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 typi- cally 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.
More general irregular sampling methods beyond pointwise uniform sampling have been extensively studied in the sampling literature, e.g. @cite_18 @cite_7 @cite_8 . One example is sampling on non-periodic quasi-crystal sets, which has been shown to be stable for bandlimited signals @cite_12 @cite_21 . These sampling approaches are of interest in some realistic situations where signals are only sampled at a nonuniformly spaced sampling set due to constraints imposed by data acquisition devices. Many sophisticated reconstruction algorithms have been developed for the class of bandlimited signals or, more generally, the class of shift-invariant signals @cite_6 @cite_27 @cite_18 . For all these nonuniform sampling methods, the Nyquist sampling rate is necessary for perfect recovery of bandlimited signals @cite_42 @cite_41 @cite_18 .
@cite_18 @cite_7 @cite_8 @cite_41 @cite_21 @cite_42 @cite_6 @cite_27 @cite_12
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This article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift-invariant spaces. It is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift-invariant subspaces by bringing together wavelet theory, frame theory, reproducing kernel Hilbert spaces, approximation theory, amalgam spaces, and sampling. Inspired by applications taken from communication, astronomy, and medicine, the following aspects will be emphasized: (a) The sampling problem is well defined within the setting of shift-invariant spaces. (b) The general theory works in arbitrary dimension and for a broad class of generators. (c) The reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms. These algorithms converge geometrically and are robust in the presence of noise. (d) To model the natural decay conditions of real signals and images, the sampling theory is developed in weighted L p-spaces. Bases in Banach Spaces - Schauder Bases Schauder's Basis for C[a,b] Orthonormal Bases in Hilbert Space The Reproducing Kernel Complete Sequences The Coefficient Functionals Duality Riesz Bases The Stability of Bases in Banach Spaces The Stability of Orthonormal Bases in Hilbert Space Entire Functions of Exponential Type The Classical Factorization Theorems - Weierstrass's Factorization Theorem Jensen's Formula Functions of Finite Order Estimates for Canonical Products Hadamard's Factorization Theorem Restrictions Along a Line - The "Phragmen-Lindelof" Method Carleman's Formula Integrability on a line The Paley-Wiener Theorem The Paley-Wiener Space The Completeness of Sets of Complex Exponentials - The Trigonometric System Exponentials Close to the Trigonometric System A Counterexample Some Intrinsic Properties of Sets of Complex Exponentials Stability Density and the Completeness Radius Interpolation and Bases in Hilbert Space - Moment Sequences in Hilbert Space Bessel Sequences and Riesz-Fischer Sequences Applications to Systems of Complex Exponentials The Moment Space and Its Relation to Equivalent Sequences Interpolation in the Paley-Wiener Space: Functions of Sine Type Interpolation in the Paley-Wiener Space: Stability The Theory of Frames The Stability of Nonharmonic Fourier Series Pointwise Convergence Notes and Comments References List of Special Symbols Index This paper introduces a filterbank interpretation of various sampling strategies, which leads to efficient interpolation and reconstruction methods. An identity, which is referred to as the interpolation identity, is developed and is used to obtain particularly efficient discrete-time systems for interpolation of generalized samples as well as a class of nonuniform samples, to uniform Nyquist samples, either for further processing in that form or for conversion to continuous time. The interpolation identity also leads to new sampling strategies including an extension of Papoulis' (1977) generalized sampling expansion. 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. Apparatus in a crop roll forming machine for improving the core formation of the crop rolls, reducing the amount of particles of crop material lost, and collecting particles of crop material lost from either the crop package or loose crop material during the roll formation process comprising stripping surfaces and an elongated tailgate with a collection pan. The particles are recycled from the collection pan back into the roll forming region by the cooperative interaction of the tailgate collection pan and the upper bale forming means as the bale forming means traverses a predetermined path imparting rotative motion to the crop material delivered to the roll forming region. A constructive solution of the irregular sampling problem for band- limited functions is given. We show how a band-limited function can be com- pletely reconstructed from any random sampling set whose density is higher than the Nyquist rate, and give precise estimates for the speed of convergence of this iteration method. Variations of this algorithm allow for irregular sampling with derivatives, reconstruction of band-limited functions from local averages, and irregular sampling of multivariate band-limited functions. In the irregular sampling problem one is asked whether and how a band- limited function can be completely reconstructed from its irregularly sam- pled values f(xi). This has many applications in signal and image processing, seismology, meteorology, medical imaging, etc. Finding constructive solutions of this problem has received considerable attention among mathematicians and engineers. The mathematical literature provides several uniqueness results (1, 2, 17, 18, 19). It is now part of the folklore that for stable sampling the sampling rate must be at least the Nyquist rate (18). These results, as deep as they are, have had little impact for the applied sciences, because they were not constructive. If the sampling set is just a perturbation of the regular oversampling, then a reconstruction method has been obtained in a seminal paper by Duffin and Schaeffer (6) (see also (29)): if for some L > 0, a > 0, and o > 0 the sampling points xk , k e Z , satisfy (a) - ok a, k ^ I, then the norm equivalence A iR (x) ) with w < n o. This norm equivalence implies that it is possible to reconstruct through an iterative procedure, the so-called frame method. Most of the later work on constructive methods consists of variations of this method (3, 21, 22, 26). The above conditions on the sampling set exclude random irregular sampling sets, e.g., sets with regions of higher sampling density. A partial, but undesirable remedy, to handle highly irregular sampling sets, would be to force the above conditions by throwing away information on part of the points and accept a very slow convergence of the iteration. This chapter presents methods for the reconstruction of bandlimited functions from irregular samples. The first part discusses algorithms for a constructive solution of the irregular sampling problem. An important aspect of these algorithms is the explicit knowledge of the constants involved and efficient error estimates. The second part discusses the numerical implementation of these algorithms and compares the performance of various reconstruction methods. Although most of the material has already appeared in print, several new results are included: (a) a new method to estimate frame bounds, (b) a reconstruction of band-limited functions from partial information, which is not just the samples; (c) a new result on the complete reconstruction of band-limited functions from local averages; (d) a systematic exposition of recent experimental results. Abstract Irregular sampling and “stable sampling” of band-limited functions have been studied by H.J. Landau [H.J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967) 37–52]. We prove that quasicrystals are sets of stable sampling. To cite this article: B. Matei, Y. Meyer, C. R. Acad. Sci. Paris, Ser. I 346 (2008).