 
A General Hilbert Space Framework for the Discretization of Continuous Signal Processing Operators
M. Unser
Proceedings of the SPIE Conference on Mathematical Imaging: Wavelet Applications in Signal and Image Processing III, San Diego CA, USA, July 914, 1995, vol. 2569, part I, pp. 5161.
We present a unifying framework for the design of discrete algorithms that implement continuous signal processing operators. The underlying continuoustime signals are represented as linear combinations of the integershifts of a generating function φ_{i} with (i = 1, 2) (continuous/discrete representation). The corresponding input and output functions spaces are V(φ_{1}) and V(φ_{2}), respectively. The principle of the method is as follows: we start by interpolating the discrete input signal with a function S_{1} ∈ V(φ_{1}). We then apply a linear operator T to this function and compute the minimum error approximation of the result in the output space V(φ_{2}). The corresponding algorithm can be expressed in terms of digital filters and a matrix multiplication. In this context, we emphasize the advantages of Bsplines, and show how a judicious use of these basis functions can result in fast implementations of various types of operators. We present design examples of differential operators involving very short FIR filters. We also describe an efficient procedure for the geometric affine transformation of signals. The present formulation is general enough to include most earlier continuous/discrete signal processing techniques (e.g., standard bandlimited approach, spline or waveletbased) as special cases.

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AUTHOR="Unser, M.",
TITLE="A General {H}ilbert Space Framework for the Discretization of
Continuous Signal Processing Operators",
BOOKTITLE="Proceedings of the {SPIE} Conference on Mathematical
Imaging: {W}avelet Applications in Signal and Image Processing
{III}",
YEAR="1995",
editor="",
volume="2569",
series="",
pages="5161",
address="San Diego CA, USA",
month="July 914,",
organization="",
publisher="",
note="{Part {I}}")
©
1995
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