applications of vector bundles to factorization of rational matrices abstract we show that wiener hopf s and forney s factorization theorems are immediate consequences of grothendieck s theorem on vector bundles

generalised autoregressive models and their trajectories the set of classical ar models is not compact so that a sequence of ar models may fail to converge to anything reasonable a compactification leads to ar models with arbitrary singularity at infinity these limit ar models are represented by pairs of polynomial and proper rational matrices satisfying certain compatibility condition it is shown that they produce trajectories in a very natural way then the classical equivalence theorem is extended to them stating that two ar models having the same behaviour are equivalent

linear systems with locally integrable trajectories abstract the locally integrable function space l 1 loc is made a module over the ring of proper rational functions using this module structure the notion of relative dimension is defined naturally for every linear subspace in l 1 loc q it is shown that the property of having finite relative dimension together with the property of having sufficiently many smooth trajectories and the evident property of differentiation invariance characterize the weak solutions sets of linear constant coefficient differential systems among all linear subspaces

on quantum detection and the square root measurement we consider the problem of constructing measurements optimized to distinguish between a collection of possibly nonorthogonal quantum states we consider a collection of pure states and seek a positive operator valued measure povm consisting of rank one operators with measurement vectors closest in squared norm to the given states we compare our results to previous measurements suggested by peres and wootters 1991 and hausladen et al 1996 where we refer to the latter as the square root measurement srm we obtain a new characterization of the srm and prove that it is optimal in a least squares sense in addition we show that for a geometrically uniform state set the srm minimizes the probability of a detection error this generalizes a similar result of ban et al see int j theor phys vol 36 p 1269 88 1997

nonuniform sampling and reconstruction in shift invariant spaces this article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift invariant spaces it is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift invariant subspaces by bringing together wavelet theory frame theory reproducing kernel hilbert spaces approximation theory amalgam spaces and sampling inspired by applications taken from communication astronomy and medicine the following aspects will be emphasized a the sampling problem is well defined within the setting of shift invariant spaces b the general theory works in arbitrary dimension and for a broad class of generators c the reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms these algorithms converge geometrically and are robust in the presence of noise d to model the natural decay conditions of real signals and images the sampling theory is developed in weighted l p spaces

relative completeness and specifiedness properties of continuous linear dynamical systems the properties of relative completeness and specifiedness are introduced for continuous linear dynamical systems it is shown then that having these two properties and the differentiation invariance property is necessary and sufficient for a continuous linear dynamical system to be represented by means of a linear constant coefficient differential equation

cutoff rate for the m ary psk modulation channel with optimal quantum detection the cutoff rate r sub 0 of an optical channel transmitting m ary phase shift keyed coherent states is calculated for a receiver using the optimum quantum detector this is shown to be nearly equivalent when m 1 to a system that measures the phase of the received signal a closed form expression yielding a good approximation to r sub 0 is derived

linear subspaces of smooth vector fields as a kernel of some linear first order partial differential equation abstract there are given sufficient conditions for linear subspaces s v c g r n r n of smooth vector fields to be written as a kernel of some linear first order partial differential equation pde in particular it implies that an overdetermined system of linear first order pde has a nontrivial solution and in addition the control system associated with s v is not controllable

transformations of linear systems under external equivalence abstract we consider systems of linear differential and algebraic equations in which some of the variables are distinguished as external variables two systems are called equivalent if the set of solutions for the external variables is the same for both systems we give an operational form for this definition of equivalence i e we describe a set of system transformations having the property that two systems are equivalent if and only if they can be taken into each other by transformations from that set next an algorithm is described to transform a given system in general form to a system in minimal state space form this algorithm differs from existing methods in that it first takes the equations to first order form so that each subsequent step can be formulated and interpreted in state space terms we also compute the structure indices in terms of a state space description in nonminimal form and use this to prove the minimality of the end result of the algorithm finally an application is shown to the problem of ill posedness of feedback connections

frame theoretic analysis of oversampled filter banks we provide a frame theoretic analysis of oversampled finite impulse response fir and infinite impulse response fir uniform filter banks fbs our analysis is based on a new relationship between the fbs polyphase matrices and the frame operator corresponding to an fb for a given oversampled analysis fb we present a parameterization of all synthesis fbs providing perfect reconstruction we find necessary and sufficient conditions for an oversampled fb to provide a frame expansion a new frame theoretic procedure for the design of paraunitary fbs from given nonparaunitary fbs is formulated we show that the frame bounds of an fb can be obtained by an eigen analysis of the polyphase matrices the relevance of the frame bounds as a characterization of important numerical properties of an fb is assessed by means of a stochastic sensitivity analysis we consider special cases in which the calculation of the frame bounds and synthesis filters is simplified finally simulation results are presented

geometrically uniform frames we introduce a new class of finite dimensional frames with strong symmetry properties called geometrically uniform gu frames that are defined over a finite abelian group of unitary matrices and are generated by a single generating vector the notion of gu frames is then extended to compound gu cgu frames which are generated by a finite abelian group of unitary matrices using multiple generating vectors the dual frame vectors and canonical tight frame vectors associated with gu frames are shown to be gu and therefore also generated by a single generating vector which can be computed very efficiently using a fourier transform ft defined over the generating group of the frame similarly the dual frame vectors and canonical tight frame vectors associated with cgu frames are shown to be cgu the impact of removing single or multiple elements from a gu frame is considered a systematic method for constructing optimal gu frames from a given set of frame vectors that are not gu is also developed finally the euclidean distance properties of gu frames are discussed and conditions are derived on the abelian group of unitary matrices to yield gu frames with strictly positive distance spectrum irrespective of the generating vector

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

representation theory for high rate multiple antenna code design multiple antennas can greatly increase the data rate and reliability of a wireless communication link in a fading environment but the practical success of using multiple antennas depends crucially on our ability to design high rate space time constellations with low encoding and decoding complexity it has been shown that full transmitter diversity where the constellation is a set of unitary matrices whose differences have nonzero determinant is a desirable property for good performance we use the powerful theory of fixed point free groups and their representations to design high rate constellations with full diversity furthermore we thereby classify all full diversity constellations that form a group for all rates and numbers of transmitter antennas the group structure makes the constellations especially suitable for differential modulation and low complexity decoding algorithms the classification also reveals that the number of different group structures with full diversity is very limited when the number of transmitter antennas is large and odd we therefore also consider extensions of the constellation designs to nongroups we conclude by showing that many of our designed constellations perform excellently on both simulated and real wireless channels

oversampled filter banks perfect reconstruction oversampled filter banks are equivalent to a particular class of frames in l sup 2 z these frames are the subject of this paper first the necessary and sufficient conditions of a filter bank for implementing a frame or a tight frame expansion are established as well as a necessary and sufficient condition for perfect reconstruction using fir filters after an fir analysis complete parameterizations of oversampled filter banks satisfying these conditions are given further we study the condition under which the frame dual to the frame associated with an fir filter bank is also fir and give a parameterization of a class of filter banks satisfying this property then we focus on non subsampled filter banks non subsampled filter banks implement transforms similar to continuous time transforms and allow for very flexible design we investigate the relations of these filter banks to continuous time filtering and illustrate the design flexibility by giving a procedure for designing maximally flat two channel filter banks that yield highly regular wavelets with a given number of vanishing moments

on multidimensional coded modulations having uniform error property for generalized decoding and flat fading channels we consider the problem of uniform error property uep for a coded modulation with constant energy multidimensional symbols transmitted over the additive white gaussian noise awgn or fading channels and received by a broad class of decoders this class includes coherent partially coherent double differential and noncoherent decoders decoders designed for fading channels decoders using one or multiple symbol observations and many more these decoders are described as special cases of a general decoder model this decoder operates by maximizing an arbitrary likelihood function that its arguments are front end correlator matched filter outputs a group code structure that guarantees uep is developed by using the theory of geometrically uniform codes and applying it to the general decoder these codes are defined over groups commonly nonbinary with isometric mapping to channel symbols we show the code construction for the specific case of lth dimensional m ary phase shift keying mpsk an additional interesting property of these general uniform error codes is related to the case of noncoherent decoding we show that when using codes of this family if a code is noncoherently catastrophic then it is also rotationally invariant then the use of preceding of the input such that the code becomes rotationally transparent will also make it noncatastrophic

behaviors defined by rational functions abstract in this article behaviors defined by differential equations involving matrices of rational functions are introduced conditions in terms of controllability observability and stabilizability for the existence of rational representations that are prime over various subrings of the field of rational functions are derived elimination of latent variables observable image like representations of controllable systems and the structure of the rational annihilators of a behavior are discussed

least squares inner product shaping we develop methods that construct an optimal set of vectors with a specified inner product structure from a given set of vectors in a complex hilbert space the optimal vectors are chosen to minimize the sum of the squared norms of the errors between the constructed vectors and the given vectors four special cases are considered in the first the constructed vectors are orthonormal in the second they are orthogonal in the third the gram matrix of inner products of the constructed vectors is a circulant matrix as we show the vectors form a cyclic set in the fourth the gram matrix has the property that the rows are all permutations of each other the constructed vectors are shown to be geometrically uniform

smooth impulsive linear systems axiomatic description abstract paper concerns with linear systems having arbitrary singularity at infinity these systems deserve to study since they are limits of familiar classical linear systems an axiomatic description of such systems is obtained

a generalization of the cellina colombo theorem for a class of non convex variational problems abstract we state a condition under which the integral functional i x t 0 l t x t x t dt attains a minimum under the assumption that x l t x x is concave

smooth impulsive linear systems controllability in this article a fairly comprehensive study of the concept of controllability for smooth impulsive linear systems is undertaken

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

rational differential operators and their kernels abstract a linear differential integral operator is associated to a rational matrix in a natural way and the behavior is defined to be the kernel of this operator conditions under which two rational matrices have the same behavior are derived

continuous and discrete wavelet transforms this paper is an expository survey of results on integral representations and discrete sum expansions of functions in l 2 bf r in terms of coherent states two types of coherent states are considered weyl heisenberg coherent states which arise from translations and modulations of a single function and affine coherent states called wavelets which arise as translations and dilations of a single function in each case it is shown how to represent any function in l 2 bf r as a sum or integral of these states most of the paper is a survey of literature most notably the work of i daubechies a grossmann and j morlet a few results of the authors are included

characterization of linear differential systems in several variables abstract we show that a set of smooth trajectories is the solution set of a linear constant coefficient partial differential equation if and only if it is linear shift invariant and complete by completeness we mean exactly what willems called jet completeness in his automatica paper in 1986

brief paper axiomatic characterization of linear differential systems and operators in his ieee trans aut contr paper in 1991 willems posed the following question given a set of smooth trajectories when does there exist a linear constant coefficient differential operator whose kernel is precisely the given set we show that the properties that are necessary and sufficient for a set of smooth trajectories to admit a kernel representation are linearity time invariance jet closedness and jet determinedness it is interesting to note that the properties of jet closedness and jet determinedness were introduced by willems himself in his automatica paper in 1986

geometrically uniform codes a signal space code c is defined as geometrically uniform if for any two code sequences in c there exists an isometry that maps one sequence into the other while leaving the code c invariant geometrical uniformity a strong kind of symmetry implies such properties as a the distance profiles from code sequences in c to all other code sequences are all the same and b all voronoi regions of code sequences in c have the same shape it is stronger than ungerboeck zehavi wolf symmetry or calderbank sloane regularity nonetheless most known good classes of signal space codes are shown to be generalized coset codes and therefore geometrically uniform including a lattice type trellis codes based on lattice partitions lambda lambda such that z sup n lambda lambda 4z sup n is a lattice partition chain and b phase shift keying psk type trellis codes based on up to four way partitions of a 2 sup n psk signal set

characterising solution sets of lti differential equations we prove that a set of smooth trajectories is ltid i e the solution set of a constant coefficient linear differential equation if and only if it is the direct sum of a linear time invariant closed controllable part and a linear time invariant finite dimensional part this characterisation does not directly involve derivatives in its formulation it solves a problem opened by willems 1991 we also characterise morphisms between ltid sets as linear time invariant maps which do not increase support

an optimal whitening approach to linear multiuser detection we propose a new linear multiuser receiver for synchronous code division multiple access cdma systems referred to as the orthogonal multiuser omu receiver unlike the linear minimum mean squared error mmse receiver the omu receiver depends only on the signature vectors and does not require knowledge of the received amplitudes or the channel signal to noise ratio snr several equivalent representations of the receiver are developed with different implications in terms of implementation in the first the receiver consists of a decorrelator demodulator followed by an optimal whitening transformation on a space formed by the signatures in the second the receiver consists of a bank of correlators with correlating vectors that are projections of a set of orthogonal vectors and are closest in a least squares sense to the decorrelator vectors and also closest in a least squares sense to the signature vectors in the third the receiver consists of a single user matched filter mf followed by an optimal whitening transformation on a space formed by the signatures we derive exact and approximate expressions for the probability of bit error as well as the asymptotic signal to interference noise ratio sinr in the large system limit the analysis suggests that over a wide range of channel parameters the omu receiver can outperform both the decorrelator and the single user mf and perform similarly to the linear mmse receiver despite not knowing the channel parameters

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

when are linear differentiation invariant spaces differential it is shown that a linear differentiation invariant subspace of a c trajectory space is differential i e can be represented as the kernel of a linear constant coefficient differential operator if and only if its mcmillan degree is finite

optimal detection of symmetric mixed quantum states we develop a sufficient condition for the least squares measurement lsm or the square root measurement to minimize the probability of a detection error when distinguishing between a collection of mixed quantum states using this condition we derive the optimal measurement for state sets with a broad class of symmetries we first consider geometrically uniform gu state sets with a possibly non abelian generating group and show that if the generator satisfies a weighted norm constraint then the lsm is optimal in particular for pure state gu ensembles the lsm is shown to be optimal for arbitrary gu state sets we show that the optimal measurement operators are gu with generator that can be computed very efficiently in polynomial time within any desired accuracy we then consider compound gu cgu state sets which consist of subsets that are gu when the generators satisfy a certain constraint the lsm is again optimal for arbitrary cgu state sets the optimal measurement operators are shown to be cgu with generators that can be computed efficiently in polynomial time

behaviors and symbols of rational matrices abstract recently we have defined rational differential equations and derived conditions when two such equations have the same solution set in this article we study the same question but present a different approach based on the new notion of symbols also we try to describe the relationship with the work behaviors defined by rational functions of willems and yamamoto

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed