minimum probability of error for asynchronous gaussian multiple access channels consider a gaussian multiple access channel shared by k users who transmit asynchronously independent data streams by modulating a set of assigned signal waveforms the uncoded probability of error achievable by optimum multiuser detectors is investigated it is shown that the k user maximum likelihood sequence detector consists of a bank of single user matched filters followed by a viterbi algorithm whose complexity per binary decision is o 2 k the upper bound analysis of this detector follows an approach based on the decomposition of error sequences the issues of convergence and tightness of the bounds are examined and it is shown that the minimum multiuser error probability is equivalent in the iow noise region to that of a single user system with reduced power these results show that the proposed multiuser detectors afford important performance gains over conventional single user systems in which the signal constellation carries the entire burden of complexity required to achieve a given performance level

asymptotic multiuser efficiencies for decision directed multiuser detectors the asymptotic multiuser efficiencies ames are derived for various classes of decision directed multiuser detectors including multistage detectors and decision feedback detectors novel classes of soft decision multistage detectors are proposed and analyzed each class is specified in part by a soft decision nonlinearity such as a symmetric quantizer or a linear clipper closed form expressions for two user ames are derived for soft decision two stage detectors and can be used as a design criterion to optimize the soft decision nonlinearities for a special case of two synchronous users the soft decision two stage detector using an optimized linear clipper with either conventional or decorrelated tentative decisions is shown to achieve optimum ame upper and lower bounds on the ame are obtained for decision feedback detectors using either conventional or decorrelated tentative decisions it is demonstrated that decision directed multiuser detectors with conventional tentative decisions have low near far resistance compared to those with decorrelated tentative decisions

idempotent relations and the conjecture of birch and swinnerton dyer this is a special case of the main theorem from this 1 follows immediately by using the results of milne 9 a secondary aim of this paper is to introduce twisted idempotent relations which generalize the idempotent relations of kani 5 and kani rosen 6 we show how to use such relations to obtain similar results involving twists of abelian varieties cf main theorem which enable us to give some interesting applications cf theorem 11 and corollaries 12 14

symmetric quadrature rules on a triangle abstract we present a class of quadrature rules on triangles in r 2 which somewhat similar to gaussian rules on intervals in r 1 have rapid convergence positive weights and symmetry by a scheme combining simple group theory and numerical optimization we obtain quadrature rules of this kind up to the order 30 on triangles this scheme essentially a formalization and generalization of the approach used by lyness and jespersen over 25 years ago can be easily extended to other regions in r 2 and surfaces in higher dimensions such as squares spheres we present example formulae and relevant numerical results

symmetric gauss legendre quadrature formulas for composite numerical integration over a triangular surface abstract this paper first presents a gauss legendre quadrature method for numerical integration of i t f x y d x d y where f x y is an analytic function in x y and t is the standard triangular surface x y 0 x y 1 x y 1 in the cartesian two dimensional x y space we then use a transformation x x y y to change the integral i to an equivalent integral s f x y x y d d where s is now the 2 square in space 1 1 we then apply the one dimensional gauss legendre quadrature rules in and variables to arrive at an efficient quadrature rule with new weight coefficients and new sampling points we then propose the discretisation of the standard triangular surface t into n 2 right isosceles triangular surfaces t i i 1 1 n 2 each of which has an area equal to 1 2 n 2 units we have again shown that the use of affine transformation over each t i and the use of linearity property of integrals lead to the result i i 1 n n t i f x y d x d y 1 n 2 t h x y d x d y where h x y i 1 n n f x i x y y i x y and x x i x y and y y i x y refer to affine transformations which map each t i in x y space into a standard triangular surface t in x y space we can now apply gauss legendre quadrature formulas which are derived earlier for i to evaluate the integral i 1 n 2 t h x y d x d y we observe that the above procedure which clearly amounts to composite numerical integration over t and it converges to the exact value of the integral t f x y d x d y for sufficiently large value of n even for the lower order gauss legendre quadrature rules we have demonstrated this aspect by applying the above explained composite numerical integration method to some typical integrals

solving a class of optimum multiuser detection problems with polynomial complexity we identify a class of optimum multiuser detection problems which can be solved with polynomial complexity in the number of users the identification is based on transforming a quadratic 0 1 programming problem into an equivalent problem in graph theory for a synchronous direct sequence code division multiple access cdma system the result translates to designing a set of pseudorandom codes with the property that the cross correlation between every pair of codes in the set over one symbol period is nonpositive we give two sets of codes with good correlation properties that fall within this class finally we derive a bound on the cardinality of a signal set in an n dimensional space having the property that the cross correlation between every pair of signals in the set is nonpositive

mimo detection by lagrangian dual maximum likelihood relaxation reinterpreting regularized lattice decoding this paper considers lattice decoding for multi input multi output mimo detection under pam constellations a key aspect of lattice decoding is that it relaxes the symbol bound constraints in the optimal maximum likelihood ml detector for faster implementations it is known that such a symbol bound relaxation may lead to a damaging effect on the system performance for this reason regularization was proposed to mitigate the out of bound symbol effects in lattice decoding however minimum mean square error mmse regularization is the only method of choice for regularization in the present literature we propose a systematic regularization optimization approach considering a lagrangian dual relaxation ldr of the ml detection problem as it turns out the proposed ldr formulation is to find the best diagonally regularized lattice decoder to approximate the ml detector and all diagonal regularizations including the mmse regularization can be subsumed under the ldr formalism we show that for the 2 pam case strong duality holds between the ldr and ml problems also for general pam we prove that the ldr problem yields a duality gap no worse than that of the well known semidefinite relaxation method to physically realize the proposed ldr the projected subgradient method is employed to handle the ldr problem so that the best regularization can be found the resultant method can physically be viewed as an adaptive symbol bound control wherein regularized lattice decoding is recursively performed to correct the decision simulation results show that the proposed ldr approach can outperform the conventional mmse based lattice decoding approach

a block alternating likelihood maximization approach to multiuser detection in this paper we address the maximum likelihood ml multiuser detection problem for asynchronous code division multiple access cdma systems with multiple receiver antennas in frequency selective fading environments multiuser ml detection mld in this case provides attractive symbol error performance but it requires the solution of a large scale combinatorial optimization problem to deal with the computational complexity of this problem we propose an efficient approximation method based on a block alternating likelihood maximization balm principle the idea behind balm is to decompose the large scale mld problem into smaller subproblems assuming binary or quaternary phase shift keying bpsk or qpsk which are often employed in cdma the combinatorial subproblems are then accurately and efficiently approximated by the semidefinite relaxation sdr algorithm an algorithm that has been recently shown to lead to quasi ml performance in synchronous cdma scenarios simulation results indicate that this balm detector provides close to optimal bit error rate ber performance the balm principle is quite flexible and we demonstrate this flexibility by extending balm to multicarrier mc multiuser systems by exploiting the special signal correlation structure of mc systems we develop a variation of balm in which dynamic programming dp is used to solve the subproblems it is shown using simulations that the ber performance of this dp based balm detector is as promising as that of the sdr based balm detector

symmetric numerical integral formulas for regular polygons the theory of interpolatory integration formulas in n dimensions is combined with group theory to classify and generate symmetric integration formulas in a systematic way the points are computed as the common zeros of a set of polynomials these polynomials constitute a canonical basis for the real ideal belonging to the formula for a symmetric formula they can be chosen to span a representation of the symmetry group for regions with high symmetry and formulas of low degree this leads to only a few distinct possibilities and the existence of the corresponding formulas is easily checked in general however the canonical basis is not completely determined by symmetry and degree and a suitable choice has to be made the application to regular polygons in two dimensions is discussed and formulas are presented for polygons with three five six seven and eight vertices with varying degrees of exactness up to order 27 for the hexagon

a multiuser receiver for code division multiple access communications over multipath channels a multiuser communication system is considered where k users share a channel with multipath propagation by using code division for multiple access data modulation is carried out by binary phase shift keying and direct sequence spread spectrum signaling the micro cellular communication media is modeled as a frequency selective fading channel with multipath propagation the multipath diversity of the received signals from the k users is exploited by a bank of k rake correlators algorithms based on the maximum likelihood rule have been developed for estimating the complex channel coefficients as well as for detection of the desired data packets from the sufficient statistics provided by the rake correlators the performance of the resulting multiuser detector is evaluated analytically and via monte carlo simulations the results indicate that the estimator of the channel coefficients has a variance close to the cramer rao lower bound and that the proposed multiuser detector is capable of eliminating the near far effect as well as processing the signals propagated through multiple paths

blind adaptive multiuser detection the decorrelating detector and the linear minimum mean square error mmse detector are known to be effective strategies to counter the presence of multiuser interference in code division multiple access channels in particular those multiuser detectors provide optimum near far resistance when training data sequences are available the mmse multiuser detector can be implemented adaptively without knowledge of signature waveforms or received amplitudes this paper introduces an adaptive multiuser detector which converges for any initialization to the mmse detector without requiring training sequences this blind multiuser detector requires no more knowledge than does the conventional single user receiver the desired user s signature waveform and its timing the proposed blind multiuser detector is made robust with respect to imprecise knowledge of the received signature waveform of the user of interest

cdma multiuser detection a nonlinear programming approach the optimum receiver to detect the bits of multiple code division multiple access cdma users has an exponential complexity in the number of active users in the system consequently many suboptimum receivers have been developed to achieve good performance with less complexity we take the approach of approximating the solution of the optimum multiuser detection problem omud using nonlinear programming relaxations first we observe that some popular suboptimum receivers indeed correspond to relaxations of the optimal detection problem in particular one proposed approximation method yields to iterative solutions which correspond to previously proposed heuristic nonlinear detectors using a nonlinear programming approach we identify the convergence properties of these iterative detectors secondly we propose a relaxation that yields a receiver which we call the generalized minimum mean squared error detector we give a simple iterative implementation of the detector its performance is evaluated and comparisons to other suboptimum detection schemes are given

speed and accuracy comparison of techniques for multiuser detection in synchronous cdma in this letter we compare the complexity and efficiency of several methods used for multiuser detection in a synchronous code division multiple access system various methods are discussed including decision feedback df detection group decision feedback gdf detection coordinate descent quadratic programming with constraints space alternating generalized em sage detection tabu search a boltzmann machine detector semidefinite relaxation probabilistic data association pda branch and bound bbd and the sphere decoding sd method the efficiencies of the algorithms defined as the probability of group detection error divided by the number of floating point computations are compared under various situations of particular interest is the appearance of an efficient frontier of algorithms primarily composed of df detector gdf detector pda detector the bbd optimal algorithm and the sd method the efficient frontier is the convex hull of algorithms as plotted on probability of error versus computational demands axes algorithms not on this efficient frontier can be considered dominated by those that are

high degree efficient symmetrical gaussian quadrature rules for the triangle gaussian quadrature is required for the computation of matrices based on the isoparametric formulztion of the finite element method a brief review of existing quadrature rules for the triangle is given and the method for the determination of high degree efficient symmetrical rules for the triangle is discussed new quadrature rules of degree 12 20 are presented and a short fortran program is included

linear multiuser detectors for synchronous code division multiple access channels under the assumptions of symbol synchronous transmissions and white gaussian noise the authors analyze the detection mechanism at the receiver comparing different detectors by their bit error rates in the low background noise region and by their worst case behavior in a near far environment where the received energies of the users are not necessarily similar optimum multiuser detection achieves important performance gains over conventional single user detection at the expense of computational complexity that grows exponentially with the number of users it is shown that in the synchronous case the performance achieved by linear multiuser detectors is similar to that of optimum multiuser detection attention is focused on detectors whose linear memoryless transformation is a generalized inverse of the matrix of signature waveform crosscorrelations and on the optimum linear detector it is shown that the generalized inverse detectors exhibit the same degree of near far resistance as the optimum multiuser detectors the optimum linear detector is obtained

mmse interference suppression for direct sequence spread spectrum cdma we consider interference suppression for direct sequence spread spectrum code division multiple access cdma systems using the minimum mean squared error mmse performance criterion the conventional matched filter receiver suffers from the near far problem and requires strict power control typically involving feedback from receiver to transmitter for acceptable performance multiuser detection schemes previously proposed mitigate the near far problem but are complex and require explicit knowledge or estimates of the interference parameters in this paper we present and analyze several new mmse interference suppression schemes which have the advantage of being near far resistant to varying degrees depending on their complexity and can be implemented adaptively when interference parameters are unknown and or time varying numerical results are provided that show that these schemes offer significant performance gains relative to the matched filter receiver we conclude that mmse detectors can alleviate the need for stringent power control in cdma systems and may be a practical alternative to the matched filter receiver

on the birch swinnerton dyer quotients modulo squares let a be an abelian variety over a number field k an identity between the l functions l a k i s for extensions k i of k induces a conjectural relation between the birch swinnerton dyer quotients we prove these relations modulo finiteness of lii and give an analogous statement for selmer groups based on this we develop a method for determining the parity of various combinations of ranks of a over extensions of k as one of the applications we establish the parity conjecture for elliptic curves assuming finiteness of iii e k e 2 6 and some restrictions on the reduction at primes above 2 and 3 the parity of the mordell weil rank of e k agrees with the parity of the analytic rank as determined by the root number we also prove the p parity conjecture for all elliptic curves over q and all primes p the parities of the p selmer rank and the analytic rank agree

generalized gaussian quadrature rules on arbitrary polygons in this paper we present a numerical algorithm based on group theory and numerical optimization to compute efficient quadrature rules for integration of bivariate polynomials over arbitrary polygons these quadratures have desirable properties such as positivity of weights and interiority of nodes and can readily be used as software libraries where numerical integration within planar polygons is required we have used this algorithm for the construction of symmetric and non symmetric quadrature rules over convex and concave polygons while in the case of symmetric quadratures our results are comparable to available rules the proposed algorithm has the advantage of being flexible enough so that it can be applied to arbitrary planar regions for the integration of generalized classes of functions to demonstrate the efficiency of the new quadrature rules we have tested them for the integration of rational polygonal shape functions over a regular hexagon for a relative error of 10 8 in the computation of stiffness matrix entries one needs at least 198 evaluation points when the region is partitioned whereas 85 points suffice with our quadrature rule copyright 2009 john wiley sons ltd

application of computational geometry to multiuser detection in cdma the maximum likelihood multiuser detection problem in code division multiple access is known to be an optimization problem with an objective function that is required to be optimized over a combinatorial decision region conventional suboptimal detectors relax the combinatorial decision region by a convex region without altering the objective function to be optimized we take an approach wherein the objective function is reduced to a form appropriate for the application of a polynomial complexity algorithm in computational geometry while keeping the decision region combinatorial the resulting detector allows a tradeoff between performance and computational complexity the bit error rate performance of the detector has been found to be better than the decorrelator and the linear minimum mean square error detectors for the same level of complexity

iterative multiuser receivers for cdma channels an em based approach maximum likelihood detection for the multiuser code division multiple access cdma channel is prohibitively complex this paper considers new iterative multiuser receivers based on the expectation maximization em algorithm and related more powerful space alternating algorithms the latter algorithms include the sage algorithm and a new missing parameter space alternating algorithm that alternately updates individual parameter components or treats them as probabilistic missing data application of these em based algorithms to the problem of discrete parameter estimation i e data detection in the gaussian multiple access channel leads to a variety of convergent receiver structures that incorporate soft decision feedback for interference cancellation and or sequential updating of iterative bit estimates convergence and performance analyses are based on well known properties of the em algorithm and on numerical simulation

designing low complexity near optimal multiple symbol detectors for impulse radio uwb systems multiple symbol detection has recently caught attention in ultrawideband uwb communications because of its high performance without requiring explicit channel estimation in this context the generalized likelihood ratio test glrt for multiple symbol detection problem is developed to jointly detect multiple symbols which exhibits considerable error performance improvement over transmitted reference transmissions methods unfortunately the glrt is a boolean quadratic programming bqp problem which is generally nondeterministic polynomial hard np hard in this paper we propose two near optimal detectors with polynomial complexity the first detector performs semidefinite relaxation to approximately solve the bqp problem called sdp msd the second detector is based on a sphere based relaxation of the bqp problem we refer to this as modified unconstrained relaxation multiple symbol detector mur msd both detectors achieve near optimal performance while the sdp msd performs slightly better than the mur msd at the price of higher computational complexity furthermore the mur msd can be treated as a further nontight relaxation of sdp msd simulations are utilized to validate our findings and to demonstrate performance robustness to multiaccess interference

on the conjecture of birch and swinnerton dyer in this paper we complete rubin s partial verification of the conjecture for a large class of elliptic curves with complex multiplication by q c

quadrature rules for triangular and tetrahedral elements with generalized functions quadrature rules are developed for exactly integrating products of polynomials and generalized functions over triangular and tetrahedral domains these quadrature rules greatly simplify the implementation of finite element methods that involve integrals over volumes and interfaces that are not coincident with the element boundaries specifically the integrands considered here consist of a quadratic polynomial multiplied by a heaviside or dirac delta function operating on a linear polynomial this form allows for exact integration of expressions obtained from linear finite elements over domains and interfaces defined by a linear level set function exact quadrature rules are derived that involve fixed quadrature point locations with weights that depend continuously on the nodal level set values compared with methods involving explicit integration over subdomains the quadrature rules developed here accommodate degenerate interface geometries without any need for special consideration and provide analytical jacobian information describing the dependence of the integrals on the nodal level set values the accuracy of the method is demonstrated for a simple conduction problem with the neumann and robin type boundary conditions copyright 2007 john wiley sons ltd

generalized gaussian quadrature rules for systems of arbitrary functions a numerical algorithm is presented for the construction of generalized gaussian quadrature rules originally introduced by s karlin and w studden over three decades ago the quadrature rules to be discussed possess most of the desirable properties of the classical gaussian integration formulae such as positivity of the weights rapid convergence mathematical elegance etc the algorithm is applicable to a wide class of functions including smooth functions not necessarily polynomials as well as functions with end point singularities such as those encountered in the solution of integral equations complex analysis potential theory and several other areas the performance of the algorithm is illustrated with several numerical examples

symbol error rate performance of box relaxation decoders in massive mimo the maximum likelihood ml decoder for symbol detection in large multiple input multiple output wireless communication systems is typically computationally prohibitive in this paper we study a popular and practical alternative namely the box relaxation optimization bro decoder which is a natural convex relaxation of the ml for independent identically distributed real gaussian channels with additive gaussian noise we obtain exact asymptotic expressions for the symbol error rate ser of the bro the formulas are particularly simple they yield useful insights and they allow accurate comparisons to the matched filter bound mfb and to linear decoders such as zero forcing and linear minimum mean square error for binary phase shift keying signals the ser performance of the bro is within 3 db of the mfb for square systems and it approaches the mfb as the number of receive antennas grows large compared to the number of transmit antennas our analysis further characterizes the empirical density function of the solution of the bro and shows that error events for any fixed number of symbols are asymptotically independent the fundamental tool behind the analysis is the convex gaussian min max theorem

asymptotic performance of discrete valued vector reconstruction via box constrained optimization with sum of ell _ 1 regularizers in this paper we analyze the asymptotic performance of convex optimization based discrete valued vector reconstruction from linear measurements we firstly propose a box constrained version of the conventional sum of absolute values soav optimization which uses a weighted sum of ell 1 regularizers as a regularizer for the discrete valued vector we then derive the asymptotic symbol error rate ser performance of the box constrained soav box soav optimization theoretically by using the convex gaussian min max theorem cgmt we also derive the asymptotic distribution of the estimate obtained by the box soav optimization on the basis of the asymptotic results we can obtain the optimal parameters of the box soav optimization in terms of the asymptotic ser moreover we can also optimize the quantizer to obtain the final estimate of the unknown discrete valued vector simulation results show that the empirical ser performance of box soav and the conventional soav is very close to the theoretical result for box soav when the problem size is sufficiently large we also show that we can obtain better ser performance by using the proposed asymptotically optimal parameters and quantizers compared to the case with some fixed parameter and a naive quantizer

semidefinite relaxation based multiuser detection for m ary psk multiuser systems because of the powerful symbol error performance of multiuser maximum likelihood ml detection recently there has been much interest in seeking effective ways of approximating multiuser ml detection mld with affordable computational costs it has been illustrated that for the synchronous code division multiple access cdma scenario the so called semidefinite relaxation sdr algorithm can accurately and efficiently approximate multiuser mld this sdr mld algorithm however can only handle binary and quadratic phase shift keying psk symbol constellations in this sequel we propose an extended sdr algorithm for mld with m ary psk mpsk constellations for the synchronous cdma scenario the proposed sdr algorithm provides an attractive polynomial time complexity order of k sup 3 5 where k is the number of users simulation results indicate that the proposed detector provides improved symbol error performance compared with several commonly used multiuser detectors