The paper investigates the stabilization problem for discrete LTI processes by realizable digital controllers. The process is described in the form of forward models, which are mostly used by engineers but proved to be unsuitable for causality considerations. The set of all stabilizing causal controllers is constructed in a parametric form. Moreover, the authors construct the set of all causal stabilizing controllers which satisfy additional constraints, i.e. the set of elementary divisors of the characteristic matrix of the closed loop has a certain structure of practical relevance. As an example, the causal structural stabilization problem is solved for the discrete forward model of the double integrator.