Given a rational matrix G with complex coefficients and a domain Γ in the closed complex plane, both arbitrary, we develop a complete theory of coprime factorizations of G over Γ, with denominators of McMillan degree as small as possible. The main tool is a general pole displacement theorem which gives conditions for an invertible rational matrix to dislocate by multiplication a part of the pol...

This note presents a state-space algorithm for the calculation of a normalized coprime factorization of continuous-time generalized dynamical systems. It will be shown that two Riccati equations have to be solved to obtain this normalized coprime factorization.

A coprime array is capable of achieving more degrees-of-freedom for direction-of-arrival (DOA) estimation than a uniform linear array when utilizing the same number of sensors. However, existing algorithms exploiting coprime array usually adopt predefined spatial sampling grids for optimization problem design or include spectrum peak search process for DOA estimation, resulting in the contradic...

In this paper, it is shown that a linear uncertain system described by a certain L = multiplicative or relative perturbation in its coprime factors that are not necessarily normalized, is the same as the one described by a gap or l-gap metric ball. Hence all the stability robustness results for gap or l-gap metric uncertainty carry over to this type of coprime factor perturbation. Uncertain sys...

The main purpose of this paper is to revisit the internal/simultaneous/robust stabilization problems without assuming the existence of doubly coprime factorizations for the transfer matrices. Indeed, it has been recently shown in the literature that an internally stabilizable does not generally admit doubly coprime factorizations. Firstly, we give new necessary and sufficient conditions for int...

We study the problem of strongly coprime factorization over H-infinity of the unit disc. We give a necessary and sufficient condition for the existence of such a coprime factorization in terms of an optimal control problem over the doubly infinite discrete-time axis. In particular, we show that an equivalent condition for the existence of such a coprime factorization is that both the control an...

This sequence was first considered as a pseudorandom number generator by D. H. Lehmer. For the power generator we are given integers e, n > 1 and a seed u = u0 > 1, and we compute the sequence ui+1 = u e i (mod n) so that ui = u ei (mod n). A popular case is e = 2, which is called the Blum–Blum–Shub (BBS) generator. Both of these generators are periodic sequences, and it is of interest to compu...

This sequence was first considered as a pseudorandom number generator by D. H. Lehmer. For the power generator we are given integers e, n > 1 and a seed u0 > 1, and we compute the sequence ui+1 = u e i (mod n) so that ui = u ei 0 (mod n). A popular case is e = 2, which is called the Blum–Blum–Shub (BBS) generator. Both of these generators are periodic sequences, and it is of interest to compute...