Sampling Moments and Reconstructing Signals of Finite Rate of Innovation: Shannon Meets Strang–Fix

By Pier Luigi Dragotti, Martin Vetterli, Thierry Blu. IEEE Transactions on Signal Processing, 2007

468 citations recorded by the source corpus.

Consider the problem of sampling signals which are not bandlimited, but still have a finite number of degrees of freedom per unit of time, such as, for example, nonuniform splines or piecewise polynomials, and call the number of degrees of freedom per unit of time the rate of innovation. Classical sampling theory does not enable a perfect reconstruction of such signals since they are not bandlimited. Recently, it was shown that, by using an adequate sampling kernel and a sampling rate greater or equal to the rate of innovation, it is possible to reconstruct such signals uniquely . These sampling schemes, however, use kernels with infinite support, and this leads to complex and potentially unstable reconstruction algorithms. In this paper, we show that many signals with a finite rate of innovation can be sampled and perfectly reconstructed using physically realizable kernels of compact support and a local reconstruction algorithm. The class of kernels that we can use is very rich and includes functions satisfying Strang-Fix conditions, exponential splines and functions with rational Fourier transform. This last class of kernels is quite general and includes, for instance, any linear electric circuit. We, thus, show with an example how to estimate a signal of finite rate of innovation at the output of an RC circuit. The case of noisy measurements is also analyzed, and we present a novel algorithm that reduces the effect of noise by oversampling

Subfields: Applied Mathematics, Computational Mechanics, Computer Vision and Pattern Recognition

View Paper: https://www.semanticscholar.org/paper/0c6f012f43c97a3fd028ddc854e23d0e3da694a4

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Citations in the corpus

Structured Compressed Sensing: From Theory to Applications (2011, methodology): “values of [7].”
A Theory for Sampling Signals From a Union of Subspaces (2008, unknown): “Our proposed sampling framework has close ties to the recent work on sampling signals with finite rate of innovation [12]–[14], which demonstrates that several classes of non-bandlimited signals can be uniformly sampled and perfectly reconstructed.”
Sparse Sampling of Signal Innovations (2008, methodology): “Importantly, this approach can be generalized to other kernels such as the (nonperiodized) sinc, the Gaussian windows [7], and more recently any window that satisfies a Strang-Fix like condition [ 8 ].”
Parameter estimation for nonincreasing exponential sums by Prony-like methods (2013, unknown): “AMS Subject Classifications: 65D10, 41A45, 65F15, 65F20, 94A12.”
Towards a Mathematical Theory of Super‐resolution (2012, background): “Leveraging ideas related to error correction codes and spectral estimation, [16] shows that it is possible to recover trains of Dirac distributions from low-pass measurements at their rate of innovation (in essence, the density of spikes per unit of time).”
Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals (2009, background): “We believe that these applications will be easier to achieve in the near future because suitable ADC and DSP technology is already available.”
Innovation Rate Sampling of Pulse Streams With Application to Ultrasound Imaging (2010, methodology): “The work in [13] introduced a general family of polynomial and exponential reproducing kernels, which can be used to solve FRI problems.”
Robust Recovery of Signals From a Structured Union of Subspaces (2008, background): “Another example of a union of subspaces is the set of finite rate of innovation signals [20], [21], tha t are modelled as a weighted sum of shifts of a given generating function, where the shifts are unknown.”