on the existence and multiplicity of positive periodic solutions for first order vector differential equation abstract in this paper we consider the existence and multiplicity of positive periodic solutions for first order vector differential equation x t f t x t 0 a e t 0 under the periodic boundary value condition x 0 x here is a positive constant and f 0 r n r n is a caratheodory function some existence and multiplicity results of positive periodic solutions are derived by using a fixed point theorem in cones

remarks on first and second order periodic boundary value problems on etudie les problemes aux valeurs limites periodiques d ordre 1 et 2 suivants u f t u u 0 u 2 et u f t u u 0 u 2 u 0 u 2 on obtient l existence de solutions extremales comme limites d iterees monotones

further analytical results on the joint detection of cochannel signals using diversity arrays we derive a fully analytical expression for the union bound on symbol error rate for the joint detection of several cochannel fading signals using a diversity antenna array putting numerical results previously published on a firm analytical base we prove that with pilot based multiuser channel estimation any number of users can enjoy diversity order equal to the number of antennas and we quantify the performance penalty relative to single user binary phase shift keying as a function of number of users constellation density and number of antennas we also demonstrate the interdependence of all users participating in the detection process

exact symbol error rate and diversity analysis of orthogonalized spatial multiplexing systems with optimal precoding orthogonalized spatial multiplexing osm systems with optimal precoding denoted by posm was recently proposed which allows a single symbol decodable maximum likelihood receiver in closed loop multiple input multiple output mimo systems it was shown that the performance of the posm is identical to that of the optimal closed loop mimo system in terms of the minimum distance in this letter we derive an exact symbol error rate ser expression of the posm for 4 qam and 16 qam in rayleigh fading channels by using polar coordinates and the singular value distributions the final expression of the ser can be evaluated with a single integral form also in the high snr regime we obtain a simple average ser expression of the posm which shows that the posm with two transmitting antennas achieves full diversity simulation results confirm the accuracy of our derived analysis

eigenvalues invariant factors highest weights and schubert calculus we describe recent work of klyachko totaro knutson and tao that characterizes eigenvalues of sums of hermitian matrices and decomposition of tensor products of representations of gln c we explain related applications to invariant factors of products of matrices intersections in grassmann varieties and singular values of sums and products of arbitrary matrices recent breakthroughs primarily by a klyachko b totaro a knutson and t tao with contributions by p belkale and c woodward have led to complete solutions of several old problems involving the various notions in the title our aim here is to describe this work and especially to show how these solutions are derived from it along the way we will see that these problems are also related to other areas of mathematics including geometric invariant theory symplectic geometry and combinatorics in addition we present some related applications to singular values of arbitrary matrices although many of the theorems we state here have not appeared elsewhere their proofs are mostly soft algebra based on the hard geometric or combinatorial work of others indeed this paper emphasizes concrete elementary arguments we do give some new examples and counterexamples and raise some new open questions we have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before however there is a very large literature particularly for linear algebra problems about eigenvalues singular values and invariant factors we have listed only a few of these articles from whose bibliographies we hope an interested reader can trace the history we apologize to the many whose work is not cited directly we begin in the first five sections by describing each of the problems together with some of their early histories and we state as theorems the new solutions to these problems in section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians sections 7 8 9 and 10 also contain variations and generalizations of some of the theorems stated in the first five sections as well as attributions of the theorems to their authors one of our fascinations with this subject even now that we have proofs of the theorems is the challenge to understand in a deeper way why all these subjects are received by the editors in july 1999 and in revised form january 3 2000 2000 mathematics subject classification primary 15a42 22e46 14m15 secondary 05e15 13f10 14c17 15a18 47b07 the author was partly supported by nsf grant dms9970435 c 2000 american mathematical society

on the error performance of the a_ n lattices we consider the root lattice a n and derive explicit recursive formulas for the moments of its voronoi cell these formulas enable accurate prediction of the error probability of lattice codes constructed from a n

the combinatorial structure of generalized eigenspaces from nonnegative matrices to general matrices abstract the perron frobenius spectral theory of nonnegative matrices motivated an intensive study of the relationship between graph theoretic properties and spectral properties of matrices while for about seventy years research focused on nonnegative matrices in the past fifteen years the study has been extended to general matrices over an arbitrary field one of the major original problems in this context is determining the relations between the matrix analytic height characteristic of a matrix and the graph theoretic level characteristic in this article the history of this problem is reviewed from its introduction for nonnegative matrices through its complete solution for nonnegative matrices to the solution of the generalized version of the problem for general matrices

the method of lower and upper solutions for systems of boundary value problems the method of lower and upper solutions for second order differential system x f t x with dirichlet boundary condition x 0 x 1 0 is used to give sufficient conditions for the existence of at least one solution

ordinary differential equations with nonlinear boundary conditions of antiperiodic type abstract the quasilinearization method is successfully applied to differential problems with nonlinear boundary conditions of antiperiodic type given are sufficient conditions under which the monotone iterations converge quadratically to a unique solution of our problem the advantage of this approach lies in the fact that the approximate solutions are obtained iteratively as solutions of the above mentioned iterations

level characteristics corresponding to peripheral eigenvalues of a nonnegative matrix in this paper we give necessary and sufficient conditions for a set of jordan blocks to correspond to the peripheral spectrum of a nonnegative matrix for each eigenvalue the level characteristic with respect to the spectral radius is defined the necessary and sufficient conditions include a requirement that the level characteristic is majorized by the height characteristic an algorithm which has been implemented in matlab is given to determine when a multiset of jordan blocks corresponds to the peripheral spectrum of a nonnegative matrix the algorithm is based on the necessary and sufficient conditions given in this paper

microdiversity on rician fading channels formats coherent phase shift keying cpsk and noncoherent frequency shift keying ncfsk results using both maximal ratio combining mrc and selection diversity combining sc are presented for comparison our results show that from a feasibility and practical tradeoffs point of view the performance of an eg combiner may be as good as that of a mr combiner the effects of gain unbalance between branches of the eg combiner on the probability distribution of snr and on the bit error rates are also investigated the rician fading model may be used to model both the microcellular environment and the mobile satellite fading channel hence the results of this paper may be useful in both of these areas furthermore in the development of the analysis we present an efficient method for computing the distribution of sums of rich random variables this may be useful for other problems involving rician fading the suitability of modeling a rician fading environment by a properly chosen nakagami model is examined a formula for determining the corresponding values of rician parameter k and nakagami parameter m is also assessed

orthogonalized spatial multiplexing for closed loop mimo systems in this paper we propose a novel spatial multiplexing sm scheme for transmission over flat fading multiple input multiple output mimo channels which allows simple maximum likelihood decoding at the receiver with small feedback information we begin with a real valued representation of the complex valued system model and show that we can achieve orthogonality between transmitted signals by applying a proper rotation to transmitted symbols compared with other closed loop methods the proposed scheme significantly reduces the processing complexity at both transmitter and receiver as well as the feedback overhead we also combine the proposed orthogonalization method with antenna selection techniques and then discuss a criterion based on the minimum euclidean distance between received vectors for selecting the optimal subset of multiple transmit antennas using geometrical properties of the receive constellations we also present a simple antenna selection metric for the proposed sm systems simulation results demonstrate that our sm system performs close to the optimum closed loop system with much reduced complexity

limited feedback unitary precoding for spatial multiplexing systems multiple input multiple output mimo wireless systems use antenna arrays at both the transmitter and receiver to provide communication links with substantial diversity and capacity spatial multiplexing is a common space time modulation technique for mimo communication systems where independent information streams are sent over different transmit antennas unfortunately spatial multiplexing is sensitive to ill conditioning of the channel matrix precoding can improve the resilience of spatial multiplexing at the expense of full channel knowledge at the transmitter which is often not realistic this correspondence proposes a quantized precoding system where the optimal precoder is chosen from a finite codebook known to both receiver and transmitter the index of the optimal precoder is conveyed from the receiver to the transmitter over a low delay feedback link criteria are presented for selecting the optimal precoding matrix based on the error rate and mutual information for different receiver designs codebook design criteria are proposed for each selection criterion by minimizing a bound on the average distortion assuming a rayleigh fading matrix channel the design criteria are shown to be equivalent to packing subspaces in the grassmann manifold using the projection two norm and fubini study distances simulation results show that the proposed system outperforms antenna subset selection and performs close to optimal unitary precoding with a minimal amount of feedback

exact method for the error probability calculation of three dimensional signal constellations the contribution of this letter is the computation of exact symbol error probability sep of three dimensional 3 d signal constellations over an additive white gaussian noise awgn channel the originality of the proposed method is that it can be applied to any arbitrary 3 d constellation whose decision regions may not meet at right angles we express the sep in a triple integral form which is further simplified the simplified form requires a single integral evaluation of standard erf function using the derived exact sep formula we plot sep for a number of selected 3 d constellations the sep obtained using the proposed formula is validated by simulation results it is also compared with the union bound approximation an upper bound for sep

generalized linear precoder and decoder design for mimo channels using the weighted mmse criterion we address the problem of designing jointly optimum linear precoder and decoder for a mimo channel possibly with delay spread using a weighted minimum mean squared error mmse criterion subject to a transmit power constraint we show that the optimum linear precoder and decoder diagonalize the mimo channel into eigen subchannels for any set of error weights furthermore we derive the optimum linear precoder and decoder as functions of the error weights and consider specialized designs based on specific choices of error weights we show how to obtain 1 the maximum information rate design 2 qos based design we show how to achieve any set of relative snrs across the subchannels and 3 the unweighted mmse and equal error design for fixed rate systems

unified analysis of switched diversity systems in independent and correlated fading channels the moment generating function mgf of the signal power at the output of dual branch switch and stay selection diversity ssd combiners is derived the first order derivative of the mgf with respect to the switching threshold is also derived these expressions are obtained for the general case of correlated fading and nonidentical diversity branches and hold for any common fading distributions e g rayleigh nakagami m rician nakagami q the mgf yields the performance bit or symbol error probability of a broad class of coherent differentially coherent and noncoherent digital modulation formats with ssd reception the optimum switching threshold in a minimum error rate sense is obtained by solving a nonlinear equation which is formed by using the first order derivative of the mgf this nonlinear equation can be simplified for several special cases for independent and identically distributed diversity branches the optimal switching threshold in closed form is derived for three generic forms of the conditional error probability for correlated rayleigh or nakagami m fading with identical branches the optimal switching threshold in closed form is derived for the noncoherent binary modulation formats we show previously published results as special cases of our unified expression selected numerical examples are presented and discussed

a generalized upper and lower solutions method and multiplicity results for nonlinear first order ordinary differential equations abstract we introduce a generalized upper and lower solutions method for the solvability of first order ordinary differential equations u t t u t u 0 u 1 in order to cover the case when the function satisfies caratheodory conditions this method is then applied to get multiplicity results when the nonlinearity interacts with the real eigenvalue of the linearized problem our proofs are based on differential inequalities and classical leray schauder degree

error probabilities of two dimensional m ary signaling in fading a new exact method for computing the average symbol error probability of two dimensional m ary signaling in slow fading is presented the method is generally applicable to polygonal decision regions the exact average symbol error rate of coherent 16 star qam is obtained and compared to that of 16 rectangular qam new results for optimum ring ratios of 16 star qam in additive white gaussian noise awgn and in slow fading are also given

a generalization of the method of upper and lower solutions for discontinuous first order problems with nonlinear boundary conditions in this paper we prove the existence of extremal solutions for a discontinuous bvp between a lower and an upper solution which may be discontinuous in a finite set of points furthermore we present a monotone method with which we can explicitly construct a pair of sequences that converge to the extremal solutions despite the discontinuities of the right hand side of the equation

accurate approximation of qam error probability on quasi static mimo channels and its application to adaptive modulation an accurate approximation for the conditional error probability on quasi static multiple input multiple output mimo antenna channels is proposed for a fixed channel matrix it is possible to accurately predict the performance of quadrature amplitude modulations qam transmitted over the mimo channel in presence of additive white gaussian noise the tight approximation is based on a simple union bound for the point error probability in the n dimensional real space instead of making an exhaustive evaluation of all pairwise error probabilities intractable in many cases a pohst or a schnorr euchner lattice enumeration is used to limit the local theta series inside a finite radius sphere the local theta series is derived from the original lattice theta series and the point position within the finite multidimensional qam constellation in particular we take into account the number of constellation facets hyperplanes that are crossing the sphere center as a direct application to the accurate approximation for the conditional error probability we describe a new adaptive qam modulation for quasi static multiple antenna channels

algebraic tools to build modulation schemes for fading channels a unified framework is presented in order to build lattice constellations matched to both the rayleigh fading channel and the gaussian channel the method encompasses the situations where the interleaving is done on the real components or on two dimensional signals in the latter case a simple construction of lattices congruent to the densest binary lattices with respect to the euclidean distance is proposed it generalizes in a sense to be clarified later the structural construction proposed by forney 1991 these constellations are next combined with coset codes the partitioning rules and the gain formula are similar to those used for the gaussian channel

performance analysis of maximum likelihood detection in a mimo antenna system we provide an analysis of the performance of maximum likelihood detection mld over flat fading channels in a wireless multiple input multiple output mimo antenna system a tight union bound with an asymptotic form on the probability of symbol error rate ser for mimo mld systems with two dimensional signal constellations such as qam and psk is introduced using this analytic bound the performance of the mimo antenna system is demonstrated quantitatively with respect to channel estimation constellation size and antenna configuration

nonlinear second order periodic boundary value problems abstract in this paper we study a periodic boundary value problem pbvp for brevity of second order it is well known that if and are upper and lower solutions respectively with then there exists a solution u such that u when nothing was known the main purpose of this paper is to consider this last case also we study the structure of the set of solutions of the pbvp in the sector when and we apply the results to the case

error performance of multidimensional lattice constellations part ii evaluation over fading channels this is the second part of a two part series of papers where the error performance of multidimensional lattice constellations with signal space diversity ssd is investigated in part i following a novel combinatorial geometrical approach which is based on parallelotope geometry we have presented an exact analytical expression and two closed form bounds for the symbol error probability sep in additive white gaussian noise awgn in the present part ii we extend the analysis and present a novel analytical expression for the frame error probability fep of multidimensional lattice constellations over nakagami m fading channels as the fep of infinite lattice constellations is lower bounded by the sphere lower bound slb we propose the sphere upper bound sub for block fading channels furthermore two novel bounds for the fep of multidimensional lattice constellations over block fading channels named multiple sphere lower bound mslb and multiple sphere upper bound msub are presented the expressions for the slb and sub are given in closed form while the corresponding ones for mslb and msub are given in closed form for unitary block length numerical and simulation results illustrate the tightness of the proposed bounds which can be efficiently used to set the performance limits on the fep of lattice constellations of arbitrary structure dimension and rank

the monotone method for first order problems with linear and nonlinear boundary conditions in this paper we develop the monotone method for the first order problem u t f t u t for a e t a b when f is a caratheodory function and u w 1 1 a b we consider the nonlinear boundary conditions l u a u b 0 with l c r 2 r nondecreasing in x or nonincreasing in y and the linear boundary conditions a 0u 0 b 0u t l 0 with a 0 b 0 and l 0 r we prove the existence of solutions and the validity of the monotone method if there exists a lower solution a and an upper solution b with either a b for the linear conditions we obtain eight new concepts of lower and upper solutions which generalize previous known cases

performance analysis and design criteria of bicm id with signal space diversity for keyhole nakagami m fading channels this paper generalizes the application bit interleaved coded modulation with iterative decoding bicm id using signal space diversity ssd over keyhole nakagami m fading channels the tight union bound on the asymptotic error performance is first analytically derived the near optimal rotation matrix with respect to both the asymptotic performance and the convergence behavior is then determined in particular it is demonstrated that the suitable rotation matrix is the one that has 1 all entries equal in magnitude 2 a high diversity order and 3 a large minimum product of the ratios between squared distances to the power m and log squared distances to the power m of the rotated constellation scaled by factors of signal to noise ratio snr and the parameter m various analytical and simulation results show that by employing ssd with a sufficiently large dimension the error performance can closely approach that over an additive white gaussian noise awgn channel even in the worst case of keyhole fading

an mgf based performance analysis of generalized selection combining over rayleigh fading channels using the notion of the spacing between ordered exponential random variables a performance analysis of the generalized selection combining gsc diversity scheme over rayleigh fading channels is presented and compared with that of the conventional maximal ratio combining and selection combining schemes starting with the moment generating function mgf of the gsc output signal to noise ratio snr we derive closed form expressions for the average combined snr outage probability and average error probability of a wide variety of modulation schemes operating over independently identically distributed i i d diversity paths because of their simple form these expressions readily allow numerical evaluation for cases of practical interest the results are also extended to the case of non i i d diversity paths

signaling constellations for fading channels the performance of various coherent 8 ary and 16 ary modulations in additive white gaussian noise awgn and slowly fading channels are analyzed new expressions for the exact symbol error rates ser in fading with diversity combining are derived for any two dimensional signaling format having polygonal decision boundaries maximal ratio combining equal gain combining and selection combining are considered the ser formulas obtained make it possible for the first time to optimize parameters of various constellations precisely and to determine which constellation has the lowest probability of error for example a star constellation such as that specified in the ccitt v 29 standard can be improved by adjusting the amplitude ratios of the points in the constellation to save about 0 63 db power in awgn without sacrificing the phase error tolerance while maintaining the same error rate the sensitivity of each constellation to phase error is presented and comparisons are made six 8 ary signal sets and 11 16 ary signal sets are examined using the new symbol error probability formulas to determine best signal sets for fading channels

on constructions of algebraic space time codes with am psk constellations satisfying rate diversity tradeoff constructions of space time codes having amplitude modulated phase shift keying am psk constellations are presented in this paper the first construction termed p radii construction is obtained by extending hammons dyadic dual radii construction to the cases when the size of the constellation is a power of a prime p p ges 2 the resultant code is optimal with respect to the rate diversity tradeoff and has an am psk constellation with signal points distributed over p concentric circles in the complex plane i e there are p radii also contained in this paper is the identification of rich classes of nontrivial subset subcodes of the newly constructed space time codes and it is shown that these subset subcodes are again all optimal finally a new generalization of the super unified construction by hammons is presented it is shown that codes obtained from several previously known constructions are subset subcodes of the one derived from this generalized construction

new algebraic constructions of rotated z sup n lattice constellations for the rayleigh fading channel in this correspondence we present various families of full diversity rotated z sup n lattice constellations based on algebraic number theory constructions we are able to give closed form expressions of their minimum product distance using the corresponding algebraic properties

good lattice constellations for both rayleigh fading and gaussian channels recent work on lattices matched to the rayleigh fading channel has shown how to construct good signal constellations with high spectral efficiency we present a new family of lattice constellations based on complex algebraic number fields which have good performance on rayleigh fading channels some of these lattices also present a reasonable packing density and thus may be used at the same time over a gaussian channel conversely we show that particular versions of the best lattice packings d sub 4 e sub 6 e sub 8 k sub 12 spl lambda sub 16 spl lambda sub 24 constructed from totally complex algebraic cyclotomic fields present better performance over the rayleigh fading channel the practical interest in such signal constellations rises from the need to transmit information at high rates over both terrestrial and satellite links some further results in algebraic number theory related to ideals and their factorization are presented and the decoding algorithm used with these lattice constellations are illustrated together with practical results

existence and approximation of solutions for discontinuous functional differential equations in this paper existence of solutions of initial value problems for discontinuous functional differential equations is investigated firstly by applying the method of upper and lower solutions which may be discontinuous some existence results of extremal solutions are obtained furthermore we also develop a monotone iterative technique for obtaining extremal solutions which are obtained as limits of monotone sequences

constellation mappings for two dimensional signaling of nonuniform sources the design of two dimensional constellation mappings for the transmission of binary nonuniform memoryless sources over additive white gaussian noise channels using standard m ary psk and qam modulation schemes is investigated the main application of this problem is the incorporation of an adaptive mapping assignment in modem devices that employ fixed psk qam modulation schemes for the transmission of heterogenous data such as multimedia information containing various levels of nonuniformity in general the optimal mapping depends on both the probability distribution of the input signals and the signal to noise ratio snr in the channel in addition to the geometry of the signal constellation we show that constellation mappings which follow the objective of minimizing the average symbol energy and given this maximizing the decoding probability of the most likely signals can yield symbol error rate and bit error rate performance that is substantially better than gray encoding maps gains as high as 3 5 db in snr e sub b n sub 0 are obtained for highly nonuniform sources finally we note that the mappings techniques result in nonzero mean constellations and briefly consider their performances when they are converted to zero mean constellations by shifting in this case we observe that the shifted zero mean gray map outperforms our shifted maps for small to medium sized constellations m spl les 32 but not for larger sizes

a general method for calculating error probabilities over fading channels signal fading is a ubiquitous problem in mobile and wireless communications in digital systems fading results in bit errors and evaluating the average error rate under fairly general fading models and multichannel reception is often required predominantly to date most researchers perform the averaging using the probability density function method or the moment generating function mgf method this paper presents a third method called the characteristic function chf method for calculating the average error rates and outage performance of a broad class of coherent differentially coherent and noncoherent communication systems with or without diversity reception in a myriad of fading environments unlike the mgf technique the proposed chf method based on parseval s theorem enables us to unify the average error rate analysis of different modulation formats and all commonly used predetection diversity techniques i e maximal ratio combining equal gain combining selection diversity and switched diversity within a single common framework the chf method also lends itself to the averaging of the conditional error probability involving the complementary incomplete gamma function and the confluent hypergeometric function over fading amplitudes which heretofore resisted to a simple form as an aside we show some previous results as special cases of our unified framework

error performance of multidimensional lattice constellations part i a parallelotope geometry based approach for the awgn channel multidimensional lattice constellations which present signal space diversity ssd have been extensively studied for single antenna transmission over fading channels with focus on their optimal design for achieving high diversity gain in this two part series of papers we present a novel combinatorial geometrical approach based on parallelotope geometry for the performance evaluation of multidimensional finite lattice constellations with arbitrary structure dimension and rank in part i we present an analytical expression for the exact symbol error probability sep of multidimensional signal sets and two novel closed form bounds named multiple sphere lower bound mlsb and multiple sphere upper bound msub part ii extends the analysis to the transmission over fading channels where multidimensional signal sets are commonly used to combat fading degradation numerical and simulation results show that the proposed geometrical approach leads to bounds that can be used for the performance evaluation and the design of multidimensional lattice constellations both in additive white gaussian noise awgn and fading channels

space time codes with am psk constellations this correspondence presents a new signal mapper that maps the maximal rank distance codes to space time st codes with amplitude modulation phase shift keying am psk constellations it is shown that this new mapper is rank distance preserving comparing to the multiradii construction proposed by hammons this new mapper has linear increase in the radii and the resulting signal constellations have larger minimum distance and lower peak to average power ratio variations of this new mapper are also given to provide st codes with rotated am psk constellations

simple iterative methods to exploit the signal space diversity signal space diversity is a power and bandwidth efficient diversity technique to exploit the signal space diversity joint maximum likelihood ml detection at the receiver is usually needed where the complexity grows exponentially with the dimension of the lattice in this letter we propose a serial concatenated scheme and two simple iterative methods to exploit the signal space diversity the simple iterative methods are based on the idea of soft interference cancellation the first iterative method is based on a vector gaussian approximation while the second one is based on a scalar gaussian approximation the complexity of the first iterative method grows cubically with the dimension of the lattice and the simulations show that its performance approaches that of the optimal maximum a posteriori detection method the complexity of the second iterative method grows linearly with the dimension of the lattice and the simulations show that when the dimension of the lattice n 32 at bit error rate 10 sup 5 the performance gap between the rayleigh fading channel and the gaussian channel is only 0 3 db

universal bound on the performance of lattice codes we present a lower bound on the probability of symbol error for maximum likelihood decoding of lattices and lattice codes on a gaussian channel the bound is tight for error probabilities and signal to noise ratios of practical interest as opposed to most existing bounds that become tight asymptotically for high signal to noise ratios the bound is also universal it provides a limit on the highest possible coding gain that may be achieved at specific symbol error probabilities using any lattice or lattice code in n dimensions in particular it is shown that the effective coding gains of the densest known lattices are much lower than their nominal coding gains the asymptotic as n spl rarr spl infin behavior of the new bound is shown to coincide with the shannon 1948 limit for gaussian channels

error probability performance of l branch diversity reception of mqam in rayleigh fading new exact expressions involving hypergeometric functions are derived for the symbol error rate ser of m ary quadrature amplitude modulation mqam for l branch diversity reception in rayleigh fading and additive white gaussian noise awgn the diversity combining techniques considered are maximum ratio combining mrc and selection combining sc mrc with identical channels and dissimilar channels are analyzed

a monotone method for the equation uxyz f comparison theorems for the nonlinear boundary value problem u x y z f x y z u u x u y u z u x y u y z u z x with the boundary conditions u o y z a y z u x 0 z b z x u x y 0 c x y a y 0 c 0 y a 0 z b z 0 b 0 x c x 0 are established under certain conditions then it is shown that there exist sequences of lower and upper solutions which converge uniformly and monotonically to minimal and maximal solutions respectively

periodic solutions of nonlinear boundary value problems one of the well known techniques in the theory of nonlinear boundary value problems bvp is the method of differential inequalities or the method of upper and lower solutions the method of alternative problems a global variant of the lyapunov schmidt method has been used in the study of problems at resonance the investigation of periodic bvp s forms an important subclass of problems at resonance our aim is to combine the two approaches to discuss nonlinear problems at resonance we restrict ourselves in this paper to the discussion of periodic boundary value problems in section 1 we shall indicate the method of upper and lower solutions the results for systems are given in a general way so as to include known results and also offer new directions section 2 deals with the abstract existence results at resonance in the desired framework in section 3 we consider the scalar periodic bvp s by combining the two techniques namely alternative methods and the method of differential inequalities

upper and lower solutions with jumps abstract we study a periodic boundary value problem for a first order differential equation from a new point of view which permits us to obtain an existence result involving upper and lower solutions with jump discontinuities we present some applications of our result with illustrative examples

signal constellations for bit interleaved coded modulation bit interleaved coded modulation bicm is a bandwidth efficient coding technique consisting of serial concatenation of binary error correcting coding bit by bit interleaving and high order modulation bicm is capable of achieving excellent error performance provided that powerful codes such as for example turbo codes or low density parity check ldpc codes are employed we address the problem of finding the signal sets that are the most suitable ones for designing power efficient bicm schemes over an additive white gaussian noise awgn channel to this end we exploit the expression of the bicm capacity limit and evaluate it for several 8 and 16 ary constellations the bit error rate ber performance of some bicm schemes made up of turbo codes and various signal sets is also investigated by computer simulations so as to illustrate the theoretical results we show that for spectral efficiencies of practical interest the most attractive signal sets are those for which gray mapping is possible provided that their symbol error rate performance is sufficiently close to the optimum this explains why some constellations having a simple structure such as 8 psk and 16 qam perform very well when combined with a powerful code at the same time the constellations displaying optimal error performance without coding are generally not of interest for bicm

a characterization of jordan canonical forms which are similar to eventually nonnegative matrices with the properties of nonnegative matrices abstract in this paper we give necessary and sufficient conditions for a matrix in jordan canonical form to be similar to an eventually nonnegative matrix whose irreducible diagonal blocks satisfy the conditions identified by zaslavsky and tam and whose subdiagonal blocks with respect to its frobenius normal form are nonnegative these matrices are referred to as seminonnegative matrices and we show that they exhibit many of the same combinatorial spectral properties as nonnegative matrices this paper extends the work on jordan forms of irreducible eventually nonnegative matrices to the reducible case

jordan chains of h cyclic matrices abstract arising from the classification of the matrix roots of a nonnegative imprimitive irreducible matrix we present results concerning the jordan chains of an h cyclic matrix we also present ancillary results applicable to nonnegative imprimitive irreducible matrices and demonstrate these results via examples

the peripheral spectrum of a nonnegative matrix abstract in this paper we provide a necessary and sufficient condition for a collection of jordan blocks to correspond to the peripheral spectrum of a nonnegative matrix we arrive at our condition by linking the tam schneider condition to the level sets with respect to the spectral radius of a nonnegative matrix

algebraic lattice constellations bounds on performance in this work we give a bound on performance of any full diversity lattice constellation constructed from algebraic number fields we show that most of the already available constructions are almost optimal in the sense that any further improvement of the minimum product distance would lead to a negligible coding gain furthermore we discuss constructions minimum product distance and bounds for full diversity complex rotated z i sup n lattices for any dimension n which avoid the need of component interleaving

existence of periodic solutions for fully nonlinear first order differential equations in this paper we establish some existence results for periodic solutions of fully nonlinear first order scalar differential equation f t x x 0 the approach here uses the viscosity solutions method developed in recent decades and the classical upper lower solutions method

extremal solutions of measure differential equations abstract we investigate the existence of the least and greatest solutions to measure differential equations as well as the relation between the extremal solutions and lower or upper solutions along the way we prove a fairly general local existence theorem and an analogue of peano s uniqueness theorem for measure differential equations finally we show that the general theory is applicable in the study of differential equations with impulses or dynamic equations on time scales

analysis of equal gain diversity on nakagami fading channels an infinite series for the complementary probability distribution function cdf of the signal to noise ratio snr at the output of l branch equal gain eg diversity combiners in nakagami 1960 fading channels is derived the bit error rate for a matched filter receiver is analyzed for the l branch eg combiner and different fading parameters both coherent phase shift keying cpsk and differential coherent phase shift keying dcpsk are considered the effects of gain unbalance between branches on the probability distribution of the snr and on the bit error rates are investigated bit error rate results are also obtained for coherent and noncoherent reception of frequency shift keying fsk the effects of gain unbalances on fsk modulations are also investigated bit error rates for eg combining on rayleigh fading channels are obtained for l 2 these results are presented as a special case of the more generalized nakagami fading model

exploiting quantized channel norm feedback through conditional statistics in arbitrarily correlated mimo systems in the design of narrowband multi antenna systems a limiting factor is the amount of channel state information csi available at the transmitter this is especially evident in multi user systems where the spatial user separability determines the multiplexing gain but it is also important for transmission rate adaptation in single user systems to limit the feedback load the unknown and multi dimensional channel needs to be represented by a limited number of bits when combined with long term channel statistics the norm of the channel matrix has been shown to provide substantial csi that permits efficient user selection linear precoder design and rate adaptation herein we consider quantized feedback of the squared frobenius norm in a rayleigh fading environment with arbitrary spatial correlation the conditional channel statistics are characterized and their moments are derived for both identical distinct and sets of repeated eigenvalues these results are applied for minimum mean square error mmse estimation of signal and interference powers in single and multi user systems for the purpose of reliable rate adaptation and resource allocation the problem of efficient feedback quantization is discussed and an entropy maximizing framework is developed where the post user selection distribution can be taken into account in the design of the quantization levels the analytic results of this paper are directly applicable in many widely used communication techniques such as space time block codes linear precoding space division multiple access sdma and scheduling

on the error probability of linearly modulated signals on frequency flat ricean rayleigh and awgn channels the method used in aghamohammadi and meyr 1990 for finding the error probability of linearly modulated signals on rayleigh frequency flat fading channels has been applied to the more general case of ricean fading a signal received on a fading channel is subject to a multiplicative distortion md and to the usual additive noise following a compensation of the md the signal provided to the detector may be thought to include only a single additive distortion term final noise which comprises the effects of the original additive noise the md and the error in md compensation an exact expression for the probability density function of the final noise is derived this allows calculation of error probability for arbitrary types of linear modulations results for many cases of interest are presented furthermore as special cases of ricean fading error probability for rayleigh fading and non fading channels are obtained which either match the results or complete the approximate derivations formerly known from the literature

on the symbol error rates for signal space diversity schemes over a rician fading channel a signal space diversity ssd scheme is one of techniques to achieve diversity gain in fading channels this method consists of two key operations constellation rotation and component wise interleaving because of these operations the decision boundaries for the ssd are no longer perpendicular and thus different coordination approaches are required for the analysis of error rates compared to conventional rectangular coordinates in this letter we derive an exact expression of the symbol error rate for the ssd scheme in rician fading channels with m qam and m psk by defining the ratio of the standard deviation of the inphase and quadrature components we introduce a new signal model for the ssd based on this signal model we can compute the exact symbol error rate using polar coordinates the computer simulation results confirm the accuracy of our analysis for fading channels

on the error probability of linearly modulated signals on rayleigh frequency flat fading channels consideration is given to optimal detection of linearly modulated signals subject to multiplicative rayleigh distributed distortion and additive white gaussian noise for coherent detection regenerated amplitude and phase references are employed at the receiver to compensate for amplitude and phase deviations from the correct values a system model is formulated under the assumption of perfect symbol timing and in the absence of intersymbol interference producing a final additive noise term applied just before the detection which contains the effects of the original additive and multiplicative distortions and of the errors in the phase and amplitude references by determining the probability density function of this final noise term for arbitrary types of linear modulation it is possible to perform exact calculations of error probabilities

error rates for nakagami m fading multichannel reception of binary and m ary signals this paper derives new closed form formulas for the error probabilities of single and multichannel communications in rayleigh and nakagami m 1960 fading closed form solutions to three generic trigonometric integrals are presented as part of the main result providing a unified method for the derivation of exact closed form average symbol error probability expressions for binary and m ary signals with l independent channel diversity reception both selection diversity and maximal ratio combining mrc techniques are considered the results are generally applicable for arbitrary two dimensional signal constellations that have polygonal decision regions operating in a slow nakagami m fading environments with positive integer fading severity index mrc with generically correlated fading is also considered the new expressions are applicable in many cases of practical interest the closed form expressions derived for a single channel reception case can be extended to provide an approximation for the error rates of binary and m ary signals that employ an equal gain combining diversity receiver

approximation of solutions for nonlinear periodic boundary value problems with discontinuous upper and lower solutions we prove that it is possible to approximate the solutions of a nonlinear periodic boundary value problem via monotone sequences starting at piecewise continuous upper and lower solutions in fact we present here two different iterative methods explicit expression of each iteration is obtained by solving some linear initial value problems

performance analysis of linear modulation schemes with generalized diversity combining on rayleigh fading channels with noisy channel estimates generalized diversity combining gdc also known as hybrid selection maximal ratio combining or generalized selection combining is a low complexity diversity combining technique by which a fixed subset of a large number of available diversity channels is chosen and then combined using the rules of maximal ratio combining in this paper we analyze the performance of gdc on time correlated rayleigh fading channels with noisy channel estimates we derive expressions for the probability of error for various linear modulation schemes with coherent detection and discuss the conditions under which the analysis can be extended to noncoherent and differentially coherent receiver structures throughout the paper using a fundamental approach to obtain the decision statistic at the combiner output a number of new expressions for the error probabilities are obtained in a rigorous way along with a presentation of their performance with channel estimation errors the final expressions have roughly the same complexity of evaluation as that for the channel with only additive gaussian noise our results correct various inaccuracies in the literature and show that coherent receivers based on imperfectly estimated channel knowledge incur a significant performance loss

exact error rate analysis of diversity 16 qam with channel estimation error the bit error rate ber performance of multilevel quadrature amplitude modulation with pilot symbol assisted modulation channel estimation in static and rayleigh fading channels is derived both for single branch reception and maximal ratio combining diversity receiver systems the effects of noise and estimator decorrelation on the received ber are examined the high sensitivity of diversity systems to channel estimation error is investigated and quantified the influence of the pilot symbol interpolation filter windowing is also considered

the influence of the marked reduced graph of a nonnegative matrix on the jordan form and on related properties a survey abstract for a nonnegative matrix p we discuss the relation of its marked reduced graph to that part of the jordan form that is associated with the perron frobenius root to the nonnegativity of the eigenvectors and generalized eigenvectors to the nonnegativity of solutions of linear equations and to the asymptotic growth of powers of the matrix results are often stated in terms of m matrices and standard results on irreducible matrices are assumed we give examples to illustrate the theorems surveyed

further results on the beaulieu series a frequent problem in digital communications is the computation of the probability density function pdf and cumulative distribution function cdf given the characteristic function chf of a random variable rv this problem arises in signal detection equalizer performance equal gain diversity combining intersymbol interference and elsewhere often it is impossible to analytically invert the chf to get the pdf and cdf in closed form beaulieu 1990 1991 has derived an infinite series for the cdf of a sum of rvs that has been widely used we rederive his series using the gil pelaez 1951 inversion formula and the poisson sum formula this derivation has several advantages including both the bridging of the well known sampling theorem with beaulieu s series and yielding a simple expression for calculating the truncation error term it is also shown that the pdf and cdf can be computed directly using a discrete fourier transform

some new twists to problems involving the gaussian probability integral using an alternate form of the gaussian probability integral discovered a number of years ago it is shown that the solution to a number of previously considered communication problems can be simplified and in some cases made more accurate i e exact rather than bounded these problems include the evaluation of 1 the bit error probability of uncoded phase shift keying psk with costas loop tracking 2 word error probability of antipodal modulation in the presence of fading 3 bit error probability of coded m ary psk mpsk over the memoryless fading channel with given channel state information 4 conditional symbol error probability of mpsk in the presence of carrier synchronization error and 5 the average error probability for the binary additive white gaussian noise awgn intersymbol interference channel also obtained is a generalization of this new alternate form to the case of a two dimensional gaussian probability integral with arbitrary correlation which can be used to evaluate the symbol error probability of mpsk with i q unbalance

real domain decoder for full rate full diversity stbc with multidimensional constellations in this letter we present a new maximum likelihood ml decoding algorithm for space time block codes stbcs that employ multidimensional constellations we start with a lattice representation for stbcs which transforms complex channel models into real matrix equations based on the lattice representation we propose a new decoding algorithm for quasi orthogonal stbcs qo stbc which allows simple ml decoding with performance identical to the conventional ml decoder multidimensional rotated constellations are constructed for the qo stbcs to achieve full diversity as a consequence for quasi orthogonal designs with an arbitrary number of transmit antennas n nges4 the proposed decoding scheme achieves full rate and full diversity while reducing the decoding complexity from o mc n 2 to o mc n 4 in a mc qam constellation

an infinite series for the computation of the complementary probability distribution function of a sum of independent random variables and its application to the sum of rayleigh random variables the properties of the series are studied for both bounded and unbounded random variables the technique is used to find efficient series for computation of the distributions of sums of uniform random variables and sums of rayleigh random variables a useful closed form expression for the characteristic function of a rayleigh random variable is presented and an efficient method for computing a confluent hypergeometric function is given an infinite series for the probability density function of a sum of independent random variables is also derived the inversion of characteristic functions a trapezoidal rule for numerical integration and the sampling theorem in the frequency domain are related to and interpreted in terms of the results

bounds for multihop relayed communications in nakagami m fading we present closed form lower bounds for the performance of multihop transmissions with nonregenerative relays over not necessarily identically distributed nakagami m fading channels the end to end signal to noise ratio is formulated and upper bounded by using an inequality between harmonic and geometric means of positive random variables rvs novel closed form expressions are derived for the moment generating function the probability density function and the cumulative distribution function of the product of rational powers of statistically independent gamma rvs these statistical results are then applied to studying the outage probability and the average bit error probability for phase and frequency modulated signaling numerical examples compare analytical and simulation results verifying the tightness of the proposed bounds

joint multiuser detection of multidimensional constellations over fading channels we investigate the error performance of multidimensional constellations in the multiple access and broadcast channels more specifically we provide closed form expressions for the pairwise error probability pep of the joint maximum likelihood detection for multiuser signaling in the presence of additive white gaussian noise and rayleigh fading arbitrary numbers of users and multidimensional signal sets are assumed while the provided formula for the pep is a function of the dimension wise distances of the multidimensional constellation furthermore a useful upper bound on the average symbol error probability is also obtained through the union bound the analysis is applied to the sparse code multiple access systems the analytical results are validated successfully through simulations and show their importance in the multidimensional constellation design

penalty of hybrid diversity for two dimensional signaling in rayleigh fading lower and upper bound expressions for the penalty of hybrid diversity used with any two dimensional 2 d signaling constellation with polygonal decision regions are derived using these bounds with well known results for maximal ratio combining the error probability performance of 2 d signaling with hybrid selection maximal ratio combining can be approximated to a high accuracy

effect of macrodiversity on average error probabilities in a rician fading channel with correlated lognormal shadowing the effect of correlated lognormal shadowing on the average probability of error performance of narrow band mobile communication systems with micro and macrodiversity reception in a rician fading channel is studied by considering a constant correlation model for the shadowed signals at the base stations the performance degradation due to correlated shadowing is illustrated by considering both coherent and differentially coherent binary phase shift keying schemes numerical results presented show that when the base stations in a cellular system are very closely spaced the effect of correlated shadowing on system performance cannot be ignored