from time series to linear system part i finite dimensional linear time invariant systems abstract in the first part of this paper the definition of a dynamical system as simply consisting of a family of time series will be developed in this context the notions of linearity time invariance and finite dimensionality will be introduced it will be shown that a given family of time series may be represented by a system of ar equations riw t l rl 1w t l 1 r0w t 0 or equivalently by a finite dimensional linear time invariant system x t 1 ax t bu t y t cx t du t w u y if and only if this family is linear shift invariant and complete or as is equivalent closed in the topology of pointwise convergence this yields a very high level and elegant set of axioms which characterize these familiar objects it is emphasized however that no a priori choice is made as to which components of w are inputs and which are outputs such a separation always exists in any specific linear time invariant model starting from these definitions the structural indices of such systems are introduced and it is shown how an ar representation of a system having a given behaviour can be constructed these results will be used in a modelling context in part ii of the paper

parallel concatenated convolutional codes from linear systems theory viewpoint abstract the aim of this work is to characterize two models of concatenated convolutional codes based on the theory of linear systems the problem we consider can be viewed as the study of composite linear system from the classical control theory or as the interconnection from the behavioral system viewpoint in this paper we provide an input state output representation of both models and introduce some conditions for such representations to be both controllable and observable we also introduce a lower bound on their free distances and the column distances

rate dependent analysis of the asymptotic behavior of channel polarization we consider the asymptotic behavior of the polarization process in the large block length regime when transmission takes place over a binary input memoryless symmetric channel w in particular we study the asymptotics of the cumulative distribution p zn z where zn is the bhattacharyya process associated with w and its dependence on the rate of transmission on the basis of this result we characterize the asymptotic behavior as well as its dependence on the rate of the block error probability of polar codes using the successive cancellation decoder this refines the original asymptotic bounds by arikan and telatar our results apply to general polar codes based on l l kernel matrices we also provide asymptotic lower bounds on the block error probability of polar codes using the maximum a posteriori map decoder the map lower bound and the successive cancellation upper bound coincide when l 2 but there is a gap for l 2

realization by inspection we investigate which first order representations can be obtained from high order representations of linear systems by inspection that is just by rearrangement of the data under quite weak conditions it is possible to obtain minimal realizations in the so called pencil form under stronger conditions one can obtain minimal realizations in standard state space form by inspection the development is based on a reformulation of the realization problem as a problem of finding a complete set of basis vectors for the nullspace of a given constant matrix since no numerical computation is needed the realization method in particular is suitable for situations in which some of the coefficients are symbolic rather than numerical

a controllability test for general first order representations we derive a new controllability rank test for general first order representations the criterion generalizes the well known controllability rank test for linear input state systems as well as a controllability rank test by mertzios et al for descriptor systems

minimality and observability of group systems abstract group systems are a generalization of willems type linear systems that are useful in error control coding it is shown that the basic ideas of willems s treatment of linear systems are easily generalized to linear systems over arbitrary rings and to group systems the interplay between systems behaviors and trellises evolution laws is discussed with respect to completeness minimality controllability and observability it is pointed out that for trellises of group systems and willems type linear systems minimality is essentially the same as observability the development is universal algebraic in nature and holds unconditionally for linear systems over the real numbers

linear systems and taylor complexes a taylor complex is a graded collection of polynomials satisfying a few simple axioms we show that there is a natural one to one correspondence between the solution sets of linear constant coefficient differential equations and taylor complexes using this correspondence we then give a characterization of the solution sets of linear constant coefficient differential equations which follows closely willems characterization of the solution sets of linear constant coefficient difference equations we also characterize linear differential operators the presentation is given for the case of several variables

operations on orbits of unimodular vectors abstract taking the m power of an entry is a well defined operation on the unimodular vectors in an modulo addition operations if n is at least 3 for an arbitrary commutative ring a and any integer m

a general realization theory for higher order linear differential equations in this note we show that the geometric quotient under a natural group action of the generalized state space systems recently considered by schumacher kuijper and geerts is algebraically isomorphic to the space of a homogeneous autoregressive system this result essentially follows from work of stromme published earlier in the algebraic geometry literature in particular these generalized state space systems represent a realization of the space of homogeneous autoregressive systems

on the control of finite support behaviors abstract in the paper the main properties of finite support behaviors are introduced and characterized in terms of the polynomial matrices involved in their representation necessary and sufficient conditions for the existence of an input output possibly non anticipating description of a finite support behavior are given the control problem is also addressed and properly restated in the behavior theory context as an interconnection problem conditions for the existence of controllers that make the resulting connected system observable as well as a complete description of the class of characteristic polynomials that the resulting controlled system possibly exhibits are given finally the problem of controlling a dynamical system in input output from by means of a controller with the same structure is investigated and solved

notes on the definition of behavioural controllability we take another look at the behavioural definition of controllability for discrete 1d systems and its extension to multidimensional nd systems defined on z n or n n we suggest that the current definition for nd systems is inappropriate for systems defined on the domain n n and that care has to be taken even with the 1d definition we propose a new definition which applies to all standard classes of discrete nd systems

the dynamics of group codes dual abelian group codes and systems fundamental results concerning the dynamics of abelian group codes behaviors and their duals are developed duals of sequence spaces over locally compact abelian lca groups may be defined via pontryagin duality dual group codes are orthogonal subgroups of dual sequence spaces the dual of a complete code or system is finite and the dual of a laurent code or system is anti laurent if c and c sup spl perp are dual codes then the state spaces of c act as the character groups of the state spaces of c sup spl perp the controllability properties of c are the observability properties of c sup spl perp in particular c is strongly controllable if and only if c sup spl perp is strongly observable and the controller memory of c is the observer memory of c sup spl perp the controller granules of c act as the character groups of the observer granules of c sup spl perp examples of minimal observer form encoder and syndrome former constructions are given finally every observer granule of c is an end around controller granule of c

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

weight preserving isomorphisms between spaces of continuous functions the scalar case abstract let f be a finite field or discrete and let a and b be vector spaces of f valued continuous functions defined on locally compact spaces x and y respectively we look at the representation of linear bijections h a b by continuous functions h y x as weighted composition operators in order to do it we extend the notion of hamming metric to infinite spaces our main result establishes that under some mild conditions every hamming isometry can be represented as a weighted composition operator connections to coding theory are also highlighted

a class of one dimensional mds convolutional codes a class of one dimensional convolutional codes will be presented they are all mds codes i e have the largest distance among all one dimensional codes of the same length n and overall constraint length furthermore their extended row distances are computed and they increase with slope at least n in certain cases of the algebraic parameters we will also derive parity check matrices of vandermonde type for these codes finally cyclicity in the convolutional sense of 8 will be discussed for our class of codes it will turn out that they are cyclic if and only if the field element used in the generator matrix has order n this can be regarded as a generalization of the block code case

a behavioral approach to estimation and dead beat observer design with applications to state space models the observer design problem is investigated in the context of linear left shift invariant discrete behaviors whose trajectories have supports on z necessary and sufficient conditions for the existence of a dead beat observer of some relevant variables from some measured ones in the presence of some unmeasured and irrelevant variables are introduced and a complete parametrization of all dead beat observers is given equivalent conditions for the existence of causal dead beat observers are then derived finally several classical problems addressed for state space models like state estimation the design of unknown input observers or the design of fault detectors and identifiers possibly in the presence of disturbances are cast in this general framework and the aforementioned equivalent conditions and parametrizations are specialized to these cases

convolutional codes over groups the basic algebraic structure theory of convolutional codes and their trellises is developed simultaneously for codes over groups rings and fields the first part which covers fundamental notions such as minimality and observability is semi tutorial in that most definitions are already standard within the modern behavioral theory as are some of the formally stated results however some of the pivotal results emphasizing the role of observability as the basic well behavedness condition for codes are new and several previous results are given simplified proofs the usefulness of the behavioral approach even for convolutional codes over fields is demonstrated by a new minimality test for encoders as well as by the straightforward derivation of some known minimality criteria for generator matrices from the basic minimality criteria for group trellises the second part of the paper deals with issues that are specific to codes over rings and groups the main result is a concise characterization the first such of those groups that can appear as the branch group of any group trellis it is further shown how such groups are presented by shift registers a new large class of noncommutative convolutional codes is also given

the effective calculation of all assignable polynomials to a single input system over a bezout domain the set of all feedback assignable polynomials to a given single input system over a bezout domain is obtained the general case is reduced to the case of weakly reachable systems for these systems a canonical form for the algebraic equivalence is obtained algorithms and examples over a euclidean domain and in particular over the ring z of rational integers are also given the problem of deciding and feedback assignment of a given polynomial to a given non reachable n dimensional linear system is solved effectively up to a cost of o n4 arithmetic operations

strongly mds convolutional codes maximum distance separable mds convolutional codes have the property that their free distance is maximal among all codes of the same rate and the same degree in this paper a class of mds convolutional codes is introduced whose column distances reach the generalized singleton bound at the earliest possible instant such codes are called strongly mds convolutional codes they also have a maximum or near maximum distance profile the extended row distances of these codes will also be discussed briefly

the dynamics of group codes state spaces trellis diagrams and canonical encoders a group code c over a group g is a set of sequences of group elements that itself forms a group under a component wise group operation a group code has a well defined state space sigma sub k at each time k each code sequence passes through a well defined state sequence the set of all state sequences is also a group code the state code of c the state code defines an essentially unique minimal realization of c the trellis diagram of c is defined by the state code of c and by labels associated with each state transition the set of all label sequences forms a group code the label code of c which is isomorphic to the state code of c if c is complete and strongly controllable then a minimal encoder in controller canonical feedbackfree form may be constructed from certain sets of shortest possible code sequences called granules the size of the state space sigma sub k is equal to the size of the state space of this canonical encoder which is given by a decomposition of the input groups of c at each time k if c is time invariant and nu controllable then mod sigma sub k mod pi sub 1

dead beat observer synthesis in the paper the observer design problem is investigated in the context of linear left shift invariant discrete behaviors whose trajectories have support on the positive axis observability and reconstructibility properties of certain manifest variables from certain others in the presence of latent variables are defined and fully characterized necessary and sufficient conditions for the existence of either a dead beat or an exact observer are introduced and a complete parametrization of all dead beat observers is given

on behaviors and convolutional codes it is well known that a convolutional code is essentially a linear system defined over a finite field in this paper we elaborate on this connection we define a convolutional code as the dual of a complete linear behavior in the sense of willems 1979 using ideas from systems theory we describe a set of generalized first order descriptions for convolutional codes as an application of these ideas we present a new algebraic construction for convolutional codes

on the algebraic parameters of convolutional codes with cyclic structure in this paper convolutional codes with cyclic structure will be investigated these codes can be understood as left principal ideals in a suitable skew polynomial ring it has been shown in 4 that only certain combinations of the algebraic parameters field size length dimension and forney indices can occur for such cyclic codes we will investigate whether all these combinations can indeed be realized by a suitable cyclic code and if so how to construct such a code a complete characterization and construction will be given for minimal cyclic codes it is derived from a detailed investigation of the units in the skew polynomial ring

algebraic aspects of two dimensional convolutional codes two dimensional 2d codes are introduced as linear shift invariant spaces of admissible signals on the discrete plane convolutional and in particular basic codes are characterized both in terms of their internal properties and by means of their input output representations the algebraic structure of the class of all encoders that correspond to a given convolutional code is investigated and the possibility of obtaining 2d decoders free from catastrophic errors as veil as efficient syndrome decoders is considered some aspects of the state space implementation of 2d encoders and decoders via finite memory 2d system are discussed

on the weight distribution of convolutional codes detailed information about the weight distribution of a convolutional code is given by the adjacency matrix of the state diagram associated with a minimal realization of the code we will show that this matrix is an invariant of the code moreover it will be proven that codes with the same adjacency matrix have the same dimension and the same forney indices and finally that for one dimensional binary convolutional codes the adjacency matrix determines the code uniquely up to monomial equivalence

linear system modelization of concatenated block and convolutional codes in this paper we characterize four models of concatenation of a block code and a convolutional code from a linear systems theory viewpoint we provide the input state output representation of these models and we give conditions in order to get a non catastrophic concatenated convolutional code with minimal representation lower bounds on the free distances of the concatenated codes are also developed

minimal grobner bases and the predictable leading monomial property abstract we focus on grobner bases for modules of univariate polynomial vectors over a ring we identify a useful property the predictable leading monomial plm property that is shared by minimal grobner bases of modules in f x q no matter what positional term order is used the plm property is useful in a range of applications and can be seen as a strengthening of the wellknown predictable degree property row reducedness a terminology introduced by forney in the 70 s because of the presence of zero divisors minimal grobner bases over a finite ring of the type z p r where p is a prime integer and r is an integer 1 do not necessarily have the plm property in this paper we show how to derive from an ordered minimal grobner basis a so called minimal grobner p basis that does have a plm property we demonstrate that minimal grobner p bases lend themselves particularly well to derive minimal realization parametrizations over z p r applications are in coding and sequences over z p r

concatenated linear systems over rings and their application to construction of concatenated families of convolutional codes abstract we present a generalization of the theory of concatenated linear systems to commutative rings with identity moreover we highlight sufficient conditions to obtain reachable and observable concatenated linear systems this approach provides us with minimal input state output representations by means of which we can construct observable concatenated families of convolutional codes with different parameters over some particular rings this work focuses on the characterization of models of serial systematic serial and parallel concatenation

subspaces of gf q ω and convolutional codes the present paper is a self contained treatment of subspaces of the space gf q of all semi infinite strings over gf q some necessary and sufficient conditions which characterize those subspaces of gf q are derived which are convolutional codes and the classes of subspaces defined by one or more of them are investigated moreover structural parameters of convolutional codes such as block length rate delay and constraint length are considered as parameters of subspaces rather than parameters of an encoding device as a conclusion it is obtained that for error control purposes none of the investigated superclasses of the class of convolutional codes is better suited than the class of convolutional codes itself

some convolutional codes whose free distances are maximal the free distance of a convolutional code of rate 1 n is bounded by n times the constant length of its encoder two classes of convolutional codes whose free distances meet this bound are studied a technique of constructing convolutional codes that meet this bound for all sufficiently large n is given

a first approximation of concatenated convolutional codes from linear systems theory viewpoint this article focuses on the characterization of two models of concatenated convolutional codes from the perspective of linear systems theory we present an input state output representation of these models and study the conditions for obtaining a minimal input state output representation and non catastrophic concatenated convolutional code we also establish conditions so that the concatenated codes are observable and give a lower bound for their free distances