strong detectability and observers conditions are given for the existence of observers that estimate unmeasured outputs on the basis of partial information on the input and the state the concepts of strong detectability and strong observability introduced before in the literature for discrete time systems only are defined and studied for continuous time systems it is shown that there are two different concepts of strong detectability which coincide for discrete time systems algebraic conditions for either concept are given it is shown that these concepts are intimately related to the existence of strong observers i e observers that only use the output of the plant

twisting the fake monster superalgebra abstract we calculate the twisted denominator identities of the fake monster superalgebra corresponding to elements of 2 aut e 8 of odd order they give many new infinite product identities and new examples of generalized kac moody superalgebras we also obtain generalizations of jacobi s identity on theta functions

controllability observability and duality in singular systems the concepts of controllability and observability for systems of the form e dot x ax bu y cx e singular are considered a theory is presented which unifies the three main approaches to this topic already existing in the literature the development includes a generalization of the duality theorem from state space theory

minimax observers for linear differential algebraic equations in this note we construct minimax observers for linear stationary differential algebraic equations dae s with bounded uncertain inputs given noisy measurements we prove a new duality principle and show that a finite infinite horizon minimax observer exists if and only if the dae is ell impulse observable ell detectable remarkably the regularity of the dae is not required

computation of the bit error rate of coherent m ary psk with gray code bit mapping efficient computation of the bit error rate ber for the coherent m ary psk signals with gray code bit mapping is considered a closed form expression for the exact ber of 8 ary psk is presented tight upper and lower bounds on ber are also obtained for m ary psk with larger m

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

observer design for descriptor systems two detectability notions are introduced for the most general class of linear descriptor systems necessary and sufficient conditions are derived for the detectabilities and for the existence of observers depending on existence conditions satisfied the observers can be either in the familiar luenberger form or in a generalized form i e in descriptor form as the original system itself is a unified procedure is provided for the design of observers in both forms

a novel approach to the design of unknown input observers a novel state estimator design scheme for linear dynamical systems driven by partially unknown inputs is presented it is assumed that there is no information available about the unknown inputs and thus no prior assumption is made about the nature of these inputs a simple approach for designing a reduced order unknown input observer uio with pole placement capability is proposed by carefully examining the dynamic system involved and simple algebraic manipulations it is possible to rewrite equations eliminating the unknown inputs from part of the system and to put them into a form where it could be partitioned into two interconnected subsystems one of which is directly driven by known inputs only this makes it possible to use a conventional luenberger observer with a slight modification for the purpose of estimating the state of the system as a result it is also possible to state similar necessary and sufficient conditions to those of a conventional observer for the existence of a stable estimator and also arbitrary placement of the eigenvalues of the observer the design and computational complexities involved in designing uios are greatly reduced in the proposed approach

observability of singular systems with commensurate time delays and neutral terms this paper deals with the observability problem of a sort of singular systems with commensurate time delays in the trajectories of the system in the time derivative of the trajectories neutral terms and in the output system by using a recursive algorithm sufficient conditions easy testable are proposed for guaranteeing the backward and the algebraic observability of the system this condition implies that the trajectories of the system can be reconstructed by using the actual and past values of the system output

observability of singular time delay systems with unknown inputs in this manuscript we give sufficient conditions guaranteeing the observability of singular linear systems with commensurable delays affected by unknown inputs appearing in both the state equation and the output equation these conditions allow for the reconstruction of the entire state vector using past and actual values of the system output the obtained conditions coincide with known necessary and sufficient conditions of singular linear systems without delays

on existence optimality and asymptotic stability of the kalman filter with partially observed inputs for linear stochastic time varying systems we investigate the properties of the kalman filter with partially observed inputs we first establish the existence condition of a general linear filter when the unknown inputs are partially observed then we examine the optimality of the kalman filter with partially observed inputs finally on the basis of the established existence condition and optimality result we investigate asymptotic stability of the filter for the corresponding time invariant systems it is shown that the results on existence and asymptotic stability obtained in this paper provide a unified approach to accommodating a variety of filtering scenarios as its special cases including the classical kalman filter and state estimation with unknown inputs

unknown input and state estimation for unobservable systems the concept of strong detectability and its relation with the concept of invariant zeros is reviewed for strongly detectable systems which includes the strongly observable systems it is proposed a hierarchical design of a robust observer whose trajectories converge to those of the original state vector furthermore it is shown that neither left invertibility is a sufficient condition nor strong detectability is a necessary condition to estimate the unknown inputs it is shown that the necessary and sufficient condition for estimating the unknown inputs is that the set of the invariant zeros that do not belong to the set of unobservable modes be within the interior of the left half plane of the complex space this shows that the unknown inputs could be estimated even if it is impossible to estimate the entire state vector of the system two numerical examples illustrate the effectiveness of the proposed estimation schemes

micro and macrodiversity mdpsk on shadowed frequency selective channels the performance of m ary differential phase shift keying mdpsk on frequency selective slow rayleigh fading lognormal shadowed channels with diversity combining is analyzed for mobile and portable applications the use of l branch equal gain postdetection microdiversity combining to mitigate the effects of fading and p port macrodiversity to alleviate the effects of shadowing are investigated four performance criteria are considered for a frequency selective multipath fading intersymbol interference channel these are the short term bit error rate ber the irreducible ber the complementary distribution over the lognormal shadowing of the average ber and the probability that the instantaneous ber exceeds a threshold value averaged over a spatial environment closed form expressions for the four performance criteria are obtained the ber and outage performance results show that diversity combining is an effective method for improving the system performance and hence system reliability when the normalized delay spread is not large it is also seen that in most cases 4dpsk gives the best performance followed by 8dpsk and 2dpsk respectively for a given information throughput

solvability controllability and observability of continuous descriptor systems in this paper we investigate the properties of the continuous descriptor system e dot x t ax t bu t 0 leqt leqb where e a and b are complex and possibly singular matrices and u t is a complex function differentiable sufficiently many times the traditional approach to such systems is to separate the state equations from the algebraic equations however such algorithms usually destroy the natural physically based sparsity and structure of the original system therefore we consider descriptor systems in their original form such systems possess numerous properties not shared by the well known state variable systems first we relate classical theories of matrix pencils to the solvability of descriptor systems then we extend the concepts of reachability controllability and observability of state variable systems to descriptor systems and describe the set of reachable states for descriptor systems

a nonlinear luenberger like observer for nonlinear singular systems this paper investigates observer design problem for a large class of nonlinear singular systems with multiple outputs we firstly regularize the singular system by injecting the derivative of outputs into the system then differential geometric method is applied to transform the regularized system into a simple normal form for which a luenberger like observer is proposed

on the kodaira dimension of orthogonal modular varieties we prove that up to scaling there are only finitely many integral lattices l of signature 2 n with n ge 21 or n 17 such that the modular variety defined by the orthogonal group of l is not of general type in particular when n ge 108 every modular variety defined by an arithmetic group for a rational quadratic form of signature 2 n is of general type we also obtain similar finiteness in n ge 9 for the stable orthogonal groups as a byproduct we derive finiteness of lattices admitting reflective modular form of bounded vanishing order which proves a conjecture of gritsenko and nikulin

causal observability of descriptor systems observability analysis for descriptor systems by using a matrix pencil approach is presented systems under consideration are the most general class of linear continuous descriptor systems a few practically relevant notions of observability particularly the causal observability which corresponds to the existence of a luenberger observer and the corresponding necessary and sufficient test conditions are provided the interpretation of various observabilities in terms of matrix pencils offers insights into the physical meaning of these concepts

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

differentiation by integration with jacobi polynomials in this paper the numerical differentiation by integration method based on jacobi polynomials originally introduced by mboup et al 19 20 is revisited in the central case where the used integration window is centered such a method based on jacobi polynomials was introduced through an algebraic approach 19 20 and extends the numerical differentiation by integration method introduced by lanczos 1956 21 the method proposed here rooted in 19 20 is used to estimate the nth email protected n order derivative from noisy data of a smooth function belonging to at least c n 1 q email protected n in 19 20 where the causal and anti causal cases were investigated the mismodelling due to the truncation of the taylor expansion was investigated and improved allowing a small time delay in the derivative estimation here for the central case we show that the bias error is o h q 2 where h is the integration window length for email protected c n q 2 in the noise free case and the corresponding convergence rate is o d q 1 n 1 q where d is the noise level for a well chosen integration window length numerical examples show that this proposed method is stable and effective

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

high order sliding mode observer for linear systems with unbounded unknown inputs a global observer is designed for strongly detectable systems with unbounded unknown inputs the design of the observer is based on three steps first the system is extended taking the unknown inputs and possibly some of their derivatives as a new state then using a global high order sliding mode differentiator a new output of the system is generated in order to fulfil what we will call the hautus condition which finally allows decomposing the system in new coordinates into two subsystems the first one being unaffected directly by the unknown inputs and the state vector of the second subsystem is obtained directly from the original system output such decomposition permits designing of a luenberger observer for the first subsystem which satisfies the hautus condition i e all the outputs have relative degree one w r t the unknown inputs this procedure enables one to estimate the state and the unknown inputs using the least number of differentiations possible simulations are given in order to show the effectiveness of the proposed observer

unknown input proportional multiple integral observer design for linear descriptor systems application to state and fault estimation in this note the problem of observer design for linear descriptor systems with faults and unknown inputs is considered first it is considered that the fault vector function f is s times piecewise continuously differentiable if the s th time derivative of f is null then s integral actions are included into a luenberger observer which is designed such that it estimates simultaneously the state the fault and its finite derivatives face to unknown inputs second when the fault is not time piecewise continuously differentiable but bounded like actuator noise or s th time derivative of fault is not null but bounded too a high gain observer is derived to attenuate the fault impact in estimation errors the considered faults may be unbounded may not be determinist and faults and unknown inputs may affect the state dynamic and plant outputs sufficient conditions for the existence of such observer are given results are illustrated with a differential algebraic power system

accurate computation of the mgf of the lognormal distribution and its application to sum of lognormals sums of lognormal random variables rvs are of wide interest in wireless communications and other areas of science and engineering since the distribution of lognormal sums is not log normal and does not have a closed form analytical expression many approximations and bounds have been developed this paper develops two computational methods for the moment generating function mgf or the characteristic function chf of a single lognormal rv the first method uses classical complex integration techniques based on steepest descent integration the saddle point of the integrand is explicitly expressed by the lambert function the steepest descent optimal contour and two closely related closed form contours are derived a simple integration rule e g the midpoint rule along any of these contours computes the mgf chf with high accuracy the second approach uses a variation on the trapezoidal rule due to ooura and mori importantly the cumulative distribution function of lognormal sums is derived as an alternating series and convergence acceleration via the epsilon algorithm is used to reduce in some cases the computational load by a factor of 106 overall accuracy levels of 13 to 15 significant digits are readily achievable

design of observers for descriptor systems in this note simple and straightforward methods to design full and reduced order observers for linear time invariant descriptor systems are presented the approach for the reduced order observer design is based on the generalized sylvester equation sufficient conditions for the existence of the observers are given an illustrative example is included

disturbance decoupled observer design for descriptor systems in this paper we study the observer design problem for descriptor systems with partly unknown inputs we give necessary and sufficient conditions for the existence of a solution to the disturbance decoupled estimation problem with or without stable error spectrum requiring at the same time that the resulting combined system is regular and of index at most one all results are proved based on a condensed form that can be computed using orthogonal matrix transformations i e transformations that can be implemented in a numerically stable way

reduced order observer design for descriptor systems with unknown inputs a new method for the design of reduced order observers for descriptor systems with unknown inputs is presented the approach is based on the generalized constrained sylvester equation sufficient conditions for the existence of the observer are given

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

observability and detectability of singular linear systems with unknown inputs in this paper the strong observability and strong detectability of a general class of singular linear systems with unknown inputs are tackled the case when the matrix pencil is non regular is comprised i e more than one solution for the differential equation is allowed it is shown that under suitable assumptions the original problem can be studied by means of a regular non singular linear system with unknown inputs and algebraic constraints thus it is shown that for purposes of analysis the algebraic equations can be included as part of an extended system output based on this analysis we obtain necessary and sufficient conditions guaranteeing the observability or detectability of the system in terms of the zeros of the system matrix corresponding algebraic conditions are given in order to test the observability and detectability a formula is provided that expresses the state as high order derivative of a function of the output which allows for the reconstruction of the actual state vector it is shown that the unknown inputs may be reconstructed also

brief paper state and input observability for structured linear systems a graph theoretic approach this paper deals with the state and input observability analysis for structured linear systems with unknown inputs the proposed method is based on a graph theoretic approach and assumes only the knowledge of the system s structure using a particular decomposition of the systems into two subsystems we express in simple graphic terms necessary and sufficient conditions for the generic state and input observability these conditions are easy to check because they are based on comparison of integers and on finding particular subgraphs in a digraph therefore our approach is suited to study large scale systems

solutions of differential algebraic equations as outputs of lti systems application to lq control problems abstract in this paper we propose a unified geometric framework for representing all solutions of a linear time invariant differential algebraic equation dae lti as outputs of classical linear time invariant ordinary differential equations ode lti the proposed framework is then used to solve an lq optimal control problem for dae ltis with rectangular matrices

probability of bit error for mpsk modulation with diversity reception in rayleigh fading and log normal shadowing channel in an additive white gaussian noise channel subject to rayleigh fading and log normal shadowing consideration is given to diversity reception with k port macroscopic selection and l branch microscopic maximal ratio combining and analytical expressions are derived for upper and lower bounds on the bit error probabilities for bpsk qpsk 8 psk and 16 psk modulations the derived expressions can be evaluated at any location in the serving cell error probability can be calculated by averaging over all possible locations within the serving cell by evaluating these bounds numerical results for the following cases are presented without diversity reception microdiversity combining macrodiversity selection and simultaneous use of macrodiversity selection and microdiversity combining for symmetrical arrangement of macrodiversity ports against the lognormal shadowing error probability at the equidistant point gives upper bounds on the error probabilities for most of the area in the serving cell error probability at the equidistant point is a good estimate of the error probability obtained by averaging over all possible locations in the cell

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

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 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

bounds on the distribution of a sum of correlated lognormal random variables and their application the cumulative distribution function cdf of a sum of correlated or even independent lognormal random variables rvs which is of wide interest in wireless communications remains unsolved despite long standing efforts several cdf approximations are thus widely used this letter derives bounds for the cdf of a sum of 2 or 3 arbitrarily correlated lognormal rvs and of a sum of any number of equally correlated lognormal rvs the bounds are single fold integrals of readily computable functions and extend previously known bounds for independent lognormal summands an improved set of bounds are also derived which are expressed as 2 fold integrals for correlated lognormal fading channels new expressions are derived for the moments of the output snr and amount of fading for maximal ratio combining mrc selection combining sc and equal gain combining egc and outage probability expressions for sc

a construction of antisymmetric modular forms for weil representations we give coefficient formulas for antisymmetric vector valued cusp forms with rational fourier coefficients for the weil representation associated to a finite quadratic module the forms we construct always span all cusp forms in weight at least three these formulas are useful for computing explicitly with theta lifts

unknown input functional observability of descriptor systems with neutral and distributed delay effects in this paper a general class of linear systems with time delays is considered which includes linear classical systems linear systems with commensurate delays neutral systems and singular systems with delays after given a formal definition of functional backward observability bo an easily testable condition is found the fulfillment of the obtained condition allows for the reconstruction of the trajectories of the system under consideration using the actual and past values of the system output and some of its derivatives the methodology we follow consists in an iterative algorithm based upon the classical silverman algorithm used for inversion of linear systems by using basic module theory we manage to prove that the proposed algorithm is convergent a direct application of studying functional observability is that a condition can be derived for systems with distributed delays also we do this as a case of study the obtained results are illustrated by two examples one is merely academic but illustrates clearly the kind of systems which the proposed methodology works for and the other is a practical system with distributed delays

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

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

observers for singular systems the following three major aspects of the design of observers for linear time invariant singular systems are resolved the necessary and sufficient conditions for the problem to have a solution the order of the minimal observer and the properties of the closed loop system separation principle an algorithm is presented for the construction of the minimal order observer

on the strong observability in linear time varying singular systems abstract in this article the strong observability conditions necessary and sufficient conditions originally stated for the state reconstruction of linear systems with unknown inputs are extended to the class of linear time varying singular systems this novel local in a time interval of interest conditions are given under a constant rank assumption a method to reconstruct the state in finite time in linear time varying singular systems despite the presence of unknown inputs is presented the proposed observer consists of two parts one part that ensures the asymptotic convergence of the estimation errors to a bounded region around the origin and another one that ensures the finite time convergence of the state estimates to their actual values

full order observers for linear systems with unknown inputs this note presents a simple method to design a full order observer for linear systems with unknown inputs the necessary and sufficient conditions for the existence of the observer are given

interval observer for a class of uncertain nonlinear singular systems this paper investigates the problem of observer design for a general class of linear singular time delay systems in which the time delays are involved in the state the output and the known input if there exists the involvement of the delay could be multiple which however is rarely studied in the literature sufficient conditions are proposed which guarantee the existence of a luenberger like observer for the general system

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

observer design for generalized state space systems with unknown inputs abstract the following four major aspects for the observer design problem for generalized state space systems with unknown inputs are resolved necessary and sufficient conditions for the problem to have a solution the order of the minimal observer the properties of the closed loop system separation principle and general analytical expressions of the minimal order observer matrices all above results are first in the field

observers for a class of nonlinear singular systems this technical note considers the observers design for a class of nonlinear singular systems the approach is based on the new parameterization of the generalized sylvester equations solutions the conditions for the existence of these observers are given and sufficient conditions for their stability are derived using linear matrix inequalities lmis formulation reduced order minimal order and full order observers are designed by a unified method a numerical example is presented to demonstrate the effectiveness of the developed method

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

automorphic products of singular weight for simple lattices we classify the simple even lattices of square free level and signature 2 n n ge 4 a lattice is called simple if the space of cusp forms of weight 1 n 2 for the dual weil representation of the lattice is trivial for a simple lattice every formal principal part obeying obvious conditions is the principal part of a vector valued modular form using this we determine all holomorphic borcherds products of singular weight arising from vector valued modular forms with non negative principal part for the simple lattices we construct the corresponding vector valued modular forms by eta products and compute expansions of the automorphic products at different cusps