Let @math be a finite abelian group and @math be a circular convolution operator on @math . The problem under consideration is how to construct minimal @math and @math such that @math is a frame for @math , where @math is the canonical basis of @math . This problem is motivated by the spatiotemporal sampling problem in discrete spatially invariant evolution processes. We will show that the cardinality of @math should be at least equal to the largest geometric multiplicity of eigenvalues of @math , and consider the universal spatiotemporal sampling sets @math for convolution operators whose eigenvalues subject to the same largest geometric multiplicity. We will give an algebraic characterization for such sampling sets and show how this problem is linked with sparse signal processing theory and polynomial interpolation theory.
Our work is closely related to @cite_35 , 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 @math that is subject to the same cardinality of @math in @cite_35 . However, we do not make sparsity assumptions on the signal space. Instead, we seek sub-Nyquist spatial sampling rate, but want to compensate the insufficient spatial sampling rate by oversampling in time.
@cite_35
2953338282
We study the compressed sensing reconstruction problem for a broad class of random, band-diagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and numerically by KrzakalaEtAl , message passing algorithms can effectively solve the reconstruction problem for spatially coupled measurements with undersampling rates close to the fraction of non-zero coordinates. We use an approximate message passing (AMP) algorithm and analyze it through the state evolution method. We give a rigorous proof that this approach is successful as soon as the undersampling rate @math exceeds the (upper) R 'enyi information dimension of the signal, @math . More precisely, for a sequence of signals of diverging dimension @math whose empirical distribution converges to @math , reconstruction is with high probability successful from @math measurements taken according to a band diagonal matrix. For sparse signals, i.e., sequences of dimension @math and @math non-zero entries, this implies reconstruction from @math measurements. For discrete' signals, i.e., signals whose coordinates take a fixed finite set of values, this implies reconstruction from @math measurements. The result is robust with respect to noise, does not apply uniquely to random signals, but requires the knowledge of the empirical distribution of the signal @math .