design of capacity approaching irregular low density parity check codes we design low density parity check ldpc codes that perform at rates extremely close to the shannon capacity the codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides our theoretical analysis of the codes is based on the work of richardson and urbanke see ibid vol 47 no 2 p 599 618 2000 assuming that the underlying communication channel is symmetric we prove that the probability densities at the message nodes of the graph possess a certain symmetry using this symmetry property we then show that under the assumption of no cycles the message densities always converge as the number of iterations tends to infinity furthermore we prove a stability condition which implies an upper bound on the fraction of errors that a belief propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution our codes are found by optimizing the degree structure of the underlying graphs we develop several strategies to perform this optimization we also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds

symmetric homotopy construction abstract homotopy continuation methods can be applied to compute all finite solutions to a given polynomial system computations will be performed more efficiently if the symmetric structure of the system can be exploited this paper presents the construction of a symmetric homotopy using this homotopy only the paths according to the generating solutions have to be traced during continuation

efficient soft input soft output tree detection via an improved path metric tree detection techniques are often used to reduce the complexity of a posteriori probability app detection in multiantenna wireless communication systems in this paper we introduce an efficient soft input soft output tree detection algorithm that employs a new type of look ahead path metric in the process of branch pruning or sorting while conventional path metrics depend only on symbols on a visited path the new path metric accounts for unvisited parts of the tree in advance through an unconstrained linear estimator and adds a bias term that reflects the contribution of as yet undecided symbols by applying the linear estimate based look ahead path metric to an algorithm that selects the best paths for each level of the tree we develop a new soft input soft output tree detector called an improved soft input soft output algorithm iss ma based on an analysis of the probability of correct path loss we show that the improved path metric offers substantial performance gain over the conventional path metric we also demonstrate through simulations that the proposed iss ma can be a promising candidate for soft input soft output detection in high dimensional systems

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

the list sequential liss algorithm and its application for iterative detection decoding turbo schemes we modify sequential decoding which contrary to a posteriori probability app bahl cocke jelinek raviv bcjr decoding enjoys a complexity almost independent of the number of states this novel list sequential liss decoder avoids most of the drawbacks of the classical sequential decoders such as variable workload and erased frames when working within a turbo scheme it uses a metric containing a priori and channel values a metric length bias term for speeding up the tree search a soft extension of paths without increasing the stack size and soft weighting to obtain a soft output we present several turbo applications using the liss including equalization single antenna interference cancellation multiuser and multiple input multiple output detection it is shown that the liss achieves the optimal genie or app performance at a bit error rate after the outer decoder of less than 10 4 but it also works with channels which have a high number of taps constellation points antennas and states where the app bcjr algorithm becomes infeasible

efficient implementation of quasi maximum likelihood detection based on semidefinite relaxation in this paper we develop two quasi maximum likelihood ml channel detectors for multiuser detection semidefinite relaxation sdr detector and phase shift keying psk detector these detectors can deliver near ml bit error rate ber performance with a polynomial worst case complexity the sdr detector for binary phase shift keying bpsk constellation is based on a convex sdr whereas the psk detector for m psk constellations is based on a nonconvex low rank sdr the sdr detector is implemented using a dual scaling interior point method while the psk detector is based on a coordinate descent strategy on a feasible region homotopy we use dynamic dimension reduction and warm start techniques to achieve signal to noise ratio snr sensitive improvements for both detectors numerical simulations of ber performance and running time indicate the effectiveness of the two quasi ml detectors when compared to the conventional sphere decoder and its variants

partial marginalization soft mimo detection with higher order constellations a new method for multiple input multiple output mimo detection with soft output the partial marginalization pm algorithm was recently proposed advantages of the method are that it is straightforward to parallelize and that it offers a fully predictable runtime pm trades performance for computational complexity via a user defined parameter in the limit of high computational complexity the algorithm becomes the map demodulator the pm algorithm also works with soft input but until now it has been unclear how to apply it for other modulation formats than binary phase shift keying bpsk per real dimension in this correspondence we explain how to extend pm with soft input to general signaling constellations while maintaining the low complexity advantage of the original algorithm

complete blow up for degenerate semilinear parabolic equations abstract let t a q and x0 be constants with a 0 q 0 and 0 x q u t u xx f u x 0 t 0 0 u x 0 u 0 x 0 0 x a u 0 t u a t 0 0 where f and u0 are given functions we show that under certain conditions u blows up in a finite time and the set of blow up points is the entire interval 0 a

a hybrid polynomial system solving method for mixed trigonometric polynomial systems mixed trigonometric polynomial systems arise in many fields of science and engineering commonly this class of systems is transformed into polynomial systems by variable substituting and adding some quadratic equations and then solved by some polynomial system solving method in this paper by exploiting the special structure of the additional quadratic equations an efficient hybrid method for solving polynomial systems coming from mixed trigonometric polynomial systems is presented it combines the homotopy method in which the homotopy is a combination of coefficient parameter homotopy and the random product homotopy with decomposition variable substitution and reduction techniques numerical tests are given to show its effectiveness and it is applied to solve a practical problem

universal binary semidefinite relaxation for ml signal detection semidefinite relaxation sdr provides a computationally efficient polynomial time approximation of the maximum likelihood detector however most of the existing works mainly focus on particular signal constellations in this paper we propose a universal binary semidefinite relaxation scheme that can handle arbitrary signal constellations in polynomial time the proposed scheme first binarizes the original signal space to a linearly constrained binary space and then solves the detection problem through sdr colorblack a specialized dual barrier method is provided to solve the sdr more efficiently in addition we propose to apply on the fly decision feedback to further reduce the computational complexity and improve the detection performance the proposed binary sdr together with on the fly decision feedback scheme can provide comparable or better solutions compared to existing sdr methods specialized to specific constellations such as 16 qam and 8 psk in terms of computational complexity and symbol error rate furthermore the proposed scheme is universal and can solve any other constellations such as 12 qam 32 qam or m psk

sumis near optimal soft in soft out mimo detection with low and fixed complexity the fundamental problem of interest here is soft input soft output multiple input multiple output mimo detection we propose a method referred to as subspace marginalization with interference suppression sumis that yields unprecedented performance at low and fixed deterministic complexity our method provides a well defined tradeoff between computational complexity and performance apart from an initial sorting step consisting of selecting channel matrix columns the algorithm involves no searching nor algorithmic branching hence the algorithm has a completely predictable run time and allows for a highly parallel implementation we numerically assess the performance of sumis in different practical settings full partial channel state information sequential iterative decoding and low high rate outer codes we also comment on how the sumis method performs in systems with a large number of transmit antennas

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

low complexity equalization of ofdm in doubly selective channels orthogonal frequency division multiplexing ofdm systems may experience significant inter carrier interference ici when used in time and frequency selective or doubly selective channels in such cases the classical symbol estimation schemes e g minimum mean squared error mmse and zero forcing zf estimation require matrix inversion that is prohibitively complex for large symbol lengths an analysis of the ici generation mechanism leads us to propose a novel two stage equalizer whose complexity apart from the fft is linear in the ofdm symbol length the first stage applies optimal linear preprocessing to restrict ici support and the second stage uses iterative mmse estimation to estimate finite alphabet frequency domain symbols simulation results indicate that our equalizer has significant performance and complexity advantages over the classical linear mmse estimator in doubly selective channels

layered tabu search algorithm for large mimo detection and a lower bound on ml performance in this letter we are concerned with low complexity detection in large multiple input multiple output mimo systems with tens of transmit receive antennas our new contributions in this letter are two fold first we propose a low complexity algorithm for large mimo detection based on a layered low complexity local neighborhood search second we obtain a lower bound on the maximum likelihood ml bit error performance using the local neighborhood search the advantages of the proposed ml lower bound are i it is easily obtained for mimo systems with large number of antennas because of the inherent low complexity of the search algorithm ii it is tight at moderate to high snrs and iii it can be tightened at low snrs by increasing the number of symbols in the neighborhood definition the proposed detection algorithm based on the layered local neighborhood search achieves bit error performances which are quite close to this lower bound for large number of antennas and higher order qam

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

global existence and blow up for degenerate and singular parabolic system with localized sources this paper deals with the blow up properties of the solution to degenerate and singular parabolic system with localized sources and homogeneous dirichlet boundary conditions the existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution exists globally and blows up in finite time are obtained furthermore under certain conditions it is proved that the blow up set of the solution is the whole domain

semidefinite relaxation approaches to soft mimo demodulation for higher order qam signaling the soft demodulator is a key component of receivers that employ iterative demodulation and decoding for multiple input multiple output systems with binary or qpsk signaling the semidefinite relaxation sdr approach to list based soft demodulation provides a desirable balance between the performance and computational cost in this paper that approach is extended to systems with higher order qam signaling for the class of list sdr demodulators the key to the extension is a quadratic approximation of the a priori information and for the more computationally efficient class of single sdr demodulators the key to the extension is an algebraic expression derived herein for the probability that the sdr randomization procedure generates each candidate symbol to implement these demodulators customized primal dual and dual scaling algorithms are developed to solve the underlying semidefinite programs the performance of the proposed demodulators is compared with that of some existing demodulators by using error rate simulations the empirical distribution of the computational cost and exit chart analyses in our comparisons the proposed single sdr demodulator provides performance that comes close to that of the list sequential demodulator at a computational cost that is close to that of the minimum mean square error parallel soft interference canceller

blow up at the boundary for degenerate semilinear parabolic equations this paper treats a superlinear parabolic equation degenerate in the time derivative it is shown that the solution may blow up in finite time moreover it is proved that for a large class of initial data blow up occurs at the boundary of the domain when the nonlinearity is no worse than quadratic various estimates are obtained which determine the asymptotic behaviour near the blow up the mathematical analysis is then extended to equations with other degeneracies

quasi maximum likelihood multiuser detection using semi definite relaxation with application to synchronous cdma the maximum likelihood ml multiuser detector is well known to exhibit better bit error rate ber performance than many other multiuser detectors unfortunately ml detection mld is a nondeterministic polynomial time hard np hard problem for which there is no known algorithm that can find the optimal solution with polynomial time complexity in the number of users in this paper a polynomial time approximation method called semi definite sd relaxation is applied to the mld problem with antipodal data transmission sd relaxation is an accurate approximation method for certain np hard problems the sd relaxation ml sdr ml detector is efficient in that its complexity is of the order of k3 5 where k is the number of users we illustrate the potential of the sdr ml detector by showing that some existing detectors such as the decorrelator and the linear minimum mean square error detector can be interpreted as degenerate forms of the sdr ml detector simulation results indicate that the ber performance of the sdr ml detector is better than that of these existing detectors and is close to that of the true ml detector even when the cross correlations between users are strong or the near far effect is significant

solving large scale sparse semidefinite programs for combinatorial optimization we present a dual scaling interior point algorithm and show how it exploits the structure and sparsity of some large scale problems we solve the positive semidefinite relaxation of combinatorial and quadratic optimization problems subject to boolean constraints we report the first computational results of interior point algorithms for approximating maximum cut semidefinite programs with dimension up to 3 000

iterative tree search detection for mimo wireless systems this paper presents a reduced complexity soft input soft output detection scheme called iterative tree search detection for multiple input multiple output wireless communication systems employing turbo processing at the receiver in this scheme a reduced search space is selected with the aid of the m algorithm and qam signal constellations with block partitionable labels are used in order to make the detection complexity per bit almost independent of the modulation order as well as asymptotically linear in the number of transmit antennas results from computer simulations are presented which demonstrate the capability of the scheme to approach optimal performance at considerably reduced complexity

an interior point method for semidefinite programming we propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices

reduced complexity soft output mimo sphere detectors part i algorithmic optimizations optimum soft output so multiple input multiple output mimo tree search detection algorithms pose significant implementation challenges due to their nondeterministic processing throughput and high computational complexity in this two part work we present extensive algorithmic and architectural optimizations of the sphere decoding algorithm targeted at achieving practical tradeoffs between desired link performance and affordable computational complexity the algorithmic optimizations in this part span the tree search traversal scheme leaf processing step internal node pruning and skipping step child enumeration based on a state machine adaptive radius scaling for llr clipping qr decomposition based on minimum cumulative residuals and multitree configurations the optimizations demonstrate that a 64 qam so mimo detector for lte is capable of attaining almost ml performance with an snr loss of only 0 85 db at 1 bler by visiting at most 200 tree nodes

nonlinear reduction for solving deficient polynomial systems by continuation methods neither continuation methods nor symbolic elimination methods can be directly applied to compute all finite solutions to polynomial systems because the amount of computational time is mostly not proportional to the dimension of the system and to the number of finite solutions the notion of s polynomials is used to developed a reduction algorithm to lower the total degree of the deficient polynomial system so that computing the solutions at infinity can be avoided applying the reduction algorithm before solving the system with continuation methods yields a reliable solution method

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

performance assessment of mimo bicm demodulators based on mutual information we provide a comprehensive performance comparison of soft output and hard output demodulators in the context of non iterative multiple input multiple output bit interleaved coded modulation mimo bicm coded bit error rate ber widely used in literature for demodulator comparison has the drawback of depending strongly on the error correcting code being used this motivates us to propose the mutual information of the equivalent modulation channel comprising modulator wireless channel and demodulator as a code independent performance measure we present extensive numerical results for spatially independent identically distributed i i d ergodic and quasi static fading channels under perfect and imperfect channel state information these results reveal that the performance ranking of mimo demodulators is rate dependent and provide new insights regarding mimo bicm system design i e the choice of antenna configuration symbol constellation and demodulator for a given target rate

reduced dimension map turbo blast detection the bell labs layered space time blast architecture is a simple and efficient multiantenna coding structure that can achieve high spectral efficiency many blast detectors require more receiver antennas than transmitter antennas we propose two novel turbo processing blast detectors that can operate in systems with fewer receiver antennas than transmitter antennas both detectors are based on the group detection strategy the first proposed detector the reduced dimension maximum a posteriori rdmap detector uses a dynamically formed group for each bit decision while the second proposed detector the group maximum a posteriori gmap uses a static grouping for both detectors a maximum a posteriori map decision is made using a group of transmitted symbols and the remaining signal contribution is treated as interference the interference is characterized as nonzero mean colored noise source that is whitened before a decision is made both proposed detectors are generalizations of the map detector and the turbo processing minimum mean squared error mmse detector in sellathurai and haykin and abe and matsumoto an uncoded bit error rate analysis for an independent rayleigh fading environment is also presented simulated results are presented which show that both the rdmap and gmap detectors have a performance improvement over the mmse detector especially in systems having an excess number of transmitter antennas

probabilistic analysis of semidefinite relaxation for binary quadratic minimization we consider semidefinite programming relaxation sdr of a binary quadratic minimization problem this np hard problem arises naturally in the maximum likelihood detection of discrete signals for digital communications we analyze the average performance of the sdr algorithm for a class of randomly generated binary quadratic minimization problems although the sdr worst case approximation ratio is unbounded for this np hard problem our analysis shows that sdr can provide in polynomial time a provably near optimal solution achieving a constant factor approximation of the optimal objective value in probability moreover this constant factor remains bounded with increasing problem size our proof is based on an asymptotic analysis of karush kuhn tucker optimality conditions using random matrix theory

on the complexity of computing mixed volumes this paper gives various positive and negative results on the complexity of the problem of computing and approximating mixed volumes of polytopes and more general convex bodies in arbitrary dimension on the negative side we present several p hardness results that focus on the difference of computing mixed volumes versus computing the volume of polytopes we show that computing the volume of zonotopes is p hard while each corresponding mixed volume can be computed easily but also give examples showing that computing mixed volumes is hard even when computing the volume is easy on the positive side we derive a randomized algorithm for computing the mixed volumes v overbrace k 1 ld k 1 m 1 overbrace k 2 dots k 2 m 2 dots overbrace k s dots k s m s of well presented convex bodies k 1 dots k s where m 1 dots m s in n 0 and m 1 geq n psi n with psi n o frac log n log log n the algorithm is an interpolation method based on polynomial time randomized algorithms for computing the volume of convex bodies this paper concludes with applications of our results to various problems in discrete mathematics combinatorics computational convexity algebraic geometry geometry of numbers and operations research

design of low rate irregular ldpc codes using trellis search a simple design method using trellis search is proposed for good low density parity check ldpc codes with relatively low code rates by applying a trellis search technique to the design of a pre assigned part of the parity check matrix that allows a simple encoding we improve the distribution of cycles formed by the entries contained in the parity check part of the parity check matrix in addition we extend the proposed algorithm to a class of structured ldpc codes which have been recently preferred in many practical applications simulation results show that the codes designed by the proposed method outperform those constructed by conventionally used greedy design algorithms

incomplete grobner basis as a preconditioner for polynomial systems precondition plays a critical role in the numerical methods for large and sparse linear systems it is also true for nonlinear algebraic systems in this paper incomplete grobner basis igb is proposed as a preconditioner of homotopy methods for polynomial systems of equations which transforms a deficient system into a system with the same finite solutions but smaller degree the reduced system can thus be solved faster numerical results show the efficiency of the preconditioner

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

existence of classical solutions for degenerate semilinear parabolic problems let q be a nonnegative real number and t be a positive real number existence of a unique classical solution is established for the following degenerate semilinear parabolic initial boundary value problem x qu t u x x f u 0