the structured controllability radii of higher order systems abstract in this paper we shall deal with the problem of calculation of the controllability radius of higher order dynamical systems of the form ex k t a k 1 x k 1 t a 1 x 1 t a 0 x t bu t where the leading coefficient matrix e need not assumed to be non singular by using multi valued linear operators we are able to derive computable formulas of the controllability radius under the assumption that the system s coefficient matrices are subjected to structured affine perturbations some examples are provided to illustrate the obtained results

the structured distance to the nearest system without property mathcal p for a system matrix m this paper explores the smallest frobenius norm additive structured perturbation delta m for which a system property mathcal p e g controllability observability stability etc fails to hold i e delta m is the structured perturbation with smallest frobenius norm such that there exists a property matrix r in mathcal p for which m delta m r drops rank the frobenius norm is used because of its direct dependence on the magnitude of each entry in the perturbation matrix necessary conditions on a locally minimum norm structured rank reducing perturbation delta m and associated property matrix r are set forth and proven an iterative algorithm is also set forth that computes a locally minimum norm structured perturbation and associated property matrix satisfying the necessary conditions algorithm convergence is proven using a discrete lyapunov function

the model matching problem for periodic discrete time systems in this paper the model matching problem via state feedback is studied for discrete time periodic systems it is shown that under some zero matching conditions it is possible to assign a target input output periodic system by means of a periodic state feedback control law furthermore this entails that a periodic system can be converted into a closed loop time invariant system via state feedback

single loop stability margins for multirate and periodic control systems it is shown how to compute the stability margin of periodic systems with respect to compatible periodic perturbations of such systems it is suggested that the nonstandard time invariant model for periodic systems introduced by the author syst contr lett vol 17 pp 9 14 1991 leads to a simpler computation although the standard model gives the same result

asymptotic properties of robust complex covariance matrix estimates in many statistical signal processing applications the estimation of nuisance parameters and parameters of interest is strongly linked to the resulting performance generally these applications deal with complex data this paper focuses on covariance matrix estimation problems in non gaussian environments and particularly the m estimators in the context of elliptical distributions first this paper extends to the complex case the results of tyler in d tyler robustness and efficiency properties of scatter matrices biometrika vol 70 no 2 p 411 1983 more precisely the asymptotic distribution of these estimators as well as the asymptotic distribution of any homogeneous function of degree 0 of the m estimates are derived on the other hand we show the improvement of such results on two applications directions of arrival doa estimation using the multiple signal classification music algorithm and adaptive radar detection based on the adaptive normalized matched filter anmf test

the structured controllability radius of linear delay systems in this article we shall deal with the problem of calculation of the controllability radius of a delay dynamical systems of the form x t a 0 x t a 1 x t h 1 a k x t h k bu t by using multi valued linear operators we are able to derive computable formulas for the controllability radius of a controllable delay system in the case where the system s coefficient matrices are subjected to structured perturbations some examples are provided to illustrate the obtained results

on multiple covariance equality testing with application to sar change detection this paper deals with the problem of testing the equality of m covariance matrices we first identify a suitable group of transformations leaving the problem invariant and obtain the corresponding maximal invariant statistic then the generalized likelihood ratio test glrt is recalled and explicit expressions for rao wald gradient and durbin tests are provided also equivalences among them and with other well known tests proposed in open literature mostly for the real valued case are analyzed and compared finally the application of the proposed framework to the relevant signal processing application of multipass coherent change detection ccd in polarimetric synthetic aperture radar is demonstrated both on simulated and on live data

real perturbation values and real quadratic forms in a complex vector space abstract a sequence of real numbers connected to a complex matrix is introduced it is shown how these real perturbation values can be computed and that they have several properties similar to the singular values the so called real pseudospectra and real stability radii can be computed using the real perturbation values the main result concerns the signature of real quadratic forms in complex vector spaces

radius of approximate controllability of linear retarded systems under structured perturbations abstract in this paper we shall deal with the problem of calculation of the radius of approximate controllability in the banach state space k n l 2 h k 0 k n for linear retarded systems of the form x t a 0 x t a 1 x t h 1 a k x t h k b u t by using multi valued linear operators we are able to derive computable formulas for this radius when the system s coefficient matrices are subjected to structured perturbations some examples are provided to illustrate the obtained results

a unifying framework for adaptive radar detection in homogeneous plus structured interference part i on the maximal invariant statistic this paper deals with the problem of adaptive multidimensional multichannel signal detection in homogeneous gaussian disturbance with unknown covariance matrix and structured deterministic interference the aforementioned problem corresponds to a generalization of the well known generalized multivariate analysis of variance gmanova in this part i of the paper we formulate the considered problem in canonical form and after identifying a desirable group of transformations for the considered hypothesis testing we derive a maximal invariant statistic mis for the problem at hand furthermore we provide the mis distribution in the form of a stochastic representation finally strong connections to the mis obtained in the open literature in simpler scenarios are underlined

on the long range propagation of sound over irregular terrain abstract the theory of sound propagation over randomly irregular nominally plane terrain of finite impedance is discussed the analysis is an extension of the theory of coherent scatter originally proposed by biot 1 for an irregular rigid surface it combines biot s approach wherein the surface irregularities are modeled by a homogeneous distribution of hemispherical bosses with more conventional analyses in which the ground is modeled as a smooth plane of finite impedance at sufficiently low frequencies it is shown that the interaction of the surface irregularities with the near field of a ground based source leads to the production of surface waves which are effective in penetrating the ground shadow zone predicted for a smooth surface of the same impedance

sound propagation above an impedance boundary an asymptotic solution of the scalar wave field due to a point source above a locally reacting plane surface is obtained by a modified saddle point method numerical calculation indicates that the relatively simple solution is more accurate than the thomasson asymptotic solution and the chien soroka solution furthermore it is shown that another solution simplified still more is also in good agreement with the exact solution

nearest pair with more nonconstant invariant factors and pseudospectrum abstract let a b c n n c n m suppose that the number of nonconstant i e 1 invariant factors of the polynomial matrix i n 0 a b is less than k for all complex number denote by n k 1 i n 0 a b the greatest n k 1 th singular value of the matrix i n 0 a b the minimum absolute value of the real function of complex variable n k 1 i n 0 a b gives the distance from a b to the set of pairs with more or equal number of nonconstant invariant factors when k 1 this specializes in the formula of eising for the distance from a controllable pair a b to the nearest uncontrollable pair the complex numbers lying in the sublevel set c n i n 0 a b e of the function n i n 0 a b are the uncontrollable modes of all the pairs that are within an e tolerance of a b all the results of this paper are an immediate consequence of the singular value decomposition of a matrix and of the interpretation of the singular values as the distances to the nearest matrices of lower ranks

a unifying framework for adaptive radar detection in homogeneous plus structured interference part ii detectors design this paper deals with the problem of adaptive multidimensional multichannel signal detection in homogeneous gaussian disturbance with unknown covariance matrix and structured unknown deterministic interference the aforementioned problem extends the well known generalized multivariate analysis of variance gmanova tackled in the open literature in part i of this paper we have obtained the maximal invariant statistic mis for the problem under consideration as an enabling tool for the design of suitable detectors which possess the constant false alarm rate cfar property herein we focus on the development of several theoretically founded detectors for the problem under consideration first all the considered detectors are shown to be function of the mis thus proving their cfarness property second coincidence or statistical equivalence among some of them in such a general signal model is proved third strong connections to well known simpler scenarios analyzed in adaptive detection literature are established finally simulation results are provided for a comparison of the proposed receivers

computation of the real controllability radius and minimum norm perturbations of higher order descriptor and time delay lti systems a linear time invariant lti system s controllability radius measures the norm of the smallest parametric perturbation such that the perturbed system is uncontrollable and is of practical importance in this note we study the real controllability radii of i higher order systems ii descriptor systems and iii time delay systems where the perturbations are restricted to the set of real values and the spectral norm is considered formulas for these radii are presented using a framework involving generalized real perturbation values which has certain computational advantages over other formulations found in the literature in particular the formulas are readily more computable especially for higher dimensional systems and a minimum norm perturbation can also easily be obtained numerical examples are presented

between controllable and uncontrollable the distance between a system a b and the set of uncontrollable systems is the minimum of the smallest singular value of i a b with respect to

multivariable multirate sampled data systems state space description transfer characteristics and nyquist criterion first a discrete time state space description is derived for multivariable multirate sampled data systems then the multirate impulse modulation mim and the symmetric coordinate multirate impulse modulation smim of a transfer matrix are defined and their relation to the discrete time state space description is clarified based on the above results four stability criteria are derived the contents of the paper will be useful to extend the modern results about continuous time and single rate sampled data control systems to the multirate sampled data case especially extensions of the ina method and of the robust stability conditions will be straightforward

optimal rank based tests for homogeneity of scatter we propose a class of locally and asymptotically optimal tests based on multivariate ranks and signs for the homogeneity of scatter matrices in m elliptical populations contrary to the existing parametric procedures these tests remain valid without any moment assumptions and thus are perfectly robust against heavy tailed distributions validity robustness nevertheless they reach semiparametric efficiency bounds at correctly specified elliptical densities and maintain high powers under all efficiency robustness in particular their normal score version outperforms traditional gaussian likelihood ratio tests and their pseudo gaussian robustifications under a very broad range of non gaussian densities including for instance all multivariate student and power exponential distributions

a formula for computation of the real stability radius this paper presents a readily computable formula for the real stability radius with respect to an arbitrary stability region in the complex plane

a technique for the identification of linear periodic state space models a non stochastic state space identification algorithm for linear time invariant systems is modified for use with periodic linear systems like its predecessor the modified algorithm forms block hankel matrices from input output data and uses the singular value decomposition of these hankel matrices to compute state vector sequences the state vector sequences are then used to compute system matrices associated with the periodic linear system by solving an overdetermined linear system the modifications to the original algorithm are as follows first the periodic system of period p is viewed as p separate time invariant period mapped systems this technique allows the structure of a periodic hankel matrix to be deduced which in turn allows a state vector sub sequence for the periodic system to be computed when a complete state vector sequence is computed it is used directly to construct the periodic state space models second similarities in the structure of each of the periodic hankel matrices ass

the structured distance to uncontrollability under multi perturbations an approach using multi valued linear operators abstract in this paper we develop a unifying approach for computing the distance to uncontrollability of linear control systems by using multi valued linear operators in representing and estimating the system s equations and matrices we are able to derive computable formulas of the distance from a controllable linear system to the nearest uncontrollable system under the assumption that the system s matrices are subjected to structured multi perturbations and measured by arbitrary operator norms in the case of spectral norms the obtained results unify and extend some previous works as well as a recent interesting result in m karrow d kressner on the structured distance to uncontrollability systems control lett 58 2009 128 132 some illustrating examples are given

survey invariant representations of discrete time periodic systems the correspondence between periodic and time invariant systems can take a variety of forms the purpose of this survey paper is to present a complete and clear picture on time invariant representations for discrete time periodic systems the core of the treatment evolves around the concept of periodic transfer function

computing the distance to an uncontrollable system algorithms for computing the distance from a particular controllable system to the closest uncontrollable system are presented the authors give an interpretation of the problem and propose solutions in the form of several linearly convergent iterative algorithms the algorithms are presented for the real and complex perturbation problems i e when the real system under investigation is subject to either real or complex perturbations

computing the radius of controllability for state space systems abstract in this paper we discuss the problem of computing the nearest uncontrollable system to a given control system represented by a matrix pair a b in order to do so we construct a sequence of structured matrices from given system matrices a and b controllability of the pair a b is equivalent to a condition on the null space dimension of an appropriate matrix in this sequence we show that the dimension of the reachability space is also related to the above condition further it is shown that the nearest structured low rank approximation slra of this structured matrix corresponds to a nearest uncontrollable system to the pair a b

distribution of likelihood ratio statistic for testing equality of covariance matrices of multivariate gaussian models summary in this paper the exact distribution of the likelihood ratio statistic for testing equality of covariance matrices of multivariate gaussian models has been derived percentage points of the test statistic have also been tabulated

state space forms for higher order discrete time models this paper presents a fundamental study of the connection between continuous and discrete time systems provided is a definition for discrete time models that is discrete time systems with a continuous time counterpart whose order can be higher than that of the continuous time system this definition is based on a comparison in a certain sense on the time responses of continuous and discrete time systems a theorem is presented for relating the higher order discrete time models to their continuous time counterparts which is an extension of a previous theorem for models with order equal to that of the continuous time system state space forms are derived for models obtained through the use of a certain class of hold elements and through the use of mapping models and these discrete time systems are shown to be valid according to the definition special cases are models obtained using first order and slewer hold devices whose convergence to a continuous time counterpart has not been shown mathematically before and mapping models corresponding to two step linear multi step methods which have not previously presented in the state space form the derived state space forms provide a convenient way to implement these models for purposes of analysis design and implementation of discrete time systems and finds applications in such areas as digital signal processing digital simulation and digital control

algebraic riccati equations and the distance to the nearest uncontrollable pair a connection is established between nearness to unstabilizability of a stabilizable pair a b of matrices and nearness to singularity of the symmetric positive definite solution to an associated algebraic riccati equation from this result computable upper and lower bounds are derived for the distance of a b to the nearest uncontrollable pair numerical tests confirm the validity of the method and potential applications are discussed

the observability radius of networks this paper studies the observability radius of network systems which measures the robustness of a network to perturbations of the edges we consider linear networks where the dynamics are described by a weighted adjacency matrix and dedicated sensors are positioned at a subset of nodes we allow for perturbations of certain edge weights with the objective of preventing observability of some modes of the network dynamics to comply with the network setting our work considers perturbations with a desired sparsity structure thus extending the classic literature on the observability radius of linear systems the paper proposes two sets of results first we propose an optimization framework to determine a perturbation with smallest frobenius norm that renders a desired mode unobservable from the existing sensor nodes second we study the expected observability radius of networks with given structure and random edge weights we provide fundamental robustness bounds dependent on the connectivity properties of the network and we analytically characterize optimal perturbations of line and star networks showing that line networks are inherently more robust than star networks

on the structured distance to uncontrollability this article is concerned with the structured distance to uncontrollability of a linear time invariant system and relates this concept to a variation of the value the developed framework is applied to derive computational expressions for the class of real perturbations as well as for hermitian symmetric and skew symmetric perturbations in a relatively simple manner examples demonstrate that the structured distance can differ from the standard unstructured distance to uncontrollability by an arbitrary amount it is also shown how systems of higher order can be addressed

on the distance to uncontrollability and the distance to instability and their relation to some condition numbers in control according to the methodology of 6 many measures of distance arising in problems in numerical linear algebra and control can be bounded by a factor times the reciprocal of an appropriate condition number where the distance is thought of as the distance between a given problem to the nearest ill posed problem in this paper four major problems in numerical linear algebra and control are further considered the computation of system hessenberg form the solution of the algebraic riccati equation the pole assignment problem and the matrix exponential the distances considered here are the distance to uncontrollability and the distance to instability

a new shift invariant representation for periodic linear systems abstract this paper presents a new family of shift invariant representations for periodic linear discrete time systems this representation has the advantages over the standard representation 5 of preserving the time step of the original system and of preserving causality under arbitrary feedback the disadvantage is that the realization is not minimal an application in attitude determination is presented

the structured distance to non surjectivity and its application to calculating the controllability radius of descriptor systems abstract the classical eckart young formula for square matrices identifies the distance to singularity of a matrix the main purpose of this paper is to get generalizations of this formula we characterize the distance to non surjectivity of a linear operator w l x y in finite dimensional normed spaces x y under the assumption that the operator w is surjective i e w x y and subjected to structured perturbations of the form w m n as an application of these results we shall derive formulas of the controllability radius for a descriptor controllable system e a b e x a x b u t 0 under the assumption that systems matrices e a b are subjected to structured perturbations and to multi perturbations

a maximal invariant framework for adaptive detection with structured and unstructured covariance matrices we introduce a framework for exploring array detection problems in a reduced dimensional space by exploiting the theory of invariance in hypothesis testing this involves calculating a low dimensional basis set of functions called the maximal invariant the statistics of which are often tractable to obtain thereby making analysis feasible and facilitating the search for tests with some optimality property using this approach we obtain a locally most powerful invariant test for the unstructured covariance case and show that all invariant tests can be expressed in terms of the previously published kelly s generalized likelihood ratio glrt and robey s adaptive matched filter amf test statistics applying this framework to structured covariance matrices corresponding to stochastic interferers in a known subspace for which the glrt is unavailable we obtain the maximal invariant and propose several new invariant detectors that are shown to perform as well or better than existing ad hoc detectors these invariant tests are unaffected by most nuisance parameters hence the variation in the level of performance is sharply reduced this framework facilitates the search for such tests even when the usual glrt is unavailable

real controllability stabilizability radius of lti systems in this note we present a simple and readily computable formula for the calculation of the real controllability stabilizability and observability detectability radii for linear time invariant systems

estimating the distance to uncontrollability a fast method and a slow one abstract for a linear control system a b the distance to uncontrollability is characterized by min ec n a i b where n a i b is the smallest singular value of the augmented matrix a i b two methods are developed to estimate the distance to uncontrollability giving both a lower bound and an upper bound one method is fast requiring only one spectral decomposition of a and computations of three smallest singular values and being used for well conditioned a the other is slow requiring computations of a large number of the smallest singular values but it produces bounds as tight as possible and also a region containing global minimizers newton s method can be used to compute the global minimizers if one really wants

periodic systems a sequence algebra approach to realisation parameterisation and topology a characterisation of discrete time linear periodic systems is presented based on a general framework using cyclic projection operators on sequences two known liftings of periodic systems to a time invariant one the monodromy and the cyclic representation are readily derived in this framework this approach also leads to the definition of the operational transfer function for such systems and more generally the operational transfer inclusion it is shown that the cyclic realisation allows the multiplexing of individual realisations of a periodic system these descriptions are useful in the realisation problem and the search for canonical forms parameterisations for the class of reachable systems are recalled and their geometric and topological features are illustrated

more on pseudospectra for polynomial eigenvalue problems and applications in control theory definitions and characterizations of pseudospectra are given for rectangular matrix poly nomials expressed in homogeneous form p da d d 1 a d 1 da 0 it is shown that problems with infinite pseudo eigenvalues are elegantly treated in this framework for such problems stereographic projection onto the riemann sphere is shown to provide a convenient way to visualize pseudospectra lower bounds for the distance to the nearest nonregular polynomial and the nearest uncontrollable dth order system with equality for standard state space systems are obtained in terms of pseudospectra showing that pseudospectra are a fundamental tool for reasoning about matrix polynomials in areas such as control theory how and why to incorporate linear structure into pseudospectra is also discussed by example

on the asymptotics of maronna s robust pca the eigenvalue decomposition evd parameters of the second order statistics are ubiquitous in statistical analysis and signal processing notably the evd of the m estimators of the scatter matrix is a popular choice to perform robust probabilistic pca or other dimension reduction related applications towards the goal of characterizing this process this paper derives new asymptotics for the evd parameters i e eigenvalues eigenvectors and principal subspace of m estimators in the context of complex elliptically symmetric distributions first their gaussian asymptotic distribution is obtained by extending standard results on the sample covariance matrix in the gaussian context second their convergence towards the evd parameters of a gaussian core wishart equivalent is derived this second result represents the main contribution in the sense that it quantifies when it is acceptable to directly rely on well established results on the evd of wishart distributed matrix for characterizing the evd of m estimators finally some examples intrinsic bias analysis rank estimation and low rank adaptive filtering illustrate where the obtained results can be leveraged

controllability radii of linear neutral systems under structured perturbations abstractin this paper we shall deal with the problem of calculation of the controllability radii of linear neutral systems of the form x t a0x t a1x t h a 1x t h bu t we will derive the definition of exact controllability radius approximate controllability radius and euclidean controllability radius for this system by using multi valued linear operators the computable formulas for these controllability radii are established in the case where the system s coefficient matrices are subjected to structured perturbations some examples are provided to illustrate the obtained results

the feedback search algorithm estimating the distance of semi explicit systems to r uncontrollability abstract we show that the distance of a special descriptor system termed semi explicit from the nearest r uncontrollable system can be estimated by the global minimum of a specific real valued univariate function with variable we study this function for both real and complex and then develop a novel global optimization algorithm called the feedback search algorithm the algorithm is initially developed for one variable and then it is extended to two variables we implement the algorithm and apply it to the above function when varies on the real axis and also on the complex plane in addition to this we apply the algorithm to another relevant function from the literature for the numerical applications several random examples as well as examples from the literature were used

decay rate estimations for linear quadratic optimal regulators let u t fx t be the optimal control of the open loop system x t ax t bu t in a linear quadratic optimization problem by using different complex variable arguments we give several lower and upper estimates of the exponential decay rate of the closed loop system x t a bf x t main attention is given to the case of a skew hermitian matrix a given an operator a for a class of cases we find a matrix b that provides an almost optimal decay rate we show how our results can be applied to the problem of optimizing the decay rate for a large finite collection of control systems a bj j 1 n and illustrate this on an example of a concrete mechanical system at the end of the article we pose several questions concerning the decay rates in the context of linear quadratic optimization and in a more general context of the pole placement problem

maximal invariants and performance of some invariant hypothesis tests for an adaptive detection problem maximal invariants for adaptive detection of a signal in unknown interference from multiple observations is derived given coherent samples from p sets of observations it is shown that a maximal invariant statistic for the detection problem is a 2p 1 dimensional vector comprising the eigenvalues of two hermitian positive definite matrices obtained from the data set two invariant detectors well known for p 1 are generalized for the case of multiple observations and closed form expressions for the probability of detection and probability of false alarm are derived along with the distributions of the signal to interference plus noise loss factors several novel invariant detectors are constructed from the maximal invariants and the receiver operating characteristics of the detectors compared

on the distance to singular descriptor dynamical systems with impulsive initial conditions in this paper we study the problem of computing the distance between a given singular descriptor system e a and a nearest descriptor system that has impulsive initial conditions the link between existence of impulsive initial conditions and zeros at infinity for the associated matrix pencil se a is well known much of the literature focusses on the case when only one of e and a is perturbed we give a closed form expression of the distance to a nearest descriptor system having impulsive solutions via rank 1 perturbations when both e and a are perturbed next for the case of perturbations without rank restrictions we propose and evaluate the bounds for the distance in the context of structured perturbations we formulate and obtain an explicit expression for the distance when e and a are hermitian and are perturbed by hermitian matrices for a suitable class of systems we also show that upper and lower bounds are within a factor of sqrt 2 we finally construct examples and compare the bounds obtained from our results with those from the literature as well as with computed values of the distance obtained via three numerical optimization techniques such as the structured low rank approximation the broyden fletcher goldfarb shanno algorithm and direct optimization tools like globalsearch

a variance equality test for two correlated complex gaussian variables with application to spectral power comparison complex valued gaussian distributions occur frequently in signal processing we derive a simple statistic independent of any complex valued correlation for testing for the equality of variances using a sample drawn from such a bivariate distribution the percentage points of the distribution are easy to compute the power of the test is determined and shown to be high even for very small sample sizes when the variables are highly correlated the new test is used to determine whether the spectral power associated with an ultralow frequency wave in the solar magnetic field is equal at two different observing spacecraft

ground effect analysis surface wave and layer potential representations abstract this paper presents two kinds of analytical exact expressions of the sound field reflected by a plane boundary as obtained by using either surface wave or layer potentials representations both solutions are first expressed as a sum of integrals which have a form suitable for numerical computation then these integrals are expanded into convergent series which provide analytical approximations of the solution numerical techniques are proposed for computing a the surface wave representation and b the approximation deduced from the layer potentials representation this approximation and the classical one sum of the surface wave and the first two terms of the asymptotic series are compared with the exact solution several examples show that the approximate formulas established here are valid on a range much wider than the validity domain of the classical ones

brief paper mpc of constrained discrete time linear periodic systems a framework for asynchronous control strong feasibility stability and optimality via periodic invariance state feedback model predictive control mpc of discrete time linear periodic systems with time dependent state and input dimensions is considered the states and inputs are subject to periodically time dependent hard convex polyhedral constraints first periodic controlled and positively invariant sets are characterized and a method to determine the maximum periodic controlled and positively invariant sets is derived the proposed periodic controlled invariant sets are then employed in the design of least restrictive strongly feasible reference tracking mpc problems the proposed periodic positively invariant sets are employed in combination with well known results on optimal unconstrained periodic linear quadratic regulation lqr to yield constrained periodic lqr control laws that are stabilizing and optimal one motivation for systems with time dependent dimensions is efficient control law synthesis for discrete time systems with asynchronous inputs for which a novel modeling framework resulting in low dimensional models is proposed the presented methods are applied to a multirate nano positioning system

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 a matrix whose dimension is not necessarily minimal among all possible period specific realizations the other is with a non constant and periodically time varying a matrix whose dimension turned out minimal among all possible period specific realizations