On the zeros of blocked linear systems with single and mixed frequency data

This paper studies properties of blocked systems resulting from blocking discrete linear systems with mixed frequency data. The focus is on the zeros of the blocked systems. We first establish results on the simpler single frequency case, where the unblocked linear systems have all data at the same frequency. In particular, an explicit relation between the system matrix of the unblocked linear systems and that of their corresponding blocked systems is derived. Based on this relation, it is shown that the blocked systems are zero free if and only if the related unblocked systems are zero free. Furthermore, it is illustrated that square systems have zeros generically, i.e. for generic parameter matrices, and the corresponding kernel is of dimension one. With the help of the results obtained for the single frequency case, we then identify a situation in which the blocked systems can be zero free.