Discrete Sampling and Interpolation: Universal Sampling Sets for Discrete Bandlimited Spaces

Abstract

We study the problem of interpolating all values of a discrete signal f of length N when d < N values are known, especially in the case when the Fourier transform of the signal is zero outside some prescribed index set J ; these comprise the (generalized) bandlimited spaces B J . The sampling pattern for f is specified by an index set I, and is said to be a universal sampling set if samples in the locations I can be used to interpolate signals from B J for any J . When N is a prime power we give several characterizations of universal sampling sets, some structure theorems for such sets, an algorithm for their construction, and a formula that counts them. There are also natural applications to additive uncertainty principles.

Universal Spatiotemporal Sampling Sets for Discrete Spatially Invariant Evolutionary Systems

“…We can find an introduction and a proof of this result in the survey paper [31]. Later [10] and [24] generalized the techniques developed by Chebotarëv and gave a characterization to the special case when d is a power of prime. We list the results in [10] here.…”

“…Related Work. Our work is closely related to [24], in which the authors have studied the universal spatial sensor locations for discrete bandlimited space; in some sense, finding universal spatiotemporal sampling sets for convolution operators with eigenvalues subject to the same largest geometric multiplicity in our problem, is analogous to finding universal spatial sampling sets for discrete bandlimited space B J that is subject to the same cardinality of J in [24]. However, we do not make sparsity assumptions on the signal space.…”

“…There is a pressing need for deterministic constructions of full spark matrix ( [10,24] and the related work in [15,16]) . In [10], the authors have considered the problem of finding Ω such that (F d ) Ω is a full spark matrix.…”

Discrete Sampling: A Graph Theoretic Approach to Orthogonal Interpolation

“…See [2] for further general properties of interpolating bases. Every B J has an interpolating basis but not every B J has an orthogonal interpolating basis, and this is the starting point of our study.…”

“…As different as these areas might seem, what is most striking to us is that they intersect in the fundamental problem of sampling and interpolation for discrete signals. This is the problem that first motivated us, and the work here is a sequel to [2]. While we will refer to some of the results there we have tried to make the present paper self-contained.…”