right limits and reflectionless measures for cmv matrices we study cmv matrices by focusing on their right limit sets we prove a cmv version of a recent result of remling dealing with the implications of the existence of absolutely continuous spectrum and we study some of its consequences we further demonstrate the usefulness of right limits in the study of weak asymptotic convergence of spectral measures and ratio asymptotics for orthogonal polynomials by extending and refining earlier results of khrushchev to demonstrate the analogy with the jacobi case we recover corresponding previous results of simon using the same approach

on the capacity of free space optical intensity channels upper and lower bounds are derived on the capacity of the free space optical intensity channel this channel has a nonnegative input representing the transmitted optical intensity which is corrupted by additive white gaussian noise to preserve the battery and for safety reasons the input is constrained in both its average and its peak power for a fixed ratio of the allowed average power to the allowed peak power the difference between the upper and the lower bound tends to zero as the average power tends to infinity and their ratio tends to one as the average power tends to zero when only an average power constraint is imposed on the input the difference between the bounds tends to zero as the allowed average power tends to infinity and their ratio tends to a constant as the allowed average power tends to zero

a trace formula for canonical differential expressions we prove a trace formula for pairs of self adjoint operators associated to canonical differential expressions an important role is played by the associated weyl function

dual capacity upper bounds for noisy runlength constrained channels binary input memoryless channels with a run length constrained input are considered upper bounds to the capacity of such noisy run length constrained channels are derived using the dual capacity method with markov test distributions satisfying the karush kuhn tucker conditions for the capacity achieving output distribution simplified algebraic characterizations of the bounds are presented for the binary erasure channel and the binary symmetric channel these upper bounds are very close to achievable rates and improve upon previously known feedback based bounds for a large range of channel parameters for the binary input additive white gaussian noise channel the upper bound is simplified to a small scale numerical optimization problem these results provide some of the simplest upper bounds for an open capacity problem that has theoretical and practical relevance

reflectionless sturm liouville equations we consider compactly supported perturbations of periodic sturm liouville equations in this context one can use the floquet solutions of the periodic background to define scattering coefficients we prove that if the reflection coefficient is identically zero then the operators corresponding to the periodic and perturbed equations respectively are unitarily equivalent in some appendices we also provide the proofs of several basic estimates e g bounds and asymptotics for the relevant m functions

on isospectral sets of jacobi operators we consider the inverse spectral problem for a class of reflectionless bounded jacobi operators with empty singularly continuous spectra our spectral hypotheses admit countably many accumulation points in the set of eigenvalues as well as in the set of boundary points of intervals of absolutely continuous spectrum the corresponding isospectral set of jacobi operators is explicitly determined in terms of dirichlet type data

scattering matrices and weyl functions for a scattering system a a 0 consisting of self adjoint extensions a and a 0 of a symmetric operator a with finite deficiency indices the scattering matrix s and a spectral shift function are calculated in terms of the weyl function associated with a boundary triplet for a and a simple proof of the krein birman formula is given the results are applied to singular sturm liouville operators with scalar and matrix potentials to dirac operators and to schrodinger operators with point interactions

discrete skew selfadjoint canonical systems and the isotropic heisenberg magnet model abstract a discrete analog of a skew self adjoint canonical zakharov shabat or akns system with a pseudo exponential potential is introduced for the corresponding weyl function the direct and inverse problem are solved explicitly in terms of three parameter matrices as an application explicit solutions are obtained for the discrete integrable nonlinear equation corresponding to the isotropic heisenberg magnet model state space techniques from mathematical system theory play an important role in the proofs

on channel estimation and detection for multicarrier signals in fast and selective rayleigh fading channels time domain channel estimation and detection techniques are presented for multicarrier signals in a fast and frequency selective rayleigh fading channel as a consequence of the time varying channel the orthogonality between subcarriers is destroyed in conventional frequency domain approaches resulting in interchannel interference which increases an irreducible error floor in proportion to the normalized doppler frequency an important feature of the proposed technique is the ability to exploit the time selective channel as a provider of time diversity this enables us to achieve performance superior to any other structure without increasing bandwidth or incorporating redundancy in order to reduce the complexity of the estimator we apply the theory of optimal low rank approximation to a minimum mean squared error channel estimator and present a theoretical calculation of mean squared error and simulations to confirm that the estimator is robust to changes in channel characteristics

em based iterative receiver design with carrier frequency offset estimation for mimo ofdm systems in this letter we study the design of expectation maximization em based iterative receivers for multiple input multiple output mimo orthogonal frequency division multiplexing systems with the presence of carrier frequency offset cfo motivated by the spirit of maximum likelihood estimation in the em algorithm we first present a pilot aided cfo estimation scheme that allows fast fourier transform based fast implementation then this cfo estimation is incorporated into the initialization step of the iterative receiver experimental results show the effectiveness of our receiver design in combating cfo

design and analysis of mmse pilot aided cyclic prefixed block transmissions for doubly selective channels this paper considers affine cyclic prefixed block based pilot aided transmission pat over the single antenna doubly selective channel where the channel is assumed to obey a complex exponential basis expansion model first a tight lower bound on the mean squared error mse of pilot aided channel estimates is derived along with necessary and sufficient conditions on the pilot data pattern that achieves this bound from these conditions novel minimum mse mmse pat schemes are proposed and upper lower bounds on their ergodic achievable rates are derived a pilot data power allocation technique is also developed a high snr asymptotic analysis of the ergodic achievable rate of affine mmse pat is then performed which suggests that the channel s spreading parameters should be taken into account when choosing among affine mmse pat schemes specifically we establish that multicarrier mmse pat achieves higher rates than single carrier mmse pat when the channel s delay spread dominates its doppler spread and vice versa

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

inverse scattering for impedance schrodinger operators i step like impedance lattice abstract we study the scattering problem for schrodinger operators in impedance form with step like impedance functions in this paper the case where the jumps are located at the vertices of periodic lattice is considered the set of reflection coefficients is completely described and the scattering map is shown to be homeomorphic also an algorithm reconstructing the impedance function from the reflection coefficient is developed

semiblind iterative data detection for ofdm systems with cfo and doubly selective channels data detection for ofdm systems over unknown doubly selective channels dscs and carrier frequency offset cfo is investigated a semiblind iterative detection algorithm is developed based on the expectation maximization em algorithm it iteratively estimates the cfo channel and recovers the unknown data using only limited number of pilot subcarriers in one ofdm symbol in addition efficient initial cfo and channel estimates are also derived based on approximated maximum likelihood ml and minimum mean square error mmse criteria respectively simulation results show that the proposed data detection algorithm converges in a few iterations and moreover its performance is close to the ideal case with perfect cfo and channel state information

simulation based computation of information rates for channels with memory the information rate of finite state source channel models can be accurately estimated by sampling both a long channel input sequence and the corresponding channel output sequence followed by a forward sum product recursion on the joint source channel trellis this method is extended to compute upper and lower bounds on the information rate of very general channels with memory by means of finite state approximations further upper and lower bounds can be computed by reduced state methods

inverse scattering on the line for the matrix sturm liouville equation abstract the inverse scattering problem is studied for the matrix sturm liouville equation on the line necessary and sufficient conditions for the scattering data are obtained

rational solutions of nonlinear evolution equations vertex operators and bispectrality abstract we prove that if q x t 2 t m and r x t 2 t m are certain rational solutions of the akns hierarchy then there are eigenfunctions x k of the akns zs operator l x q r x satisfying a differential equation in the spectral parameter k of the form b k k x where b k k is a matrix differential operator independent of x and is a nonconstant function of x we also discuss the relation between this result and a similar one for the rational solutions of the schrodinger operator with potentials in the manifold of rational solutions of the kdv hierarchy

inverse spectral problem for differential operators with rational scattering matrix functions in this paper we study the inverse scattering problem for linear canonical differential equations of a special type in the case when the scattering function is a rational matrix valued function the main result here is the form of the potential in terms of a realisation of the scattering function the difference between this publication and the previous one 3 consists in the fact that here is used the marchenko method instead of kre n s method for rational scattering functions we also prove the equivalence of the two methods

exact solutions of nonlinear equations and the method of operator identities abstract exact solutions and wave functions of integrable nonlinear equations are constructed by the method of operator identities

on a theorem of kac and gilbert we prove a general operator theoretic result that asserts that many multiplicity two selfadjoint operators have simple singular spectrum

the feedback capacity of the binary erasure channel with a no consecutive ones input constraint the input constrained erasure channel with feedback is considered where the binary input sequence contains no consecutive ones i e it satisfies the 1 infty rll constraint we derive the capacity for this setting which can be expressed as c epsilon max 0 leq p leq 0 5 frac 1 epsilon h b p 1 1 epsilon p where epsilon is the erasure probability and h b cdot is the binary entropy function moreover we prove that a priori knowledge of the erasure at the encoder does not increase the feedback capacity the feedback capacity was calculated using an equivalent dynamic programming dp formulation with an optimal average reward that is equal to the capacity furthermore we obtained an optimal encoding procedure from the solution of the dp leading to a capacity achieving zero error coding scheme for our setting dp is thus shown to be a tool not only for solving optimization problems such as capacity calculation but also for constructing optimal coding schemes the derived capacity expression also serves as the only non trivial upper bound known on the capacity of the input constrained erasure channel without feedback a problem that is still open

kotani theory for one dimensional stochastic jacobi matrices we consider families of operators h on l2 given by h u n u n 1 u n 1 v n u n wherev is a stationary bounded ergodic sequence we prove analogs of kotani s results including that for a e ac h is the essential closure of the set ofe where e the lyaponov index vanishes and the result that ifv is non deterministic then ac is empty

perturbations of orthogonal polynomials with periodic recursion coefficients the results of denisov rakhmanov szeg shohat nevai and killip simon are extended from asymptotically constant orthogonal polynomials on the real line oprl and unit circle opuc to asymptotically periodic oprl and opuc the key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations

stability of the procedure of explicit recovery of skew selfadjoint dirac systems from rational weyl matrix functions abstract procedures to recover explicitly discrete and continuous skew selfadjoint dirac systems on semi axis from rational weyl matrix functions are considered their stability is shown some new facts on asymptotics of pseudo exponential potentials i e of explicit solutions of inverse problems are proved as well the gbdt version of backlund darboux transformation methods from system theory and results on algebraic riccati equations are used for this purpose

on the compatibility condition for linear systems and a factorization formula for wave functions the well known compatibility condition for linear systems wx gw and wt fw is considered and new results are obtained in this way a factorization formula for wave functions which is basic in the inverse spectral transform approach to initial boundary value problems is proved in greater generality than before some applications follow

equality of the spectral and dynamical definitions of reflection for full line jacobi matrices schrodinger operators and cmv matrices we show that being reflectionless in the sense of the well known property of m functions is equivalent to a lack of reflection in the dynamics in the sense that any state that goes entirely to x as t goes entirely to x as t this allows us to settle a conjecture of deift and simon from 1983 regarding ergodic jacobi matrices

reflectionless herglotz functions and jacobi matrices we study several related aspects of reflectionless jacobi matrices first we discuss the singular part of the corresponding spectral measures we then show how to identify sets on which measures are reflectionless by looking at the logarithmic potentials of these measures

on doubly selective channel estimation using superimposed training and discrete prolate spheroidal sequences channel estimation and data detection for frequency selective time varying channels are considered using superimposed training we employ a discrete prolate spheroidal basis expansion model dps bem to describe the time varying channel a periodic nonrandom training sequence is arithmetically added superimposed at low power to the information sequence at the transmitter before modulation and transmission therefore there is no loss in data transmission rate compared to time multiplexed tm training we first estimate the channel using dps bem and only the first order statistics of the observations in this estimator the unknown information sequence acts as interference resulting in a poor signal to noise and interference ratio snir for channel estimation we then apply a data dependent superimposed training sequence to either totally or partially cancel out the effects of the unknown information sequence at the receiver on channel estimation in total cancellation at certain frequencies the information bearing components are nulled to compensate for this information loss we investigate a partially data dependent pdd superimposed training scheme where a tradeoff is made between interference cancellation and frequency integrity design of certain parameters for pdd superimposed training is also investigated finally a deterministic maximum likelihood dml approach is used iteratively to enhance channel estimation and data detection computer simulation examples show that the proposed approaches are competitive with the conventional tm training without incurring data rate loss

capacity bounds via duality with applications to multiple antenna systems on flat fading channels a technique is proposed for the derivation of upper bounds on channel capacity it is based on a dual expression for channel capacity where the maximization of mutual information over distributions on the channel input alphabet is replaced with a minimization of average relative entropy over distributions on the channel output alphabet we also propose a technique for the analysis of the asymptotic capacity of cost constrained channels the technique is based on the observation that under fairly mild conditions capacity achieving input distributions escape to infinity the above techniques are applied to multiple antenna flat fading channels with memory where the realization of the fading process is unknown at the transmitter and unknown or only partially known at the receiver it is demonstrated that for high signal to noise ratio snr the capacity of such channels typically grows only double logarithmically in the snr to better understand this phenomenon and the rates at which it occurs we introduce the fading number as the second order term in the high snr asymptotic expansion of capacity and derive estimates on its value for various systems it is suggested that at rates that are significantly higher than the fading number communication becomes extremely power inefficient thus posing a practical limit on practically achievable rates upper and lower bounds on the fading number are also presented for single input single output siso systems the bounds coincide thus yielding a complete characterization of the fading number for general stationary and ergodic fading processes we also demonstrate that for memoryless multiple input single output miso channels the fading number is achievable using beam forming and we derive an expression for the optimal beam direction this direction depends on the fading law and is in general not the direction that maximizes the snr on the induced siso channel using a new closed form expression for the expectation of the logarithm of a noncentral chi square distributed random variable we provide some closed form expressions for the fading number of some systems with gaussian fading including siso systems with circularly symmetric stationary and ergodic gaussian fading the fading number of the latter is determined by the fading mean fading variance and the mean squared error in predicting the present fading from its past it is not directly related to the doppler spread for the rayleigh ricean and multiple antenna rayleigh fading channels we also present firm nonasymptotic upper and lower bounds on channel capacity these bounds are asymptotically tight in the sense that their difference from capacity approaches zero at high snr and their ratio to capacity approaches one at low snr

dual capacity upper bounds for binary input single tap isi channels the capacity of noisy channels with memory and finite input has been difficult to characterize explicitly in many cases in this paper we consider single tap binary input gaussian channels with inter symbol interference isi the dual capacity method is used to obtain upper bounds using markov test distributions the bound expressed as a single letter optimization problem is solved numerically for higher memory the notion of cycle basis is used to provide an exponential reduction in the computational complexity of the optimization problem bounds were obtained for the important case of the dicode channel and these are better than the previously known upper bounds and are close to achievable rates over a wide range of snrs

pseudo canonical systems with rational weyl functions explicit formulas and applications abstract the paper extends earlier results of the authors for canonical systems with spectral functions of which the absolutely continuous part has a rational derivative to a class of differential systems with skew selfadjoint potentials the corresponding direct and inverse spectral problems are solved explicitly using state space methods from mathematical system theory applications to nonlinear integrable partial differential equations are included

a method for computing the partial singular value decomposition a method for computing the partial singular value decomposition of a matrix is described the method is appropriate to problems where the matrix is known to be of low rank and only the principal singular vectors are of interest the technique is simple easy to implement in integer arithmetic and places modest memory requirements the convergence properties of the algorithm are investigated analytically and by simulation

the trace formula for schrodinger operators on the line this paper discusses certain aspects of the spectral and inverse spectral problems for the schrodinger operator l q frac d 2 dx 2 q x for q x c the space of bounded continuous functions the trace formula of the title is the relation q 0 lambda 0 sum limits j 1 infty lambda 2j lambda 2j 1 2 mu j with appropriate choices of j j 0 and j j 1 which is a familiar relation in the theory of hill s equation we characterize the set subseteq c of potentials for which this holds further extensions of the theory of hill s equation are also obtained from the spectrum l q a torust q is constructed which is in general infinite dimensjional everyq x can be mapped to a continuous path ont q described by the auxiliary spectrum j j 1 under certain geometrical conditions on l q this path is the orbit of ac 1 vector field ont q and the mapping extends to one from the hull q overline q x xi xi in mathbb r to the closure of this orbit in particular q is compact these results have applications in the theory of schrodinger operators with ergodic potentials

dirichlet to neumann maps abstract weyl titchmarsh m functions and a generalized index of unbounded meromorphic operator valued functions abstract we introduce a generalized index for certain meromorphic unbounded operator valued functions the class of functions is chosen such that energy parameter dependent dirichlet to neumann maps associated to uniformly elliptic partial differential operators particularly non self adjoint schrodinger operators on bounded lipschitz domains and abstract operator valued weyl titchmarsh m functions and donoghue type m functions corresponding to closed extensions of symmetric operators belong to it the principal purpose of this paper is to prove index formulas that relate the difference of the algebraic multiplicities of the discrete eigenvalues of robin realizations of non self adjoint schrodinger operators and more abstract pairs of closed operators in hilbert spaces with the generalized index of the corresponding energy dependent dirichlet to neumann maps and abstract weyl titchmarsh m functions respectively

almost periodicity of some random potentials in this paper we study some random potentials with purely absolutely continuous spectrum in one dimension there are no examples of random potentials with absolutely continuous spectrum that are not almost periodic though it is known from kotani 3 that such potentials are deterministic hence one could conjecture that any random potential with purely absolutely continuous spectrum is almost periodic we show this result for random potentials with infinitely many bands in their spectrum such that the gaps behave well in the finite bands case concini and johnson l discussed the same problem a precise definition of the behavior of the gaps could be found in condition 2 3 we make use of the results of levitan 7 kotani 3 4 and moser 9 for doing this we also discuss the behavior of the lyapunov exponent asymptotically and its relation to the smoothness of the potentials

uniqueness theorems in inverse spectral theory for one dimensional schrodinger operators new unique characterization results for the potential v x in connection with schrodinger operators on r and on the half line 0 are proven in terms of appropriate krein spectral shift functions particular results obtained include a generalization of a well known uniqueness theorem of borg and marchenko for schrodinger operators on the half line with purely discrete spectra to arbitrary spectral types and a new uniqueness result for schrodinger operators with confining potentials on the entire real line

scattering for general type dirac systems on the semi axis reflection coefficients and weyl functions abstract we show that for general type self adjoint and skew self adjoint dirac systems on the semi axis weyl functions are unique analytic extensions of the reflection coefficients new results on the extension of the weyl functions to the real axis and on the existence in the skew self adjoint case of the weyl functions follow important procedures to recover general type dirac systems from the weyl functions are applied to the recovery of dirac systems from the reflection coefficients we explicitly recover dirac systems from the rational reflection coefficients as well