We will introduce Weyl-Heysenberg ensembles, a large class of window-dependent Random Processes, and a new methodology for the study of the large N behaviour of their discrete counterparts, using methods from operator theory and analysis on the phase space. By selecting windows within the family of Hermite polynomials, we obtain the Ginibre ensemble and variations of Haimi-Hedenmalm polyanalytic ensembles. Most of the talk will include joint work with K. Gröchenig, J. M. Pereira, J. L. Romero and S. Torquato.