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 s = f * mu, where f is the input signal, mu 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 mu. If the function mu 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, s(i) = f * mu(i), i = 1,..., n, the problem of solving for f is well-posed if the set of convolvers {mu(i)} satisfies the condition of being what is called strongly coprime. In this case, there exist compactly supported distributions (deconvolvers) nu(i), i = 1,..., n such that delta = mu1 * nu1 + ... + mu(n) * nu(n), which in turn gives f = s1 * nu1 + ... + s(n) * nu(n). The authors describe the strongly coprime condition and give examples of sets of strongly coprime sensors and their deconvolutions for functions in one dimension. It is then shown how the theory can work in conjunction with the Shannon Sampling Theorem. In particular, if it is assumed that f is band-limited, then the analog signal f may be reconstructed from the sampled outputs of the sensors {f * mu(i)} if the sampling rate is greater than or equal to Nyquist. To do this, new interpolating functions are created by combining sinc functions with the deconvolvers. Since the purpose of deconvolution is greater signal resolution, it is shown how data compression and analysis using the wavelet and Gabor transforms are implemented in coordination with the deconvolution methods described above. Under the proposed procedure, wavelet or Gabor coefficients of the signal f may be recovered from {f * mu(i)} in a single processing step. Finally, the authors close by citing several applications of the theory.