It has been almost thirty years since Shannon introduced the sampling theorem to communications theory. In this review paper we will attempt to present the various contributions made for the sampling theorems with the necessary mathematical details to make it self-contained. We will begin by a clear statement of Shannon's sampling theorem followed by its applied interpretation for time-invariant systems. Then we will review its origin as Whittaker's interpolation series. The extensions will include sampling for functions of more than one variable, random processes, nonuniform sampling, nonband-limited functions, implicit sampling, generalized functions (distributions), sampling with the function and its derivatives as suggested by Shannon in his original paper, and sampling for general integral transforms. Also the conditions on the functions to be sampled will be summarized. The error analysis of the various sampling expansions, including specific error bounds for the truncation, aliasing, jitter and parts of various other errors will be discussed and summarized. This paper will be concluded by searching the different recent applications of the sampling theorems in other fields, besides communications theory. These include optics, crystallography, time-varying systems, boundary value problems, spline approximation, special functions, and the Fourier and other discrete transforms.

The Shannon sampling theorem&#8212;Its various extensions and applications: A tutorial review

The well-known Whittaker-Kotel'nikov-Shannon sampling theorem for frequency-bandlimited functions of time is extended to functions of multidimensional arguments. It is shown that a function whose spectrum is restricted to a finite region of wave-number space may be reconstructed from its samples taken over a periodic lattice having suitably small repetition vectors. The most efficient lattice (i.e., requiring minimum sampling points per unit hypervolume) is not in general rectangular, nor is a unique reconstruction function associated with a given sampling lattice. The above results also apply to homogeneous wave-number-limited stochastic processes in the sense of a vanishing mean-square error. It is also found that, given a particular sampling lattice, the optimum (mean-square) presampling filter for nonwave-number-limited processes effects an ideal wave-number cutoff appropriate to the specified sampling lattice. Particular attention is paid to isotropic processes: minimum sampling lattices are specified up to eight-dimensional spaces, and a number of typical reconstruction functions are calculated.

Sampling and reconstruction of wave-number-limited functions in N-dimensional euclidean spaces

This paper describes the TELIAMADE system, a new indoor positioning system based on time-of-flight (TOF) of ultrasonic signal to estimate the distance between a receiver node and a transmitter node. TELIAMADE system consists of a set of wireless nodes equipped with a radio module for communication and a module for the transmission and reception of ultrasound. The access to the ultrasonic channel is managed by applying a synchronization algorithm based on a time-division multiplexing (TDMA) scheme. The ultrasonic signal is transmitted using a carrier frequency of 40 kHz and the TOF measurement is estimated by applying a quadrature detector to the signal obtained at the A/D converter output. Low sampling frequencies of 17.78 kHz or even 12.31 kHz are possible using quadrature sampling in order to optimize memory requirements and to reduce the computational cost in signal processing. The distance is calculated from the TOF taking into account the speed of sound. An excellent accuracy in the estimation of the TOF is achieved using parabolic interpolation to detect of maximum of the signal envelope at the matched filter output. The signal phase information is also used for enhancing the TOF measurement accuracy. Experimental results show a root mean square error (rmse) less than 2 mm and a standard deviation less than 0.3 mm for pseudorange measurements in the range of distances between 2 and 6 m. The system location accuracy is also evaluated by applying multilateration. A sub-centimeter location accuracy is achieved with an average rmse of 9.6 mm.

https://www.mdpi.com/1424-8220/13/3/3501/pdf

Ultrasound Indoor Positioning System Based on a Low-Power Wireless Sensor Network Providing Sub-Centimeter Accuracy

A new series for the interpolation of band-limited functions is found by using an approximation to tho prolate spheroidal wave function as a convergence factor. A truncation error bound is found for the central interpolation of band-limited functions using the new series. It is shown that this bound is lower than other known bounds in many cases of interest.

Interpolation of band-limited functions using the approximate prolate series (Corresp.)

Sampling is one of the fundamental topics in the signal processing community. Theorems proposed under this topic form the bridge between the continuous-time signals and discrete-time signals. Several sampling theorems, which aid in the reconstruction of signals in the linear canonical transform (LCT) domain, have been proposed in the literature. However, two main practical issues associated with the sampling of the LCT still remain unresolved. The first one relates to the reconstruction of the original signal from nonuniform samples and the other issue relates to the fact that only a finite number of samples are available practically. Focusing on these issues, this paper seeks to address the above from the LCT point of view. First, we extend several previously developed theorems for signals band-limited in the Fourier domain to signals band-limited in the LCT domain, followed by the derivation of the reconstruction formulas for finite uniform or recurrent nonuniform sampling points associated with the LCT. Simulation results and the potential applications of the theorem are also proposed.

On Sampling of Band-Limited Signals Associated With the Linear Canonical Transform

Upper bounds are obtained for the error, termed truncation error, which arises in reconstituting a band-limited function by summing over only a finite number (instead of the requisite infinite number) of samples of this function in an appropriate sampling-theorem expansion. Upper bounds are given for the truncation errors of the Cardinal and Fogel sampling expansions and for "self-truncating" versions of these two sampling expansions; these latter sampling expansions are "self-truncating" in the sense that the upper bounds on their truncation errors are almost always much lower than the upper bounds on the truncation errors of their prototype sampling expansions. All of the upper bounds are given as functions of three parameters: M, the maximum magnitude of the band-limited function; q, the per unit guard band (assuming that the band-limited function is sampled at a rate greater than the Nyquist rate); and N, the measure of the number of samples in the finite summation of terms of the sampling expansion.

Truncation Error of Sampling-Theorem Expansions

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Truncation Error of Sampling-Theorem Expansions