Sampling-50 years after Shannon

By Michael A. Unser. Proceedings of the IEEE, 2000

1,484 citations recorded by the source corpus.

This paper presents an account of the current state of sampling, 50 years after Shannon's formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefitted from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbert-space formulation, we reinterpret Shannon's sampling procedure as an orthogonal projection onto the subspace of band-limited functions. We then extend the standard sampling paradigm for a presentation of functions in the more general class of "shift-in-variant" function spaces, including splines and wavelets. Practically, this allows for simpler-and possibly more realistic-interpolation models, which can be used in conjunction with a much wider class of (anti-aliasing) prefilters that are not necessarily ideal low-pass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned.

Subfields: Applied Mathematics, Computer Vision and Pattern Recognition, Signal Processing

View Paper: https://www.semanticscholar.org/paper/92be1e4e41ad577945fbcd83d590fd74056b6073

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Citations in the corpus

ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems (2004, background): “Other sampling kernels can also be used in (7);2 see [17] and [18] for excellent tutorials on sampling.”
Foundations of Signal Processing (2014, methodology): “We also recommend the review papers by Jerri [49] and by Unser [103]; the latter develops sampling in shiftinvariant subspaces.”
Convex Optimization in Signal Processing and Communications (2009, neutral): “functions [21].”
Technical foundations of TANDEM-STRAIGHT, a speech analysis, modification and synthesis framework (2011, background): “A new formulation of sampling theory, called consistent sampling (Unser 2000), provides the basis for this process.”
Shannon sampling II: Connections to learning theory (2005, background): “The rich data requirement is reasonable for function reconstruction such as sampling theory [13].”
Sampling Expansions in Reproducing Kernel Hilbert and Banach Spaces (2009, background): “For sampling and reconstruction of signals in a shift-invariant space, the reader may refer to [2, 4, 21, 23] and references therein.”
Temporally variable multi-aspect N-way morphing based on interference-free speech representations (2013, background): “An integration variable ξ is a point on the time axis of A.”
Sampling Moments and Reconstructing Signals of Finite Rate of Innovation: Shannon Meets Strang–Fix (2007, positive): “SAMPLING theory plays a central role in modern signal processing and communications and has experienced a recent revival thanks, in part, to the recent advances in wavelet theory [12], [27].”
A chronology of interpolation: from ancient astronomy to modern signal and image processing (2002, background): “…functions , that is to say, functions for which is finite, the norm of the difference between and the approximation obtained from (22) is given by [253], [274], [276], [277] as (50) In view of the practical need in many applications to optimize the cost-performance tradeoff of interpolation, this…”
A Theory for Sampling Signals From a Union of Subspaces (2008, neutral): “For example, the classical Kotelnikov-Shannon- Whittaker sampling theorem can be presented as follow [3].”
Antileakage Fourier transform for seismic data regularization (2005, methodology): “It assures that on a regular grid (k, ∈ Z), the sinc function has the weight 1 on the original data position and zero weight on all other integer locations (Unser, 2000).”
CheapTrick, a spectral envelope estimator for high-quality speech synthesis (2015, background): “Spectral recovery is carried out on the basis of consistent sampling (Unser, 2000).”
Sparse Sampling of Signal Innovations (2008, background): “Bandlimitedness, however, is not necessary as is well known but only rarely taken advantage of [1].”
Optical Imaging and Spectroscopy (2009, background): “Concepts of discrete representation and sampling have evolved considerably in the half-century since Shannon presented the sampling theorem [234].”