A period-specific realization of linear continuous-time systems

It is a well-known fact that a weighting pattern matrix has a periodic realization if and only if the matrix is separable and periodic. This fact, however, does not cope with a reasonable question of period-specific realization problem  : for a given weighting pattern matrix and a given positive real number, find a periodic realization corresponding to the given matrix and with a period equal to the given number. This paper answers the question by showing that the period-specific realization problem has a solution if and only if the given weighting pattern matrix is separable and has a period equal to the given positive real number. To this end, two types of period-specific realizations are constructed. One is with a constant AA-matrix whose dimension is not necessarily minimal among all possible period-specific realizations. The other is with a non-constant and periodically time-varying AA-matrix whose dimension turned out minimal among all possible period-specific realizations.