering and mathematics communities, see, e.g., [1,3,10,12,29] and the recent volumes [13,23,30].…”

“…However, the sampling theory for the case of stochastic signals is much less developed comparing to its deterministic counterpart. The publications [1,9,12,26,27] and references therein present an almost exhaustive survey of key approaches in stochastic sampling theory.…”