In this work, we propose a characterization of the moments of a linear system for generalized filters that do not have an implicit representation, i.e., they do not satisfy a differential equation. We devote particular attention to the case in which the filter has a non-smooth convolution kernel. The
notion of moment is extended using an integral matrix equation. We present a family of reduced-order models that achieves moment matching based on this generalized notion of moment. Finally, the developed results are demonstrated by means of a numerical example with smooth and non-smooth kernels.