on the kz reduction the korkine zolotareff kz reduction is one of the often used reduction strategies for lattice decoding in this paper we first investigate some important properties of kz reduced matrices specifically we present a linear upper bound on the hermit constant which is around 7 8 times of the existing sharpest linear upper bound and an upper bound on the kz constant which is polynomially smaller than the existing sharpest one we also propose upper bounds on the lengths of the columns of kz reduced matrices and an upper bound on the orthogonality defect of kz reduced matrices which are even polynomially and exponentially smaller than those of boosted kz reduced matrices respectively then we derive upper bounds on the magnitudes of the entries of any solution of a shortest vector problem svp when its basis matrix is lll reduced these upper bounds are useful for analyzing the complexity and understanding numerical stability of the basis expansion in a kz reduction algorithm finally we propose a new kz reduction algorithm by modifying the commonly used schnorr euchner search strategy for solving svps and the basis expansion method proposed by zhang et al simulation results show that the new kz reduction algorithm is much faster and more numerically reliable than the kz reduction algorithm proposed by zhang et al especially when the basis matrix is ill conditioned

a theoretical and numerical analysis of the measured equation of invariance the measured equation of invariance mei has been previously introduced to efficiently and accurately handle the boundary truncation for finite methods the present authors give a theoretical analysis that provides several important insights into the capabilities of the mei from the numerical study they can explain why the mei works better than one would expect both the theoretical and the numerical analyses demonstrate that the accuracy of the solution is dependent on the electrical size of the geometry as well as the distance between the mesh boundary and the geometry from the analysis the authors propose a new set of metrons that is less sensitive to the excitation than the previously proposed sinusoidal metrons

new bounds in some transference theorems in the geometry of numbers the aim of this paper is to give new bounds in certain inequalities concerning mutually reciprocal lattices in r to formulate the problem we have to introduce some notation and terminology we shall treat ir as an n dimensional euclidean space with the norm ii ii and metric d the inner product of vectors u v will be denoted by uv we shall usually write u 2 instead of uu the closed and open unit balls in f will be denoted by b and b respectively if a c ir then span a and a z denote the linear subspace spanned over a and the orthogonal complement of a in p a lattice in ir is an additive subgroup of ir generated by n linearly independent vectors the family of all lattices in ir will be denoted by a given a lattice l e a we define the polar dual reciprocal lattice l in the usual way l u u v z for each v l

determining the small solutions to s unit equations in this paper we generalize the method of wildanger for finding small solutions to unit equations to the case of s unit equations the method uses a minor generalization of the lll based techniques used to reduce the bounds derived from transcendence theory followed by an enumeration strategy based on the fincke pohst algorithm the method used reduces the computing time needed from mips years down to minutes

trellis search based soft input soft output mimo detector algorithm and vlsi architecture in this paper we propose a trellis search based soft input soft output detection algorithm and its very large scale integration vlsi architecture for iterative multiple input multiple output mimo receivers we construct a trellis diagram to represent the search space of a transmitted mimo signal with the trellis model we evenly distribute the workload of candidates searching among multiple trellis nodes for parallel processing the search complexity is significantly reduced because the number of candidates is greatly limited at each trellis node by leveraging the trellis structure we develop an approximate log map algorithm by using a small list of largest exponential terms to compute the llr log likelihood ratio values the trellis search based detector has a fixed complexity and is very suitable for parallel vlsi implementation as a case study we have designed and synthesized a trellis search based soft input soft output mimo detector for a 4 4 16 qam system using a 1 08 v tsmc 65 nm technology the detector can achieve a maximum throughput of 1 7 gb s with a core area of 1 58 mm2

faster recursions in sphere decoding most of the calculations in standard sphere decoders are redundant in the sense that they either calculate quantities that are never used or calculate some quantities more than once a new method which is applicable to lattices as well as finite constellations is proposed to avoid these redundant calculations while still returning the same result pseudocode is given to facilitate immediate implementation simulations show that the speed gain with the proposed method increases linearly with the lattice dimension at dimension 60 the new algorithms avoid about 75 of all floating point operations

approximate maximum likelihood period estimation from sparse noisy timing data the problem of estimating the period of a series of periodic events is considered under the condition where the measurements of the times of occurrence are noisy and sparse the problem is common to bit synchronisation in telecommunications and pulse train parameter estimation in electronic support among other applications two new algorithms are presented which represent different compromises between computational and statistical efficiency the first extends the separable least squares line search sls2 algorithms of sidiropoulos et al having very low computational complexity while attaining good statistical accuracy the second is an approximate maximum likelihood algorithm based on a low complexity lattice search and is found to achieve excellent accuracy

measured equation of invariance a new concept in field computations numerical computations of frequency domain field problems or elliptical partial differential equations may be based on differential equations or integral equations the new concept of field computation presented in this paper is based on the postulate of the existence of linear equations of the discretized nodal values of the fields different from the conventional equations but leading to the same solutions the postulated equations are local and invariant to excitation it is shown how the equations can be determined by a sequence of measures the measured equations are particularly useful at the mesh boundary where the finite difference methods fail the measured equations do not assume the physical condition of absorption so they are also applicable to concave boundaries using the measured equations one can terminate the finite difference mesh very close to the physical boundary and still obtain robust solutions it will definitely make a great impact on the way one applies finite difference and finite element methods in many problems computational results using the measured equations of invariance in two and three dimensions are presented

an analytical characterization of the error in the measured equation of invariance in previous publications the authors have numerically shown that the measured equation of invariance mei is not invariant to excitation the major implication is that an appropriate set of metrons should be selected for each geometry and excitation in question an argument however can be raised against these findings one can claim that the mei is indeed invariant and that any discrepancies are entirely due to the mesh discretization error the authors disprove this claim by a counterexample they perform an analytical study of the mei as applied to a perfectly conducting circular cylinder with a fixed choice of metrons they then investigate the behavior of the mei when the electrical radius of the cylinder becomes large and when the nodal separation goes to zero they prove that even as the mei residual goes to zero the error in the mei solution remains finite and cannot be reduced below a certain limit

mixed q linear space time codes a new modulation method for linear space time codes is proposed based on using constellations of different sizes for different symbols it is shown that the proposed method significantly reduces the complexity of the sphere decoding algorithm the complexity reduction is more pronounced in high rate codes where each code matrix carries a large number of symbols we also show that the choice of constellation size provides a tradeoff between performance and complexity using this some guidelines for choosing constellation size are presented as one introduces more constellation disparity in the code the complexity is further reduced while the performance loss grows typically a complexity reduction of one to two orders of magnitude can be achieved at the expense of about 3 db coding gain we suggest a simple modification in our design to reduce this loss to about 2 db

on the complexity of sphere decoding in digital communications sphere decoding has been suggested by a number of authors as an efficient algorithm to solve various detection problems in digital communications in some cases the algorithm is referred to as an algorithm of polynomial complexity without clearly specifying what assumptions are made about the problem structure another claim is that although worst case complexity is exponential the expected complexity of the algorithm is polynomial herein we study the expected complexity where the problem size is defined to be the number of symbols jointly detected and our main result is that the expected complexity is exponential for fixed signal to noise ratio snr contrary to previous claims the sphere radius which is a parameter of the algorithm must be chosen to ensure a nonvanishing probability of solving the detection problem this causes the exponential complexity since the squared radius must grow linearly with problem size the rate of linear increase is however dependent on the noise variance and thus the rate of the exponential function is strongly dependent on the snr therefore sphere decoding can be efficient for some snr and problems of moderate size even though the number of operations required by the algorithm strictly speaking always grows as an exponential function of the problem size

cayley differential unitary space time codes one method for communicating with multiple antennas is to encode the transmitted data differentially using unitary matrices at the transmitter and to decode differentially without knowing the channel coefficients at the receiver since channel knowledge is not required at the receiver differential schemes are ideal for use on wireless links where channel tracking is undesirable or infeasible either because of rapid changes in the channel characteristics or because of limited system resources although this basic principle is well understood it is not known how to generate good performing constellations of unitary matrices for any number of transmit and receive antennas and for any rate this is especially true at high rates where the constellations must be rapidly encoded and decoded we propose a class of cayley codes that works with any number of antennas and has efficient encoding and decoding at any rate the codes are named for their use of the cayley transform which maps the highly nonlinear stiefel manifold of unitary matrices to the linear space of skew hermitian matrices this transformation leads to a simple linear constellation structure in the cayley transform domain and to an information theoretic design criterion based on emulating a cauchy random matrix moreover the resulting cayley codes allow polynomial time near maximum likelihood ml decoding based on either successive nulling canceling or sphere decoding simulations show that the cayley codes allow efficient and effective high rate data transmission in multiantenna communication systems without knowing the channel

comments on a theoretical and numerical analysis of the measured equation of invariance in the original paper see ibid vol 42 no 8 p 1097 1105 1994 the authors claimed to have found theoretical insights into the measured equations of invariance mei method their first insight was a proof that the postulate of invariance was wrong and their second insight led to the discovery of an optimum set of metrons metrons are considered to be possible induced current densities due to some unknown incident fields they also presented a series of computational results to highlight their theories this article points out the defects in the analyses and conclusions presented by the authors there are two things in the paper which are basically incorrect one is that the authors consider a zero in a numerical formulation to be an absolute zero the other is that they assume the invariance to excitations to be the same as the invariance to metrons based on these assumptions they have reached conclusions which are actually contradictory to their own calculations this article shows where the defects of their analyses occur and why their two insights are contradictory to each other

on further reduction of complexity in tree pruning based sphere search in this letter we propose an extension of the probabilistic tree pruning sphere decoding ptp sd algorithm that provides further improvement of the computational complexity with minimal extra cost and negligible performance penalty in contrast to the ptp sd that considers the tightening of necessary conditions in the sphere search using per layer radius adjustment the proposed method focuses on the sphere radius control strategy when a candidate lattice point is found for this purpose the dynamic radius update strategy depending on the lattice point found as well as the lattice independent radius selection scheme are jointly exploited as a result while maintaining the effectiveness of the ptp sd further reduction of the computational complexity in particular for high snr regime can be achieved from simulations in multiple input and multiple output mimo channels it is shown that the proposed method provides a considerable improvement in complexity with near ml performance

multiuser detection with an unknown number of users this correspondence studies the problem of separating a signal of interest from co channel interference for the situation when the number of co channel users and their channels are unknown we devise a novel approach to this problem based on a trained statistical mixture model numerical results illustrate that the new method can outperform conventional training based multiuser detection

qr decomposition of laurent polynomial matrices sampled on the unit circle we consider laurent polynomial lp matrices defined on the unit circle of the complex plane qr decomposition of an lp matrix a s yields qr factors q s and r s that in general are neither lp nor rational matrices in this paper we present an invertible mapping that transforms q s and r s into lp matrices furthermore we show that given qr factors of sufficiently many samples of a s it is possible to obtain qr factors of additional samples of a s through application of this mapping followed by interpolation and inversion of the mapping the results of this paper find applications in the context of signal processing for multiple input multiple output mimo wireless communication systems that employ orthogonal frequency division multiplexing ofdm

orbits and lattices for linear random number generators with composite moduli in order to analyze certain types of combinations of multiple recursive linear congruential generators mrgs we introduce a generalized spectral test we show how to apply the test in large dimensions by a recursive procedure based on the fact that such combinations are subgenerators of other mrgs with composite moduli we illustrate this with the well known ranmar generator we also design an algorithm generalizing the procedure to arbitrary random number generators

algorithms for interpolation based qr decomposition in mimo ofdm systems detection algorithms for multiple input multiple output mimo wireless systems based on orthogonal frequency division multiplexing ofdm typically require the computation of a qr decomposition for each of the data carrying ofdm tones the resulting computational complexity will in general be significant motivated by the fact that the channel matrices arising in mimo ofdm systems result from oversampling of a polynomial matrix we formulate interpolation based qr decomposition algorithms an in depth complexity analysis based on a metric relevant for very large scale integration vlsi implementations shows that the proposed algorithms for a sufficiently large number of data carrying tones and sufficiently small channel order provably exhibit significantly smaller complexity than brute force per tone qr decomposition

mimo detection for high order qam based on a gaussian tree approximation this paper proposes a new detection algorithm for mimo communication systems employing high order qam constellations the factor graph that corresponds to this problem is very loopy in fact it is a complete graph hence a straightforward application of the belief propagation bp algorithm yields very poor results our algorithm is based on an optimal tree approximation of the gaussian density of the unconstrained linear system the finite set constraint is then applied to obtain a cycle free discrete distribution simulation results show that even though the approximation is not directly applied to the exact discrete distribution applying the bp algorithm to the cycle free factor graph outperforms current methods in terms of both performance and complexity the improved performance of the proposed algorithm is demonstrated on the problem of mimo detection

electrostatic solution for three dimensional arbitrarily shaped conducting bodies using finite element and measured equation of invariance differential equation techniques such as the finite element fe and finite difference fd have the advantage of sparse system matrices that have relatively small memory requirements for storage and relatively short central processing unit cpu time requirements for solving electrostatic problems however these techniques do not lend themselves as readily for use in open region problems as the method of moments mom because they require the discretization of the space surrounding the object where the mom only requires discretization of the surface of the object a relatively new mesh truncation method known as the measured equation of invariance mei is investigated augmenting the fe method for the solution of electrostatic problems involving three dimensional 3 d arbitrarily shaped conducting objects this technique allows truncation of the mesh as close as two node layers from the object the mei views sparse matrix numerical techniques as methods of determining the weighting coefficients between neighboring nodes and finds those weights for nodes on the boundary of the mesh by assuming viable charge distributions on the surface of the object and using green s function to measure the potentials at the nodes problems in the implementation of the fe mei are discussed and the method is compared against the mom for a cube and a sphere

the icosian code and the e_8 lattice a new 4 times 4 space time code with nonvanishing determinant this paper introduces a new rate 2 full diversity space time code for four transmit antennas and one receive antenna the 4times4 codeword matrix consists of four 2times2 alamouti blocks with entries from q i radic5 and these blocks can be viewed as quaternions which in turn represent rotations in r 3 the alamouti blocks that appear in a codeword are drawn from the icosian ring consisting of all linear combinations of 120 basic rotations corresponding to symmetries of the icosahedron this algebraic structure is different from the golden code but the complex entries are taken from a common underlying field the minimum determinant is bounded below by a constant that is independent of the signal constellation and the new code admits a simple decoding scheme that makes use of a geometric correspondence between the icosian ring and the e 8 lattice

low complexity sphere decoding algorithm for quasi orthogonal space time block codes space time codes can be decoded by the sphere decoding sd algorithm to reduce the complexity and retain maximum likelihood ml performance in this letter the ml metric of quasi orthogonal space time block codes is written into two independent euclidean norms thus sd can be applied to each function independently the new scheme reduces the complexity by at least 85 for systems with four or more transmit antennas compared with the conventional sd algorithm

applications of convex optimization in signal processing and digital communication in the last two decades the mathematical programming community has witnessed some spectacular advances in interior point methods and robust optimization these advances have recently started to significantly impact various fields of applied sciences and engineering where computational efficiency is essential this paper focuses on two such fields digital signal processing and communication in the past the widely used optimization methods in both fields had been the gradient descent or least squares methods both of which are known to suffer from the usual headaches of stepsize selection algorithm initialization and local minima with the recent advances in conic and robust optimization the opportunity is ripe to use the newly developed interior point optimization techniques and highly efficient software tools to help advance the fields of signal processing and digital communication this paper surveys recent successes of applying interior point and robust optimization to solve some core problems in these two fields the successful applications considered in this paper include adaptive filtering robust beamforming design and analysis of multi user communication system channel equalization decoding and detection throughout our emphasis is on how to exploit the hidden convexity convex reformulation of semi infinite constraints analysis of convergence complexity and performance as well as efficient practical implementation

a unified framework for tree search decoding rediscovering the sequential decoder we consider receiver design for coded transmission over linear gaussian channels we restrict ourselves to the class of lattice codes and formulate the joint detection and decoding problem as a closest lattice point search clps here a tree search framework for solving the clps is adopted in our framework the clps algorithm is decomposed into the preprocessing and tree search stages the role of the preprocessing stage is to expose the tree structure in a form matched to the search stage we argue that the forward and feedback matrix filters of the minimum mean square error decision feedback equalizer mmse dfe are instrumental for solving the joint detection and decoding problem in a single search stage it is further shown that mmse dfe filtering allows for solving underdetermined linear systems and using lattice reduction methods to diminish complexity at the expense of a marginal performance loss for the search stage we present a generic method based on the branch and bound bb algorithm and show that it encompasses all existing sphere decoders as special cases the proposed generic algorithm further allows for an interesting classification of tree search decoders sheds more light on the structural properties of all known sphere decoders and inspires the design of more efficient decoders in particular an efficient decoding algorithm that resembles the well known fano sequential decoder is identified the excellent performance complexity tradeoff achieved by the proposed mmse dfe fano decoder is established via simulation results and analytical arguments in several multiple input multiple output mimo and intersymbol interference isi scenarios

validity of the measured equation of invariance the measured equation of invariance mei is derived without any postulates it is shown that the coefficients of the mei are invariant to the field of excitation however the accuracy of the mei solution is closely related to the number of nodes in the mei coupling more nodes improves progressively the accuracy of the mei solution with increasing nodes the matrix problem for the determination of the mei coefficients becomes seriously ill conditioned and generally must be solved using multiple precision arithmetic the consequences of the ill conditioning phenomenon are discussed

a simplex algorithm for rational cp factorization in this paper we provide an algorithm similar to the simplex algorithm which determines a rational cp factorization of a given matrix whenever the matrix allows such a factorization this algorithm can be used to show that every integral completely positive 2 times 2 matrix has an integral cp factorization

soft input soft output single tree search sphere decoding soft input soft output siso detection algorithms form the basis for iterative decoding the computational complexity of siso detection often poses significant challenges for practical receiver implementations in particular in the context of multiple input multiple output mimo wireless communication systems in this paper we present a low complexity siso sphere decoding algorithm based on the single tree search paradigm proposed originally for soft output mimo detection in studer soft output sphere decoding algorithms and vlsi implementation ieee j sel areas commun vol 26 no 2 pp 290 300 feb 2008 the new algorithm incorporates clipping of the extrinsic log likelihood ratios llrs into the tree search which results in significant complexity savings and allows to cover a large performance complexity tradeoff region by adjusting a single parameter furthermore we propose a new method for correcting approximate llrs resulting from sub optimal detectors which often significantly improves detection performance at low additional computational complexity

individual versus social optimization in the allocation of customers to alternative servers customers arrive at a service area according to a poisson process an arriving customer must choose one of k servers without observing present congestion levels the only available information about the kth server is the service time distribution with expected duration k 1 and the cost per unit time of waiting at the kth server hk although service distributions may differ from server to server and need not be exponential it is assumed that they share the same coefficient of variation individuals acting in self interest induce an arrival rate pattern 1 2 k in contrast the social optimum is the arrival rate pattern 1 2 k which minimizes long run average cost per unit time for the entire system the main result is that k s and k s differ systematically individuals overload the servers with the smallest hk k values for an exponential service case with pre emptive lifo service an alternative charging scheme is presented which confirms that differences between individual and social optima occur precisely because individuals fail to consider the inconvenience that they cause to others

open queueing networks in heavy traffic this paper presents heavy traffic limit theorems for the queue length and sojourn time processes associated with open queueing networks these limit theorems state that properly normalized sequences of queue length and sojourn time processes converge weakly to a certain diffusion as the network traffic intensity converges to unity the limit diffusion is reflected brownian motion on the nonnegative orthant this process behaves like brownian motion on the interior of its state space and reflects instantaneously on the boundaries the reflection direction is a constant for each boundary hyperplane

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

reduced complexity soft mimo detection based on causal and noncausal decision feedback we present a reduced complexity soft detection rcsd scheme geared to multiple input multiple output mimo systems with spatial domain multiplexing leading to layered space time or space frequency architecture the proposed algorithm relies on a trellis representation of the mimo signals and a subsequent formulation of constrained depth maximum a posterior map detection in conjunction with soft decision feedback sdf decision feedback is broken into causal and noncausal parts in an effort to maximize the observation window while maintaining a reasonable computational load two variations of rcsd are proposed to utilize the noncausal information in different ways analysis based on mean squared error mse and log likelihood ratio llr associated with symbols in different stages of the trellis is done to develop insight into error propagation in the rcsd algorithm as well as to compare the quality of the soft information obtained by the two rcsd variants error rate simulations are conducted in the context of turbo like iterative demapping and decoding idd the resulting performance and required complexity are compared with those of maximum a posterior map detection soft sphere detection sd as well as the turbo blast processing scheme we observe favorable performance complexity tradeoffs with the proposed soft detection scheme for a number of modulation channel scenarios

lattice decoding for joint detection in direct sequence cdma systems a new joint detection method based on sphere packing lattice decoding is presented in this paper the algorithm is suitable for both synchronous and asynchronous multiple access direct sequence code division multiple access ds cdma systems and it may jointly detect up to 64 users with a reasonable complexity the detection complexity is independent of the modulation size and large m pam or m qam constellations can be used furthermore a theoretical gain analysis is performed in which the multiple access system performance is derived from the lattice parameters

closest point search in lattices in this semitutorial paper a comprehensive survey of closest point search methods for lattices without a regular structure is presented the existing search strategies are described in a unified framework and differences between them are elucidated an efficient closest point search algorithm based on the schnorr euchner 1995 variation of the pohst 1981 method is implemented given an arbitrary point x spl isin spl ropf sup m and a generator matrix for a lattice spl lambda the algorithm computes the point of spl lambda that is closest to x the algorithm is shown to be substantially faster than other known methods by means of a theoretical comparison with the kannan 1983 1987 algorithm and an experimental comparison with the pohst 1981 algorithm and its variants such as the viterbo boutros see ibid vol 45 p 1639 42 1999 decoder modifications of the algorithm are developed to solve a number of related search problems for lattices such as finding a shortest vector determining the kissing number computing the voronoi 1908 relevant vectors and finding a korkine zolotareff 1873 reduced basis

efficient linear programming algorithm to generate the densest lattice sphere packings finding the densest sphere packing in d dimensional euclidean space r d is an outstanding fundamental problem with relevance in many fields including the ground states of molecular systems colloidal crystal structures coding theory discrete geometry number theory and biological systems numerically generating the densest sphere packings becomes very challenging in high dimensions due to an exponentially increasing number of possible sphere contacts and sphere configurations even for the restricted problem of finding the densest lattice sphere packings in this paper we apply the torquato jiao packing algorithm which is a method based on solving a sequence of linear programs to robustly reproduce the densest known lattice sphere packings for dimensions 2 through 19 we show that the tj algorithm is appreciably more efficient at solving these problems than previously published methods indeed in some dimensions the former procedure can be as much as three orders of magnitude faster at finding the optimal solutions than earlier ones we also study the suboptimal local density maxima solutions inherent structures or extreme lattices to gain insight about the nature of the topography of the density landscape

algorithmic enumeration of ideal classes for quaternion orders we provide algorithms to count and enumerate representatives of the right ideal classes of an eichler order in a quaternion algebra defined over a number field we analyze the run time of these algorithms and consider several related problems including the computation of two sided ideal classes isomorphism classes of orders connecting ideals for orders and ideal principalization we conclude by giving the complete list of definite eichler orders with class number at most 2

joint data detection and channel estimation for ofdm systems we develop new blind and semi blind data detectors and channel estimators for orthogonal frequency division multiplexing ofdm systems our data detectors require minimizing a complex integer quadratic form in the data vector the semi blind detector uses both channel correlation and noise variance the quadratic for the blind detector suffers from rank deficiency for this we give a low complexity solution avoiding a computationally prohibitive exhaustive search we solve our data detectors using sphere decoding sd and v blast and provide simple adaptations of the sd algorithm we consider how the blind detector performs under mismatch generalize the basic data detectors to nonunitary constellations and extend them to systems with pilots and virtual carriers simulations show that our data detectors perform well

near optimum soft decision equalization for frequency selective mimo channels in this paper we develop soft decision equalization sde techniques for frequency selective multiple input multiple output mimo channels in the quest for low complexity equalizers with error performance competitive to that of maximum likelihood ml sequence detection we demonstrate that decision feedback equalization dfe based on soft decisions expressed via the posterior probabilities associated with feedback symbols is able to outperform hard decision dfe with a low computational cost that is polynomial in the number of symbols to be recovered and linear in the signal constellation size building on the probabilistic data association pda multiuser detector we present two new mimo equalization solutions to handle the distinctive channel memory the first sde algorithm adopts a zero padded transmission structure to convert the challenging sequence detection problem into a block by block least square formulation it introduces key enhancement to the original pda to enable applications in rank deficient channels and for higher level modulations the second sde algorithm takes advantage of the toeplitz channel matrix structure embodied in an equalization problem it processes the data samples through a series of overlapping sliding windows to reduce complexity and at the same time performs implicit noise tracking to maintain near optimum performance with their low complexity simple implementations and impressive near optimum performance offered by iterative soft decision processing the proposed sde methods are attractive candidates to deliver efficient reception solutions to practical high capacity mimo systems simulation comparisons of our sde methods with minimum mean square error mmse based mimo dfe and sphere decoded quasi ml detection are presented

improvements in closest point search based on dual hkz bases in this paper we review the technique to solve the cvp based on dual hkz bases by j blomer 4 the technique is based on the transference theorems given by banaszczyk 3 which imply some necessary conditions on the coefficients of the closest vectors with respect to a basis whose dual is hkz reduced recursively starting with the last coefficient intervals of length i can be derived for the i th coefficient of any closest vector this leads to n candidates for closest vectors in this paper we refine the necessary conditions derived from the transference theorems giving an exponential reduction of the number of candidates the improvement is due to the fact that the lengths of the intervals are not independent in the original algorithm the candidates for a coefficient pair v i v i 1 correspond to the integer points in a rectangle of volume email protected i 1 in our analysis we show that the candidates for v i v i 1 in fact lie in an ellipse with transverse and conjugate diameter i 1 respectively i this reduces the expected number of points to be enumerated by an exponential factor of about 0 886 n we further show how a choice of the coefficients v n v i 1 influences the interval from which v i can be chosen numerical computations show that these considerations allow to bound the expected number of points to be enumerated by n 0 7 5 n for 10

on the complexity of sphere decoding for differential detection multiple symbol differential detection msdd for multiple input multiple output rayleigh fading channels is considered msdd which jointly processes blocks of n received symbols to detect n 1 data symbols allows for power efficient transmission without requiring channel state information at the receiver in previous work the authors showed that computational efficient sphere decoding algorithms can be used to accomplish msdd in this correspondence the computational complexity of this sphere decoding based msdd is analyzed in particular it is proven by means of a lower bound that the complexity of the fincke pohst multiple symbol differential sphere decoder fp msdsd while being very low over wide ranges of n and signal to noise ratios is exponential in n in principle furthermore both exact and simple approximate expressions for the complexity of fp msdsd are derived which allow for quick assessment of ranges of useful window sizes n of fp msdsd and show that the exponential rate of growth of the complexity of fp msdsd is asymptotically equal to that of brute force msdd

improved methods for calculating vectors of short length in a lattice including a complexity analysis the standard methods for calculating vectors of short length in a lattice use a reduction procedure followed by enumerating all vectors of z in a suitable box however it suffices to consider those x e z which lie in a suitable ellipsoid having a much smaller volume than the box we show in this paper that searching through that ellipsoid is in many cases much more efficient if combined with an appropriate reduction procedure our method allows to do computations in lattices of much higher dimensions several randomly constructed numerical examples illustrate the superiority of our new method over the known ones

performance of sphere decoding of block codes a sphere decoder searches for the closest lattice point within a certain search radius the search radius provides a tradeoff between performance and complexity we focus on analyzing the performance of sphere decoding of linear block codes we analyze the performance of soft decision sphere decoding on awgn channels and a variety of modulation schemes a hard decision sphere decoder is a bounded distance decoder with the corresponding decoding radius we analyze the performance of hard decision sphere decoding on binary and q ary symmetric channels an upper bound on the performance of maximum likelihood decoding of linear codes defined over fq e g reed solomon codes and transmitted over q ary symmetric channels is derived and used in the analysis we then discuss sphere decoding of general block codes or lattices with arbitrary modulation schemes the tradeoff between the performance and complexity of a sphere decoder is then discussed

reduced complexity soft decision multiple symbol differential sphere detection unlike a generic psk qam detector which may visit a constellation diagram only once a depth first sphere decoder sd has to re visit the same constellation diagram multiple times therefore in order to prevent the sd from repeating the detection operations the schnorr euchner search strategy of schnorr and euchner may be invoked for optimizing the nodes search order where the ideal case is for the sd to visit the constellation nodes in a zigzag fashion however when the hard decision multiple symbol differential sphere detection msdsd of lampe et al is invoked for using multiple receive antennas n r ge 1 the schnorr euchner search strategy has to visit and sort all the m textrm psk constellation points a similar situation is encountered for the soft decision msdsd of pauli et al when the a priori llrs gleaned from the channel decoder are taken into account in order to tackle these open problems in this paper we propose a correlation process for the hard decision msdsd of lampe et al and a reduced complexity design for the soft decision msdsd of pauli et al so that the schnorr euchner search strategy always opts for visiting the m textrm psk constellation points in a zigzag fashion our simulation results demonstrate that a substantial complexity reduction is achieved by our reduced complexity design without imposing any performance loss explicitly up to 88 7 complexity reduction is attained for msdsd n w 4 aided d16psk this complexity reduction is quite substantial especially when the msdsd is invoked several times during turbo detection furthermore in order to offer an improved solution and a comprehensive study for the soft decision msdsd we also propose to modify the output of the sd to harmonize its operation with the near optimum approx log map then the important subject of coherent versus noncoherent is discussed in the context of coded systems which suggests that msdsd aided dpsk is an eminently suitable candidate for turbo detection assisted coded systems operating at high doppler frequencies

distance estimation using wrapped phase measurements in noise measuring a distance using phase measurements is a common practice in many areas of engineering almost inevitably these measurements are accompanied by noise and are always subject to ambiguity resulting from the phase of modulo 2 pi in the presence of phase ambiguity where for instance the unknown distance is far longer than the wavelength of the signal carrying the phase measurement the distance cannot be uniquely determined one way to resolve this phase ambiguity is to measure the signal phase at multiple frequencies converting the phase ambiguity problem into one of solving a family of diophantine equations typically under some reasonable assumptions the diophantine problems can be solved using the chinese reminder theorem as documented in the literature however the existing algorithms can experience significant computational overhead for a given application because an exhaustive search is required in this paper a novel method addressing the phase ambiguity issue using lattice theoretic ideas is proposed and a closed form algorithm is presented for the estimation of the number of wavelengths in the unknown distance using the phase measurements taken at multiple wavelengths the algorithm is extremely efficient as the diophantine equations are solved without searching the unknown distance can then be estimated via a maximum likelihood method using the unwrapped phase measurement a statistical bound of the measurement noise which ensures that the number of whole wavelengths in the unknown distance can be found with a probability close to unity is derived the robustness efficiency and estimation accuracy of the proposed method are demonstrated by the simulated results presented

analysis and design of short block ofdm spreading matrices for use on multipath fading channels we consider the use of block spreading in a multicarrier system to gain diversity advantage when employed over multipath fading channels the main idea is to split the full set of subcarriers into smaller blocks and spread the data symbols across these blocks via unitary spreading matrices in order to gain multipath diversity across each block at the receiver we pose the problem of designing the spreading matrix as a finite dimensional optimization problem in which the asymptotic error is minimized this formulation allows us to find explicit solutions for the optimal spreading matrices the performance is validated for the uncoded channel as well as for the coded channel employing turbo iterative decoding we further demonstrate that suboptimal linear complexity equalization strategies for spread orthogonal frequency division multiplexing ofdm do not gain any diversity advantage over traditional diagonal ofdm

soft quasi maximum likelihood detection for multiple antenna wireless channels the paper addresses soft maximum likelihood ml detection for multiple antenna wireless communication channels we propose a soft quasi ml detector that maximizes the log likelihood function by deploying a semi definite relaxation sdr given perfect channel state information at the receiver the quasi ml sdr detector closely approximates the performance of the optimal ml detector in both coded and uncoded multiple input multiple output mimo channels with quadrature phase shift keying qpsk modulation and frequency flat rayleigh fading the complexity of the quasi ml sdr detector is much less than that of the optimal ml detector thus offering more favorable performance complexity characteristics in contrast to the existing sphere decoder the new quasi ml detector enjoys guaranteed polynomial worst case complexity the two detectors exhibit quite comparable performance in a variety of ergodic qpsk mimo channels but the complexity of the quasi ml detector scales better with increasing number of transmit and receive antennas especially in the region of low signal to noise ratio snr

application of discrete periodic wavelets to measured equation of invariance recently the wavelet expansions have been applied in field computations in the frequency domain the application is focused on the thinning of matrices arising from the method of moment mom the thinning of matrices can best be done by the measured equation of invariance mei which provides sparsity almost without sacrificing accuracy in that the boundary equation it entails is convertible to that of the mom the real power of the wavelet expansions is to give high resolution in convolution integrals high resolution is also needed in the process of finding the mei coefficients which are obtained via an integration process almost identical to that of the mom in this paper it is shown that when the fast discrete periodic wavelets fdpw are used as metron currents in the mei method the resolutions of the mei coefficients are improved at high frequency computations or at geometric extremities the level of sparsity of the mei is much more favorable than that achievable by the thinning of mom matrix using the wavelet expansions the role of fdpw in the mei happens to be more fitting than its place in the mom

leveraging coherent space time codes for noncoherent communication via training training codes are introduced for the multiple antenna noncoherent multiple block rayleigh fading channel in which the fading coefficients which are constant over a fixed number of dimensions coherence interval for each block and then change independently to a new realization are known neither at the transmitter nor at the receiver each codeword of a training code consists of a part known to the receiver used to form a minimum mean squared error mmse estimate of the channel and a part that contains codeword s of a space time block or trellis code designed for the coherent channel in which the receiver has perfect knowledge of the channel the channel estimate is used as if it were error free for decoding the information bearing part of the training codeword training codes are hence easily designed to have high rate and low decoding complexity by choosing the underlying coherent code to have high rate and to be efficiently decodable conditions for which the estimator detector e d receiver is equivalent to the optimal noncoherent receiver are established a key performance analysis result of this paper is that the training codes when decoded with the e d receiver achieve a diversity order of the error probability that is equal to the diversity order of the underlying coherent code in some cases the performance of training codes can be measured relative to coherent reception via training efficiency which is then optimized over the energy allocation between the training and data phases in the limit of increasing block lengths training codes always achieve the performance of coherent reception the examples of training codes provided in this work have polynomial complexity in rate but an error rate comparable to the best performing unitary designs available even though the latter require exponential decoding complexity

sphere constrained ml detection for frequency selective channels the maximum likelihood ml sequence detection problem for channels with memory is investigated the viterbi algorithm va provides an exact solution its computational complexity is linear in the length of the transmitted sequence but exponential in the channel memory length on the other hand the sphere decoding sd algorithm also solves the ml detection problem exactly and has expected complexity which is a low degree polynomial often cubic in the length of the transmitted sequence over a wide range of signal to noise ratios we combine the sphere constrained search strategy of sd with the dynamic programming principles of the va the resulting algorithm has the worst case complexity determined by the va but often significantly lower expected complexity

minkowski s convex body theorem and integer programming the paper presents an algorithm for solving integer programming problems whose running time depends on the number n of variables as non this is done by reducing an n variable problem to 2n5i 2 problems in n i variables for some i greater than zero chosen by the algorithm the factor of on5 2 per variable improves the best previously known factor which is exponential in n minkowski s convex body theorem and other results from geometry of numbers play a crucial role in the algorithm several related algorithms for lattice problems are presented the complexity of these problems with respect to polynomial time reducibilities is studied

reduced and fixed complexity variants of the lll algorithm for communications the lenstra lenstra lovasz lll algorithm is a popular lattice reduction algorithm in communications in this paper variants of the lll algorithm with either reduced or fixed complexity are proposed and analyzed specifically the use of effective lll reduction for lattice decoding is presented where size reduction is only performed for pairs of consecutive basis vectors its average complexity measured by the number of floating point operations and averaged over i i d standard normal lattice bases is shown to be o n 3 log n where n is the lattice dimension this average complexity is an order lower than previously thought to address the issue of variable complexity of the lll algorithm two fixed complexity approximations are proposed one is fixed complexity effective lll for which the first vector of the basis is proven to be bounded in length the other is fixed complexity lll with deep insertion which is shown to be closely related to the well known v blast algorithm such fixed complexity structures are much desirable in hardware implementation since they allow straightforward constant throughput implementation

factorization properties of lattices over the integers abstract let a be a nonsingular n n matrix over the integers l l a denotes the lattice whose elements are combinations with integer coefficients of the rows of a l is cyclic if it can be defined in the modular form l x x i a i x i 0 mod d where the a i s and d are integers and 0 a i d let l l 1 l 2 b be lattices over the integers l l 1 l 2 is a factorization of l if every element of l is a combination of the rows of b such that the vector of combination coefficients is in l 1 and b is a nonsingular n n matrix the following results are proved every lattice can be expressed as a product of cyclic factors in polynomial time every cyclic lattice can be factored into simple term explained in the text factors in polynomial time every simple lattice can be factored into prime factors in polynomial time if a prime factorization of the determinant of its basis is given in addition we provide polynomial algorithms for the following problems transform a cyclic lattice given by a basis into a modular form and vice versa find a basis of a finite modular lattice given in modular form

the computation of sextic fields with a cubic subfield and no quadratic subfield we describe six tables of sixth degree fields k containing a cubic subfield k and no quadratic subfield one for totally real sextic fields one for sextic fields with four real places two for sextic fields with two real places and two for totally imaginary sextic fields depending on whether the cubic subfield is totally real or not the tables provide for each possible discriminant dg of k a quadratic polynomial which defines k k the discriminant of the cubic subfield and the galois group of a galois closure at q of k q

improvements in the analysis of kannan s cvp algorithm finding a closest lattice vector to a given target vector is a widely studied problem in computational mathematics and computer science the classical enumeration algorithm was given by kannan in 1983 together with an algorithm for the shortest vector problem the time complexity of the closest vector enumeration algorithm based on hkz reduced bases was proven to within 2 o n n n 2 by hanrot and stehle in 2007 using their point counting method the constant hidden in o n is strictly bigger than 1 9 in this paper we give a new counting method which we call sequence counting method to bound the number of loop iterations and show that the time complexity is actually upper bounded by 2 0 401 n o n n n 2 our paper revisits kannan s cvp algorithm and points out a new method sequence counting to upper bound the time complexity by 2 0 401 n o n n n 2 we connects the loops of the enumeration algorithm with lattice points by a one to one mapping using the sequence as a tool we use different methods to upper bound the number of lattice points near a target vector according to the dimension of the lattice and then combine them to conclude our result

a parallel search for good lattice points using lll spectral tests we present results from an extensive parallel search for rank 1 lattice rules using the lll spectral test with a new normalization strategy which is proposed in k entacher et al monte carlo methods appl 10 3 4 2004 341 366 we introduce the main concepts concerning lattice rules and the spectral test and motivate the new normalization strategy for this test the concepts for parallelization are explained and some results conclude the article

on the diversity multiplexing tradeoff under groupwise successive interference cancelation in this paper we study the diversity multiplexing tradeoff assuming groupwise successive interference cancelation gsic gsic with layered codes in multiple input multiple output mimo can vary from the plain sic to the maximum likelihood ml decoding according to the group sizes thus is flexible in both complexity and performance the maximum group size is a dominating factor for the computation complexity of gsic when the processing complexity is polynomially dependent on the group size with a high order in this paper we show the tradeoff among the maximum group size diversity gain and multiplexing gain gdmt under the gsic and under the gsic combined with antenna selection gsicas for a given group size bound c we show that the number of groups achieving the optimal dmt under the gsic with selected transmit antennas is equal to c and a larger group should have a higher priority in decoding order the optimal grouping is obtained in a systematic way based on these results we find a limited set of groupings that contains the grouping achieving the optimal gdmt under gsic and under gsicas the optimal multiplexing gain allocation for the optimal grouping is then found systematically and efficiently within the limited set

quasi maximum likelihood detector based on geometrical diversification greedy intensification this letter proposes a quasi optimum maximum likelihood detection technique based on geometrical diversification and greedy intensification gdgi the presented detector scheme is shown to achieve almost optimal performance for all signal to noise ratio snr values and a cubic computation complexity in the problem dimension it possesses a regular structure well suited for hardware implementation simulation results show that for a system with a high dimension of n 60 the loss is approximately 0 35 db at ber 10 5 compared to an optimal decoding

a universal lattice code decoder for fading channels we present a maximum likelihood decoding algorithm for an arbitrary lattice code when used over an independent fading channel with perfect channel state information at the receiver the decoder is based on a bounded distance search among the lattice points falling inside a sphere centered at the received point by judicious choice of the decoding radius we show that this decoder can be practically used to decode lattice codes of dimension up to 32 in a fading environment

low complexity decoding via reduced dimension maximum likelihood search in this paper we consider a low complexity detection technique referred to as a reduced dimension maximum likelihood search rd mls rd mls is based on a partitioned search which approximates the maximum likelihood ml estimate of symbols by searching a partitioned symbol vector space rather than that spanned by the whole symbol vector the inevitable performance loss due to a reduction in the search space is compensated by 1 the use of a list tree search which is an extension of a single best searching algorithm called sphere decoding and 2 the recomputation of a set of weak symbols i e those ignored in the reduced dimension search for each strong symbol candidate found during the list tree search through simulations on m quadrature amplitude modulation qam transmission in frequency nonselective multi input multioutput mimo channels we demonstrate that the rd mls algorithm shows near constant complexity over a wide range of bit error rate ber 10 1 10 4 while limiting performance loss to within 1 db from ml detection

cryptanalytic attacks on the multiplicative knapsack cryptosystem and on shamir s fast signature scheme the basic merkle hellman additive trapdoor knapsack public key cryptosystem was recently shown to be insecure and attacks have also been developed on stronger variants of it such as the graham shamir system and the iterated knapsack cryptosystem it is shown that some simple variants of another merkle hellman system the multiplicative knapsack cryptosystem are insecure it is also shown that the shamir fast signature scheme can be broken quickly similar attacks can also be used to break the schobi massey authentication scheme these attacks have not been rigorously proved to succeed but heuristic arguments and empirical evidence indicate that they work on systems of practical size

interpolation extrapolation and application of the measured equation of invariance to scattering by very large cylinders using the conventional method of moment mom calculations a cylinder of circumferential dimension of 100 wavelengths is considered to be large using the measured equation of invariance mei approach a cylinder of 10000 wavelengths is within the storage capacity and numerical tolerance of a workstation although the mei has greatly reduced the storage and solution time of the matrix its overhead to generate the matrix elements is about the same order as that of the mom when the target is very large that overhead can be very time consuming this paper presents an interpolation and extrapolation technique such that the boundary equations of the mei for high frequencies may be predicted from those of low frequencies it is demonstrated that in the optical limit the same set of coefficients may be used for all frequencies which is consistent with the concept of geometric optics where the same rule is applied to all frequencies

statistical pruning for near maximum likelihood decoding in many communications problems maximum likelihood ml decoding reduces to finding the closest skewed lattice point in n dimensions to a given point xisin cn in its full generality this problem is known to be np complete recently the expected complexity of the sphere decoder a particular algorithm that solves the ml problem exactly has been computed an asymptotic analysis of this complexity has also been done where it is shown that the required computations grow exponentially in n for any fixed snr at the same time numerical computations of the expected complexity show that there are certain ranges of rates snrs and dimensions n for which the expected computation counted as the number of scalar multiplications involves no more than n3 computations however when the dimension of the problem grows too large the required computations become prohibitively large as expected from the asymptotic exponential complexity in this paper we propose an algorithm that for large n offers substantial computational savings over the sphere decoder while maintaining performance arbitrarily close to ml we statistically prune the search space to a subset that with high probability contains the optimal solution thereby reducing the complexity of the search bounds on the error performance of the new method are proposed the complexity of the new algorithm is analyzed through an upper bound the asymptotic behavior of the upper bound for large n is also analyzed which shows that the upper bound is also exponential but much lower than the sphere decoder simulation results show that the algorithm is much more efficient than the original sphere decoder for smaller dimensions as well and does not sacrifice much in terms of performance

maximum likelihood sequence estimation from the lattice viewpoint considers the problem of data detection in multilevel lattice type modulation systems in the presence of intersymbol interference and additive white gaussian noise the conventional maximum likelihood sequence estimator using the viterbi algorithm has a time complexity of o m sup spl nu 1 operations per symbol and a space complexity of o spl delta m sup spl nu storage elements where m is the size of input alphabet spl nu is the length of channel memory and spl delta is the truncation depth by revising the truncation scheme and viewing the channel as a linear transform the authors identify the problem of maximum likelihood sequence estimation with that of finding the nearest lattice point from this lattice viewpoint the lattice sequence estimator for pam systems is developed which has the following desired properties 1 its expected time complexity grows as spl delta sup 2 as snr spl rarr spl infin 2 its space complexity grow as spl delta and 3 its error performance is effectively optimal for sufficiently large m a tight upper bound on the symbol error probability of the new estimator is derived and is confirmed by the simulation results of an example channel it turns out that the estimator is effectively optimal for m spl ges 4 and the loss in signal to noise ratio is less than 0 5 db even for m 2 finally limitations of the proposed estimator are also discussed