Identification of bandlimited pseudo-differential operators

Abstract

To understand whether a pseudo-differential operator with unit area spreading support is identifiable on a single input signal, we combined and improved existing approaches, and based on that, considered what we call the "pass to limit" method.We enhanced the rectification method, and by using the adjoint relations, we decomposed the identification map on rectifiable spreading support into the composition of an identification procedure on rectangle spreading support plus the action of a Gabor matrix.We analyzed the group structure behind discrete time-frequency shifts, and also classified unitary Gabor matrices on prime dimensions, that is, we showed there exist choices of window vectors for a Gabor matrix to be unitary, if and only if its support set is isomorphic to the quotient of proper non-trivial subgroups in ZN x ZN (N is a prime number).We explored properties of periodically weighted delta trains in Wiener-Amalgam spaces, and studied how these properties carry on to the identification map when using these delta trains as identifiers, in particular, we constructed a discretely supported delta train which is a universal identifier that identifies all pseudo-differential operators with rectifiable spreading support (i.e., it identifies all currently known identifiable pseudo-differential operators).We looked at weak* convergence of periodically weighted delta trains as identifiers, and demonstrated that such convergence itself, even combined with innerapproximation of the spreading support, can pass onto the weak* convergence of the identification map. However, it is not conclusive whether the limit remains as an identifier, since bounds do not pass along the weak* convergence.We also gave geometric insights in aforementioned parts, and briefly discussed identifiability of overspread pseudo-differential operators with liner correlation constraints on the values of their spreading functions.