Stephen D. Casey

Systems of convolution equations, deconvolution, Shannon Sampling, and the wavelet and Gabor transforms

Linear translation invariant systems (e.g., sensors, linear filters) are modeled by the convolution equation

Modifications of the Euclidean algorithm for isolating periodicities from a sparse set of noisy measurements

s = f *\mu

On periodic pulse interval analysis with outliers and missing observations

, where f is the input signal,

Detecting regularity in minefields using collinearity and a modified Euclidean algorithm

\mu

Self-similar fractal sets: theory and procedure

is the system impulse response function (or, more generally, impulse response distribution), and s is the output signal. In many applications, the output s is an inadequate approximation of f, which motivates solving the convolution equation for f, i.e., deconvolving f from

Frequency estimation via sparse zero crossings

\mu

Residue and Sampling Techniques in Deconvolution

. If the function

A modified Euclidean algorithm for isolating periodicities from a sparse set of noisy measurements

\mu

UWB signal processing: Projection, B-splines, and modified Gegenbauer bases

is time-limited (compactly supported) and nonsingular, it is proven that this deconvolution problem is ill-posed.A theory of solving such equations has been developed by Berenstein et al. It circumvents ill-posedness by using a multichannel system. If the signal f is overdetermined by using a system of convolution equations,

Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach

s_i = f * \mu _i ,\,i = 1, \ldots ,n