Systems of convolution equations, deconvolution, and the Shannon Sampling Theorem

In this paper we will discuss the convolution equation $s = f*\mu$, which is the model for many of the linear systems used in signal and image processing, including linear filters and remote sensors. We wish to deconvolve the equation, i.e. solve for f given $\mu$ and $f*\mu$. However, this problem is ill-posed for all physically realizable systems modeled by a single convolution equation. We will present a theory developed by Berenstein and others that circumvents ill-posedness by using a system of convolution equations (multichannel system) to overdetermine the signal. We present solutions to the deconvolution problem by using the Berenstein theory. We also will show how the deconvolution theory can be used with the Shannon Sampling Theorem. If the signal is assumed to be band-limited, then the analog signal can be reconstructed from the sampled outputs of the sensors if the sampling rate is above the Nyquist rate. We close by discussing our results on computerized simulations of the theory.