Two methods to deconvolve: L1-method using simplex algorithm and L2-method using least squares and parameter - Drachman, B. (IEEE transactions on antennas and propagation, 1984)
If r(t) is the linear scattering response of an object to an excitation waveform e(t) , then r(t) = (e * 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 · H = R by the following problem. Minimize \|h_{1}\|+ … + \|h_{n}\| subject to R - λ ≤ E · H ≤ R + λ where λ 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 · H - R \parallel^{2} +λ \parallel H \parallel^{2} where again λ 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.