Biomedical Imaging GroupSTI
English only   BIG > Publications > Generalized Sampling

 Home Page
 News & Events
 Tutorials and Reviews
 Download Algorithms

 All BibTeX References

A Generalized Sampling Theory without Band-Limiting Constraints

M. Unser, J. Zerubia

IEEE Transactions on Circuits and Systems—II: Analog and Digital Signal Processing, vol. 45, no. 8, pp. 959-969, August 1998.

We consider the problem of the reconstruction of a continuous-time function f(x)H from the samples of the responses of m linear shift-invariant systems sampled at 1 ⁄ m the reconstruction rate. We extend Papoulis' generalized sampling theory in two important respects. First, our class of admissible input signals (typ. H = L2) is considerably larger than the subspace of bandlimited functions. 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. Since we have enlarged the class of admissible input functions, we have to give up Shannon and Papoulis' principle of an exact reconstruction. Instead, we seek an approximation fV(φ) that is consistent in the sense that it produces exactly the same measurements as the input of the system. This leads to a generalization of Papoulis' sampling theorem and a practical reconstruction algorithm that takes the form of a multivariate filter. In particular, we show that the corresponding system acts as a projector from H onto V(φ). We then propose two complementary polyphase and modulation domain interpretations of our solution. The polyphase representation leads to a simple understanding of our reconstruction algorithm in terms of a perfect reconstruction filterbank. The modulation analysis, on the other hand, is useful in providing the connection with Papoulis' earlier results for the bandlimited case. Finally, we illustrate the general applicability of our theory by presenting new examples of interlaced and derivative sampling using splines.

AUTHOR="Unser, M. and Zerubia, J.",
TITLE="A Generalized Sampling Theory without Band-Limiting
JOURNAL="{IEEE} Transactions on Circuits and Systems---{II}:
        {A}nalog and Digital Signal Processing",

© 1998 IEEE. 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 IEEE.
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.