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An Extension of Papoulis' Sampling Theorem for Non-Bandlimited Functions

M. Unser

Proceedings of the International Workshop on Sampling Theory and Applications, Aveiro, Portuguese Republic, June 16-19, 1997, pp. 10-15.


We investigate the problem of the reconstruction of a continuous-time function f(x)∈H from the responses of m linear shift-invariant systems sampled at 1/m the reconstruction rate, extending Papoulis' generalized sampling theory in two important respects. First, we allow for arbitrary (non-bandlimited) input signals (typ. H=L2). Second, we use a more general specification of the reconstruction subspace V(φ), so that the output of the system can take the form of a bandlimited function, a spline, or a wavelet expansion. The system that we describe yields an approximation f∈V(φ) that is consistent with the input f(x) in the sense that it produces exactly the same measurements. We show that this solution can be computed by multivariate filtering. We also characterize the stability of the system (condition number). Finally, we prove that the generalized sampling solution is essentially equivalent to the optimal minimum error approximation (orthogonal projection) which is generally not accessible. We also present some illustrative examples using splines.

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AUTHOR="Unser, M.",
TITLE="An Extension of {P}apoulis' Sampling Theorem for
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BOOKTITLE="Proceedings of the International Workshop on Sampling
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pages="10--15",
address="Aveiro, Portuguese Republic",
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© 1997 IWSTA. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from IWSTA. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.
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