Decomposition into minimum and maximum phase components

Abstract

Abstract In this paper we give a new solution to the problem of factoring p'(t) = a(t) * b(-t) * delta (t - tau ) into its maximum-phase, minimum-phase, and pure-delay components. Our method is based on the 'winding number' and the solution of three Toeplitz systems of equations. First we use the winding number from complex analysis to estimate the pure-delay component, which we subsequently remove using p(t) = p'(t) * delta (t + tau ). Then we solve the symmetrical Toeplitz system of equations, considering only t > or = 0, p(t) * p(-t) * [a(t) * b(t)] (super -1) = a(-t) * b(-t) to get [a(t) * b(t)] (super -1) . Finally, we similarly solve the two asymmetrical Toeplitz systems of equations, p(-t) * [a(t) * b(t)] (super -1) * a(t) = a(-t) for a(t) and p(t) * [a(t) * b(t)] (super -1) * b(t) = b(-t) for b(t).The practical effectiveness of the method is confirmed by application to a synthetic waveform and to an air-gun signature. When stabilization of the Toeplitz solution is required, we find that it is important to be frugal with the addition of white noise. The computation of the winding number is robust.