Two methods to deconvolve: L<inf>1</inf>-method using simplex algorithm and L<inf>2</inf>-method using least squares and parameter

Abstract

If r(t) is the linear scattering response of an object to an excitation waveform e(t) , then r(t) = (e \ast h) (t) . One would like to deconvolve and solve for h(t) , the impulse response. It is well-known that this is often an ill-conditioned problem. Two methods are discussed. The first method replaces the discretized matrix form E \cdot H = R by the following problem. Minimize \|h_{1}\|+ \ldots + \|h_{n}\| subject to R - \lambda \leq E \cdot H \leq R + \lambda where \lambda is a column vector chosen sufficiently small to yield acceptable residuals, yet large enough to make the problem well-conditioned. This problem is converted to a linear programming problem so that the simplex algorithm can be used. The second method is to minimize \parallel E \cdot H - R \parallel^{2} +\lambda \parallel H \parallel^{2} where again \lambda is chosen small enough to yield acceptable residuals and large enough to make the problem well-conditioned. The method will be demonstrated with a Hilbert matrix inversion problem, and also by the deconvolution of the impulse response of a simple target from measured data.