With the aim of treating nonlinear systems with inputs being discrete and outputs being generalized functions, generalized Poisson functionals are defined and analysed, where the U-transforms and the renormalizations play essential roles. For Poisson functionals, the differential operators with respect to a Poisson white noise Ṗ(t), their adjoint operators and the multiplication operators by Ṗ(t) are defined. Since these operators involve the time parameter explicitly, they can be used to obtain information concerning the Poisson functionals at each point in time. As an example, a new method for measuring the Wiener kernels of such functionals is outlined.