universal space time coding a universal framework is developed for constructing full rate and full diversity coherent space time codes for systems with arbitrary numbers of transmit and receive antennas the proposed framework combines space time layering concepts with algebraic component codes optimized for single input single output siso channels each component code is assigned to a thread in the space time matrix allowing it thus full access to the channel spatial diversity in the absence of the other threads diophantine approximation theory is then used in order to make the different threads transparent to each other within this framework a special class of signals which uses algebraic number theoretic constellations as component codes is thoroughly investigated the lattice structure of the proposed number theoretic codes along with their minimal delay allow for polynomial complexity maximum likelihood ml decoding using algorithms from lattice theory combining the design framework with the cayley transform allows to construct full diversity differential and noncoherent space time codes the proposed framework subsumes many of the existing codes in the literature extends naturally to time selective and frequency selective channels and allows for more flexibility in the tradeoff between power efficiency bandwidth efficiency and receiver complexity simulation results that demonstrate the significant gains offered by the proposed codes are presented in certain representative scenarios

bicm id with signal space diversity over cascaded rayleigh fading channels transactions letters bit interleaved coded modulation with iterative decoding bicm id using signal space diversity ssd is considered for cascaded rayleigh fading channels a tight bound on the asymptotic error probability is derived to determine the optimal rotation matrix for ssd design and to identify the key parameters that influence the system performance it is shown that for small modulation constellation a cascaded rayleigh fading causes a much more severe performance degradation than a conventional rayleigh fading however bicm id employing ssd with a sufficiently large constellation can close the performance gap between the conventional and cascaded rayleigh fading channels and their performance can closely approach that over an awgn channel illustrative simulation results for various scenarios are in a good agreement with analytical derivations

on optimal multilayer cyclotomic space time code designs high rate and large diversity product or coding advantage or coding gain or determinant distance or minimum product distance are two of the most important criteria often used for good space time code designs in recent linear lattice based space time code designs more attention is paid to the high rate criterion but less to the large diversity product criterion in this paper we consider these two criteria together for multilayer cyclotomic space time code designs in a previous paper we recently proposed a systematic cyclotomic diagonal space time code design over a general cyclotomic number ring that has infinitely many designs for a fixed number of transmit antennas where diagonal codes correspond to single layer codes in this paper in this paper we first propose a general multilayer cyclotomic space time codes we present a general optimality theorem for these infinitely many cyclotomic diagonal or single layer space time codes over general cyclotomic number rings for a general number of transmit antennas we then present optimal multilayer full rate cyclotomic space time code designs for two and three transmit antennas we also present an optimal two layer cyclotomic space time code design for three and four transmit antennas the optimality here is in the sense that for a fixed mean transmission signal power its diversity product is maximized or equivalently for a fixed diversity product its mean transmission signal power is minimized it should be emphasized that all the optimal multilayer cyclotomic space time codes presented in this paper have the nonvanishing determinant property

bandwidth efficient linear modulations for multiple antenna transmission some m spl times t modulation matrices for m transmit antennas and t symbol periods with m 2 3 4 and t 2 and m t 4 are studied a transmission rate of m symbols per channel use and a transmit diversity order of min m t are achieved over a quasi static fading channel when using rotated versions of a multidimensional quadratic amplitude modulation with spectral efficiency 2 bits symbol extension to input constellations with higher spectral efficiencies is then considered the modulations are then generalized to any number of transmit antennas m and any number of symbol periods t such that a transmission rate of m symbols per channel use and a transmit diversity of t are achieved under fast fading ergodic scenario by means of signal space diversity the proposed modulations exploit the degrees of freedom of multiantenna channels and have moderate detection complexity at moderate and large signal to noise ratios snrs

rotated multi d constellations in rayleigh fading mutual information improvement and pragmatic approach for near capacity performance in high rate regions this paper studies the mutual information improvement attained by rotated multidimensional multi d constellations via a unitary precoder g in rayleigh fading at first based on the symmetric cut off rate of the n d signal space we develop a design criterion with regard to the precoder g it is then demonstrated that the use of rotated constellations in only a reasonably low dimensional signal space can significantly increase the mutual information in high rate regimes based on parameterizations of unitary matrices we then construct good unitary precoder g in 4 d signal space using a simple optimization problem which involves only four real variables and it is applicable to any modulation scheme to further illustrate the potential of multi d constellation and to show the practical use of mutual information improvement we propose a simple yet powerful bit interleaved coded modulation bicm scheme in which a multi d mapping technique employed in a multi d rotated constellation is concatenated with a short memory high rate convolutional code by using extrinsic information transfer exit charts it is shown that the proposed technique provides an exceptionally good error performance in particular both exit chart analysis and simulation results indicate that a turbo pinch off and a bit error rate around 10 6 happen at a signal to noise ratio that is well below the coded modulation and bicm capacities using traditional signal sets for example with code rates ranging from 2 3 to 7 8 the proposed system can operate 0 82 db 2 93 db lower than the bicm capacity with qpsk and gray labeling the mutual information gain offered by rotated constellations can be therefore utilized to design simple yet near shannon limit systems in the high rate regions

full diversity blind space time block codes in this paper the systematic design of a space time block code is considered for a wireless communication system with multiple transmitter receiver antennas and flat fading in which channel state information is completely unknown from the viewpoint of blind signal processing a necessary and sufficient condition is given for the unique identification of the multi input multi output flat fading channel and transmitted signal then some novel unique factorizations for a pair of coprime p ary and q ary phase shift keying psk constellations are established with this and currently available coherent space time block code designs a method is developed to systematically construct full diversity blind nonunitary space time block codes as well as unitary codes by just performing the qr decomposition of the nonunitary codes it is proved that under a noise free environment the code design proposed in this paper can guarantee that the transmitted signals and the channel coefficients are uniquely identified and that under a complex gaussian noise environment in which pth order and q th order statistics p and q are co prime of the received signals are available the channel coefficients can be still uniquely identified in addition a closed form solution to determine the channel coefficients is obtained

the jackknife estimate of variance tukey s jackknife estimate of variance for a statistic s x 1 x 2 cdots x n which is a symmetric function of i i d random variables x i is investigated using an anova like decomposition of s it is shown that the jackknife variance estimate tends always to be biased upwards a theorem to this effect being proved for the natural jackknife estimate of operatorname var s x 1 x 2 cdots x n 1 based on x 1 x 2 cdots x n

inverse determinant sums and connections between fading channel information theory and algebra this work considers inverse determinant sums which arise from the union bound on the error probability as a tool for designing and analyzing algebraic space time block codes a general framework to study these sums is established and the connection between asymptotic growth of inverse determinant sums and the diversity multiplexing gain tradeoff is investigated it is proven that the growth of the inverse determinant sum of a division algebra based space time code is completely determined by the growth of the unit group this reduces the inverse determinant sum analysis to studying certain asymptotic integrals in lie groups using recent methods from ergodic theory a complete classification of the inverse determinant sums of the most well known algebraic space time codes is provided the approach reveals an interesting and tight relation between diversity multiplexing gain tradeoff and point counting in lie groups

the properties of the cross match estimate and split sampling by noting the connection with k sample u statistics we find a simple decomposition of the variance of the cross match estimate which can be regarded as a generalization of efron and stein we apply the decomposition in assessing efficiencies of several plans of using the weighted samples from an importance scheme the applications of the formula to multiple imputations lead to a method of crossing jointly imputed data to gain more accuracy

trace orthonormal full diversity cyclotomic space time codes in this paper we consider the design of full diversity space time codes for a coherent multiple input multiple output mimo communication system starting from both the information theoretic and detection error viewpoints we first establish that a desirable property for general linear dispersion ld codes is to have an interunitary as well as an intraunitary structure a structure we call trace orthonormality by imposing the trace orthonormal structure on an ld code and applying cyclotomic number theory we establish for an arbitrary number of transmitter and receiver antennas a systematic and simple method to jointly design a unitary cyclotomic matrix the diophantine number and the corresponding constellation for an ld code as a result this enables us to construct full diversity rectangular cyclotomic ld codes with any symbol transmission rate less than or equal to the number of transmitter antennas in addition for the case when the number of transmitter antennas is greater than the number of receiver antennas by taking advantage of the delay we also arrive at the design of a special trace orthonormal full diversity cyclotomic space time block code which for the number of transmitter antenna being equal to 2m can be proved to minimize the worst case pairwise error probability of a maximum likelihood ml detector for a q ary quadrature amplitude modulation qam signal constellation and therefore achieves optimal coding gain computer simulations show that these codes have bit error performance advantages over currently available codes

an optimal two transmit antenna space time code and its stacked extensions a space time code is proposed that exhibits the highest coding gain among competing full rate full transmit diversity space time codes for the two transmit and receive antenna coherent quasi static fading channel the proposed code is derived from a layered architecture with real rotation of quadrature amplitude modulation qam information symbols in two dimensions the existing codes of similar architecture concentrate on application of complex full modulation diversity rotations or asymmetric real rotations an analytic evaluation illustrates the significant improvement in coding gain achieved with the proposed code moreover the coding gain of the proposed code is independent of its rate this implies that the proposed code achieves the optimal diversity multiplexing tradeoff curve for the two transmit antenna system a stacked extension of the proposed code offers a reduced complexity capacity optimal alternative to the full diversity codes for larger number of transmit antennas performance enhancement in several scenarios is verified through simulations

product construction of codes for rayleigh fading channels coded modulation scheme of high coding gain is presented for flat fading channels the scheme introduces a product construction that employs two types of codes standard good codes and codes that are more suitable for the fading environment short blocklength codes of unique features are thus obtained for the bandwidth limited regime using this approach codes of bandwidth efficiencies 1 2 and 3 bits s hz are detailed an iterative decoding algorithm for this product construction is devised it relies on separate columns rows decoding linked by reliability information transfer simulation results reveal that these codes compare favorably with known codes of short block length when perfect channel state information is available to the receiver the proposed approach is easily extended for the frequency selective fading channel with multicarrier signaling

some designs and normalized diversity product upper bounds for lattice based diagonal and full rate space time block codes in this paper we first present two tight upper bounds for the normalized diversity products or product distances of 2ntn2 diagonal space time block codes from quadratic extensions on bbq i and bbq mmbzeta6 where i radic 1 and mmbzeta6 exp i 2pi 6 two such codes are shown to reach the tight upper bounds and therefore have the maximal normalized diversity products we present two new diagonal space time block codes from higher order algebraic extensions on bbq i and bbq mmbzeta6 for three and four transmit antennas we also present a nontight upper bound for normalized diversity products of 2ntn2 diagonal space time block codes with qam information symbols i e in bbz i from general 2ntn2 complex valued generating matrices we then present an nntnn diagonal space time code design method directly from 2n real integers based on extended complex lattices of generating matrix size nntn2n that are shown to have better normalized diversity products than the optimal diagonal cyclotomic codes do we finally use the optimal 2ntn2 diagonal space time codes from the optimal quadratic extensions to construct two 2ntn2 full rate space time block codes and find that both of them have better normalized diversity products than the golden code does

probabilistic and worst case analyses of classical problems of combinatorial optimization in euclidean space the classical problems reviewed are the traveling salesman problem minimal spanning tree minimal matching greedy matching minimal triangulation and others each optimization problem is considered for finite sets of points in d and the feature of principal interest is the value of the associated objective function special attention is given to the asymptotic behavior of this value under probabilistic assumptions but both probabilistic and worst case analyses are surveyed

a new representation of tast codes a simple and general form of threaded algebraic space time tast codes and constellations for arbitrary numbers of transmit and receive antennas and arbitrary input alphabets is given and analyzed this new form gives revealing insights on the tast framework elucidates the connection between space time constellation expansion and the peak to average power ratio par and establishes the equivalence between a certain class of tast constellations and constellations derived from division algebras

khatri rao space time codes space time st coding techniques exploit the spatial diversity afforded by multiple transmit and receive antennas to achieve reliable transmission in scattering rich environments st block codes are capable of realizing full diversity and spatial coding gains at relatively low rates st trellis codes can achieve better rate diversity tradeoffs at the cost of high complexity on the other hand v blast supports high rates but has no built in spatial coding and does not work well with fewer receive than transmit antennas we propose a novel linear block coding scheme based on the khatri rao matrix product the proposed scheme offers flexibility for achieving full rate or full diversity or a desired rate diversity tradeoff and it can handle any transmit receive antenna configuration or signal constellation the proposed codes are shown to have numerous desirable properties including guaranteed unique linear decodability built in blind channel identifiability and efficient near maximum likelihood decoding

a central limit theorem for generalized quadratic forms random variables of the form w n mathop sum limits 1 leqq i leqq n mathop sum limits text 1 leqq j leqq n w ijn x i x j are considered with x i independent not necessarily identically distributed and w ijn borel functions such that w ijn x i x j is square integrable and has vanishing conditional expectations e w ijn x i x j x i e w ijn x i x j x j 0 text a text s text a central limit theorem is proved under the condition that the normed fourth moment tends to 3 under some restrictions the condition is also necessary finally conditions on the individual tails of w ijn x i x j and an eigenvalue condition are given that ensure asymptotic normality of w n

on random rotations diversity and minimum mse decoding of lattices we establish a simple relation between high diversity multidimensional rotations obtained from totally complex cyclotomic fields and the discrete fourier transform the diversity distribution of an hadamard like random rotation is derived analytically it is shown that a random multidimensional rotation exhibits an excellent diversity distribution and can be combined to quadrature amplitude modulation qam constellations to combat channel fading we also describe a mean square error mse universal lattice decoder suitable for large dimensions up to 1024 the mse criterion treats the lattice structure as intersymbol interference the universal decoder is applied to both gaussian and rayleigh fading channels to decode dense lattice sphere packings and rotated cubic constellations respectively

the design of trellis coded mpsk for fading channels performance criteria it has been well established that the appropriate criterion for optimum trellis coded modulation design on the additive white gaussian noise channel is maximization of the free euclidean distance it is shown that when the trellis coded modulation is used on a rician fading channel with interleaving deinterleaving the design of the code of optimum performance is guided by other factors in particular the length of the shortest error event path and the product of branch distances possibly normalized by the euclidean distance of the path along the path although maximum free distance d sub free is still an important consideration it plays a less significant role the more severe the fading is on the channel these considerations lead to the definition of a new distance measure of optimization of trellis codes transmitted over rician fading channels if no interleaving deinterleaving is used then once again the design of the trellis code is guided by maximizing d sub free

a berry esseen bound for symmetric statistics the rate of convergence of the distribution function of a symmetric function of n independent and identically distributed random variables to its normal limit is investigated under appropriate moment conditions the rate is shown to be o left n frac 1 2 right this theorem generalizes many known results for special cases and two examples are given possible further extensions are indicated

diversity comparison of spreading transforms for multicarrier spread spectrum transmission multicarrier spread spectrum transmission methods based on orthogonal frequency division multiplexing employ a linear transform to spread energy of transmitted symbols over statistically independent rayleigh fading subcarriers in order to enable a diversity gain at the receiver we jointly treat the mapping from bits into transmit symbols and the spreading transform as a code in euclidean space we describe criteria to identify good and bad spreading transforms in terms of the asymptotic bit error probability bep at high signal to noise ratio upper and lower bounds on the bep are derived it is shown that the bep performance of the commonly used hadamard transform is asymptotically bad alternative orthogonal transforms with better asymptotic performance are proposed simulation results are given for the example of 2 phase shift keying block length 8

optimal normalized diversity product of 2 times 2 lattice based diagonal space time codes from qam signal constellations in this correspondence we prove that the optimal normalized diversity product of lattice based diagonal space time block codes with gaussian integer or qam signal constellations i e and any generating matrices of complex entries not necessarily algebraic extensions of as commonly used is this result implies that lattice based diagonal space time block codes with gaussian integer signal constellations and generating matrices of entries from quadratic algebraic extensions of have already reached the optimal normalized diversity product

analysis and optimization of diagonally layered lattice schemes for mimo fading channels embodiments of the diagonal bell laboratories layered space time d blast architecture for multiple input multiple output mimo communication are developed wherein information symbol vectors are encoded using codewords from a lattice code called a diagonally layered lattice dll code which are formatted onto the diagonals of a space time frame decoding is done using a sphere decoder for each diagonal based on soft statistics obtained after zero forcing zf or minimum mean square error mmse filtering and decision feedback these operations give rise to an effective parallel channel model with channel gains with nonidentical statistics and additive noise which is gaussian in the zf filtering case and non gaussian in the mmse filtering case the so called full modulation diversity fmd property is nevertheless shown to yield the maximum achievable diversity orders over the mimo channel for both the zf and the mmse filtering based decoders respectively for any arbitrary fading distribution in the case of the independent identically distributed i i d rayleigh fading mimo channel with k transmit and n receive antennas with ngesk these diversity orders are nk k k 1 2 and nk for zf and mmse filtering based decoding respectively the error probability analysis also yields a design criterion for optimizing transmit power allocations several lattice design methods are proposed for the effective parallel channel models two methods are proposed to achieve high coding gain in the rayleigh fading mimo channel a third method is proposed that minimizes the exact symbol error probability sep and can be tailored for any given fading distribution a novel soft decision feedback decoder is also proposed based on the list sphere decoder to mitigate error propagation due to hard decision feedback the salient feature of the proposed dll schemes is that they have nearly full rate and full or high diversity order and yet a much lower decoding complexity than other existing full rate full diversity space time block codes stbcs the frame error probability fep performance of the optimized dll schemes for moderate to high spectral efficiencies and a wide range of signal to noise ratios snrs can be quite close to the performance of the best performing but more complex to decode stbcs moreover the proposed dll schemes significantly outperform other existing mimo systems of comparable decoding complexity

distributed qam based space time block codes for efficient cooperative multiple access communication distributed space time block coding schemes based on quadrature amplitude modulation qam symbols are introduced that enable users in a wireless multiple access relay channel to cooperate with each other in order to improve the reliability of their information in a fair rate efficient manner the distributed coding schemes are designed for a system that consists of m users that need to send their information to a common destination each user is equipped with only one antenna and a half duplex transceiver so that it can transmit to the destination or any other user at one time and also receive signals from any other user at another time the destination is equipped with one or more receive antennas the goal is to design cooperative schemes which achieve the maximum diversity order which is equal to the diversity order of the outage probability while being fair to the users in terms of rate allocation two cooperative coding schemes are proposed that meet these requirements a two phase cooperative scheme and a single phase self information canceling linear scl scheme the two phase scheme is of lower decoding complexity and it requires the users to only transmit qam symbols the scl scheme is of higher complexity with the users transmitting linear combination of qam symbols but it also achieves better performance both schemes incorporate new classes of cooperation rules for deciding whether or not a user acts as a relay for another user the usual outage based cooperation rule whereby a user cooperates with another user provided the mutual information of the channel between them is greater than the rate is sufficient for deriving information theoretic limits but it cannot be used directly for analysis of a particular coding scheme even though the decoder used by the destination in the proposed coding schemes assumes that the inter user communication is always successful our performance analysis does not make this assumption in fact it rigorously accounts for the decoding errors arising from the information exchange between users consequently it sheds light on precisely what cooperation rules among the class of rules analyzed lead to maximal diversity

energy spreading transform approach to achieve full diversity and full rate for mimo systems full diversity full rate fdfr space time codes achieve both high data rate and good reliability at the cost of high decoding complexity in this paper we propose a low complexity mimo scheme that achieves both full diversity and full rate over flat fading channels for a sufficiently large number of transmit and receive antennas the proposed scheme is constructed by applying energy spreading transforms est s to multiple data streams and spatially multiplexing the streams to multiple transmit antennas simulation results show that the proposed fdfr scheme outperforms the threaded algebraic space time tast code which is a fdfr code based on maximum likelihood ml detection when the number of transmit antennas with the same number of receive antennas are three and four however its detection complexity is only that of a decision feedback detector

the amplify and forward half duplex cooperative system pairwise error probability and precoder design in this paper an exact asymptotic pairwise error probability pep is derived for a half duplex cooperative system employing an amplify and forward af protocol when compared with the pep of a traditional multiple input multiple output mimo system the diversity gain for the cooperative system is no longer just a simple exponential function of the signal to noise ratio snr rather it involves the logarithm of the snr the term diversity gain function is used to designate this characteristic of the pep the coding gain on the other hand is found similar to that for the mimo system and is proportional to the determinant of the autocorrelation of the error matrix based on our analysis and observations we propose a design of unitary precoder for the cooperative system to achieve the full diversity gain function for the case of a 4 qam signal being transmitted we further optimize the coding gain and arrive at a closed form optimum precoder simulations indicate that our proposed precoder designs greatly improve the performance of the cooperative system

positive solutions of second order systems of boundary value problems abstract the existence of nonnegative solutions of boundary value problems of the form x t a t x t y t 0 y t b t g x t y t 0 x 0 x 1 y 0 y 1 0 is investigated where and g are functions from r p r q into r p and r q respectively and a b are functions defined on 0 1 with values in the spaces of diagonal matrices of order p and q respectively all functions are assumed to be continuous and nonnegativity as well as monotonicity and growth conditions are imposed on and g the existence results hold for a bounded interval of values of if and g are assumed to grow fast and have only nonzero values while if 0 0 0 g 0 0 0 it is shown that under reasonable conditions there exists a nontrivial nonnegative solution for every 0

full rate full diversity space frequency codes with optimum coding advantage a general space frequency sf block code structure is proposed that can guarantee full rate one channel symbol per subcarrier and full diversity transmission in multiple input multiple output orthogonal frequency division multiplexing mimo ofdm systems the proposed method can be used to construct sf codes for an arbitrary number of transmit antennas any memoryless modulation and arbitrary power delay profiles moreover assuming that the power delay profile is known at the transmitter we devise an interleaving method to maximize the overall performance of the code we show that the diversity product can be decomposed as the product of the intrinsic diversity product which depends only on the used signal constellation and the code design and the extrinsic diversity product which depends only on the applied interleaving method and the power delay profile of the channel based on this decomposition we propose an interleaving strategy to maximize the extrinsic diversity product extensive simulation results show that the proposed sf codes outperform the previously existing codes by about 3 5 db and that the proposed interleaving method results in about 1 3 db performance improvement compared to random interleaving

tensor analysis of anova decomposition abstract the analogy between anova for general n factor crossed layouts and the anova type decomposition of square integrable statistics is demonstrated using the notions and notations of tensor analysis and multilinear algebra a theory of tensors is developed in such a way that a it can be immediately applied in computer programs and b it can be easily generalized to l 2 spaces

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

on diagonal algebraic space time block codes theoretical and practical aspects of diagonal algebraic space time block codes over n transmit and m receive antennae are examined these codes are obtained by sending a rotated version of the information symbols over the principal diagonal of the n spl times n space time matrix over n transmit antennae and n symbol periods the output signal to noise ratios of two predecoding filters and two decoding algorithms are derived analysis of the information loss incurred by using the codes considered is used to clarify their structures and the expected performances different algebraic real and complex rotations presented in the literature are analyzed and compared as regards the achieved coding gains the complexities performances and peak to mean envelope power ratios

diagonal lattice space time codes from number fields and asymptotic bounds in this paper we reformulate some constructions of real and complex diagonal lattice space time codes from number fields which have been given explicitly or implicitly by other researchers these constructions establish a connection between good diagonal lattice space time codes and number fields with small absolute values of discriminants we present two tables for diversity products of some lattice space time codes from these constructions the maximal rank of diagonal lattice space time codes with positive diversity product is determined we also discuss the asymptotic problem of lattice space time codes by using an infinite tower of hilbert class field and a tamely ramified class field tower we obtain asymptotically good sequences of lattice space time codes some asymptotic upper bounds are given in the paper as well

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

achieving the optimal diversity versus multiplexing tradeoff for mimo flat channels with qam space time spreading and dfe equalization the use of multiple transmit tx and receive rx antennas allows to transmit multiple signal streams in parallel and hence to increase communication capacity we have previously introduced simple convolutive linear precoding schemes that spread transmitted symbols in time and space involving spatial spreading delay diversity and possibly temporal spreading in this paper we show that the use of the classical multiple input multiple output mimo decision feedback equalizer dfe but with joint detection for this system allows to achieve the optimal diversity versus multiplexing tradeoff introduced in zheng and tse diversity and multiplexing a fundamental tradeoff in multiple antenna channels ieee trans inf theory may 2003 when a minimum mean squared error mmse design is used one of the major contributions of this work is the diversity analysis of a mmse equalizer without the gaussian approximation furthermore the tradeoff is discussed for an arbitrary number of transmit and receive antennas we also show the tradeoff obtained for a mmse zero forcing zf design so another originality of this paper is to show that the mimo optimal tradeoff can be attained with a suboptimal receiver in this case a dfe as opposed to optimal maximum likelihood sequence estimation mlse

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

space time block code designs based on quadratic field extension for two transmitter antennas space time block code designs based on algebraic field extension for full rate large diversity product and nonvanishing minimum determinant of codewords have received great attention there are many different types of codes available for two transmitter antennas such as cyclotomic space time block codes the golden space time block code and rotation based space time block codes in this paper a more general space time block code design scheme which is called quadratic space time block coding is proposed for the two transmitter antennas using quadratic field extension the optimal design of the quadratic space time block codes in terms of a diversity product criterion is also presented it is shown that the optimal quadratic space time block codes designed in this paper do not belong to the existing space time block code family such as the cyclotomic golden and rotation based space time block codes the simulation results demonstrate that the average codeword error rate of the optimal quadratic space time block code attains about 0 5 db signal to noise ratio gain over those of the optimal cyclotomic and golden space time block codes

generalized entropy power inequalities and monotonicity properties of information new families of fisher information and entropy power inequalities for sums of independent random variables are presented these inequalities relate the information in the sum of n independent random variables to the information contained in sums over subsets of the random variables for an arbitrary collection of subsets as a consequence a simple proof of the monotonicity of information in central limit theorems is obtained both in the setting of independent and identically distributed i i d summands as well as in the more general setting of independent summands with variance standardized sums

coset codes i introduction and geometrical classification practically all known good constructive coding techniques for bandlimited channels including lattice codes and various trellis coded modulation schemes can be characterized as coset codes a coset code is defined by a lattice partition lambda lambda and by a binary encoder c that selects a sequence of cosets of the lattice lambda the fundamental coding gain of a coset code as well as other important parameters such as the error coefficient the decoding complexity and the constellation expansion factor are purely geometric parameters determined by c lambda lambda the known types of coset codes as well as a number of new classes that systematize and generalize known codes are classified and compared in terms of these parameters

systematic construction of full diversity algebraic constellations a simple and systematic approach for constructing full diversity m dimensional constellations carved from lattices over a number ring r is proposed for an arbitrary dimension m when r z wn the nth cyclotomic number ring all the possible dimensions that allow for achieving the optimal minimum product distances using the proposed approach are determined it turns out that one can construct optimal unitary transformations using our construction if and only if m factors into a power of 2 and powers of the primes dividing n for m not satisfying these conditions a method based on diophantine approximation theory is proposed to optimize the minimum product distance a lower bound on the product distance is given in this case thus ensuring full diversity with good minimum product distances furthermore the proposed approach subsumes the optimal unitary transformations proposed by giraud et al over r z w4 and r z w3 while giving optimal unitary transformations for infinitely many new values of n and m

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

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

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

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

maximal diversity algebraic space time codes with low peak to mean power ratio the design requirements for space time coding typically involves achieving the goals of good performance high rates and low decoding complexity in this paper we introduce a further constraint on space time code design in that the code should also lead to low values of the peak to mean envelope power ratio pmepr for each antenna towards that end we propose a new class of space time codes called the low pmepr space time lpst codes the lpst codes are obtained using the properties of certain cyclotomic number fields the lpst codes achieve a performance identical to that of the threaded algebraic space time tast codes but at a much smaller pmepr with m antennas and a rate of one symbol per channel use the lpst codes lead to a decrease in pmepr by at least a factor of m relative to a hadamard spread version of the tast code for rates beyond one symbol per channel use and up to a guaranteed amount the lpst codes have provably smaller pmepr than the corresponding tast codes additionally with the concept of punctured lpst codes proposed in this paper significant performance improvement is obtained over the full diversity tast schemes of comparable complexity numerical examples are provided to illustrate the advantage of the proposed codes in terms of pmepr reduction and performance improvement for very high rate wireless communications

on the existence of positive solutions for semilinear elliptic equations in the annulus abstract we study the existence of positive radial solutions of u g x u 0 in annuli with dirichlet dirichlet neumann boundary conditions we prove that the problems have positive radial solutions on any annulus if is sublinear at 0 and

a batch means methodology for estimation of a nonlinear function of a steady state mean we study the estimation of steady state performance measures from an d valued stochastic process y y t t 0 representing the output of a simulation in many applications we may be interested in the estimation of a steady state performance measure that cannot be expressed as a steady state mean r e g the variance of the steady state distribution the ratio of steady state means and steady state conditional expectations these examples are particular cases of a more general problem the estimation of a nonlinear function f r of r we propose a batch means based methodology that allows us to use jackknifing to reduce the bias of the point estimator asymptotically valid confidence intervals for f r are obtained by combining three different point estimators classical batch means and jackknife with two different variability estimators classical and jackknife the performances of the point estimators are discussed by considering asymptotic expansions for their biases and mean squared error

trellis coded modulation with multidimensional constellations trellis coded modulation schemes using four eight or 16 dimensional constellations have a number of potential advantages over the usual two dimensional schemes a smaller constituent two dimensional constellation easier tolerance to phase ambiguities and a better trade off between complexity and coding gain a number of such schemes are presented and evaluated starting with a variety of multidimensional lattices we show how to select multidimensional constellations how to partition them into subsets how to construct trellis codes using those subsets and how to map hits to constellation points simplifications of the viterbi decoding algorithm are presented we conclude that there are multidimensional trellis coded modulation schemes that perform better for the same complexity than do two dimensional schemes

v ofdm on performance limits over multi path rayleigh fading channels as a bridge of connecting orthogonal frequency division multiplexing ofdm with single carrier frequency domain equalization sc fde techniques vector ofdm v ofdm provides significant flexibility in system design this paper presents an analytical study of v ofdm over multi path fading channels our goal is to investigate the diversity gain and coding gain of each vector block vb in v ofdm so as to ultimately reveal its performance limits over fading channel by using algebraic number theory tools we rigorously prove for the first time that a majority of vbs in v ofdm can surely realize the diversity gain of min m g where m is the length of each vb and g is the total number of channel taps furthermore some specific vbs whose length equals the total number of channel taps can not only harvest the maximum diversity gain but also achieve the maximum coding gain it is further demonstrated that even though vbs fail to benefit from additional diversity gain when m exceeds g they can enjoy significantly increased coding gains our analysis concludes that it is preferable to choose the length of vbs to be equal to the number of channel taps in consideration of both overall system performance and computational complexity

orthogonal space time block codes with sphere packing orthogonal designs have received considerable attention in the development of efficient modulation and coding methods for future multi antenna wireless communication systems due to their special properties in this paper we propose a class of space time block codes constructed by combining orthogonal designs with sphere packing for an arbitrary number of transmit antennas the structure of the orthogonal designs is exploited to guarantee full diversity and sphere packing is used to improve the coding advantage space time block code construction from block orthogonal designs is also considered the full diversity property is ensured by rotating the sphere packing underlying the code and the optimal rotation angle is determined for a class of sphere packing code design examples are provided for two and four transmit antennas and various transmission rates the simulation results show that by jointly designing the symbols in the orthogonal designs the performance of the block codes can be significantly increased

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

on performance of vector ofdm with linear receivers vector orthogonal frequency division multiplexing ofdm for single transmit antenna systems is a general transmission scheme where ofdm and single carrier frequency domain equalization sc fde can be treated as two special extreme cases due to its flexibility it has drawn more and more attention recently so far all the studies about vector ofdm assume the maximum likelihood ml receiver in this paper we investigate the performance of vector ofdm with linear receivers i e the zero forcing zf and minimum mean square error mmse receivers we first show that the detection snr gap between the mmse and zf receivers increases with both channel snr and the vector blocks vb size defined in vector ofdm then it is proved that for both zf and mmse receivers all the transmitted symbols have equal performance this is different from the vector ofdm with ml receiver where different vbs may have different coding gain and thus may have different performances we analyze the diversity order for vector ofdm with mmse receiver and show that regardless of the vector ofdm symbol length n the diversity order can be represented as min m2 r d 1 where m is the vb size r is the spectrum efficiency in bits symbol and d is the maximum delay of the multipath channel for vector ofdm with zf receiver we show that the diversity order equals 1 and the performance is the same as the conventional ofdm at high snr

systematic construction of linear transform based full diversity rate one space time frequency codes in this paper we generalize the existing rate one space frequency sf and space time frequency stf code constructions the objective of this exercise is to provide a systematic design of full diversity stf codes with high coding gain under this generalization stf codes are formulated as linear transformations of data conditions on these linear transforms are then derived so that the resulting stf codes achieve full diversity and high coding gain with a moderate decoding complexity many of these conditions involve channel parameters like delay profile dp and temporal correlation when these quantities are not available at the transmitter design of codes that exploit full diversity on channels with arbitrary dp and temporal correlation is considered complete characterization of a class of such robust codes is provided and their bit error rate ber performance is evaluated on the other hand when channel dp and temporal correlation are available at the transmitter linear transforms are optimized to maximize the coding gain of full diversity stf codes ber performance of such optimized codes is shown to be better than those of existing codes

hoeffding decompositions and urn sequences let x x1 x2 be a nondeterministic infinite exchangeable sequence with values in 0 1 we show that x is hoeffding decomposable if and only if x is either an i i d sequence or a polya sequence this completes the results established in peccati ann probab 32 2004 1796 1829 the proof uses several combinatorial implications of the correspondence between hoeffding decomposability and weak independence our results must be compared with previous characterizations of i i d and polya sequences given by hill lane and sudderth ann probab 15 1987 1586 1592 and diaconis and ylvisaker ann statist 7 1979 269 281 the final section contains a partial characterization of hoeffding decomposable sequences with values in a set with more than two elements

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

geometrically uniform codes a signal space code c is defined as geometrically uniform if for any two code sequences in c there exists an isometry that maps one sequence into the other while leaving the code c invariant geometrical uniformity a strong kind of symmetry implies such properties as a the distance profiles from code sequences in c to all other code sequences are all the same and b all voronoi regions of code sequences in c have the same shape it is stronger than ungerboeck zehavi wolf symmetry or calderbank sloane regularity nonetheless most known good classes of signal space codes are shown to be generalized coset codes and therefore geometrically uniform including a lattice type trellis codes based on lattice partitions lambda lambda such that z sup n lambda lambda 4z sup n is a lattice partition chain and b phase shift keying psk type trellis codes based on up to four way partitions of a 2 sup n psk signal set

systematic and optimal cyclotomic lattices and diagonal space time block code designs in this correspondence a new and systematic design of cyclotomic lattices with full diversity is proposed by using some algebraic number theory this design provides infinitely many full diversity cyclotomic lattices for a given lattice size based on the packing theory and the concrete form of the design optimal cyclotomic lattices are presented by minimizing the mean transmission signal power for a given minimum diversity product or equivalently maximizing the minimum product for a given mean transmission signal power the newly proposed cyclotomic lattices can be applied to both space time code designs for multi antenna systems and linear precode design for signal space diversity in single antenna systems over fast rayleigh fading channels although there are some cyclotomic lattices space time codes existing in the literature most of them are not optimal

signal constellations for quasi orthogonal space time block codes with full diversity space time block codes stbcs from orthogonal designs proposed by alamouti and tarokh jafarkhani calderbank have attracted considerable attention lately due to their fast maximum likelihood ml decoding and full diversity however the maximum symbol transmission rate of an stbc from complex orthogonal designs for complex signals is only 3 4 for three and four transmit antennas and it is difficult to construct complex orthogonal designs with rate higher than 1 2 for more than four transmit antennas recently jafarkhani tirkkonen boariu hottinen and papadias foschini proposed stbcs from quasi orthogonal designs where the orthogonality is relaxed to provide higher symbol transmission rates with the quasi orthogonal structure the quasi orthogonal stbcs still have a fast ml decoding but do not have the full diversity the performance of these codes is better than that of the codes from orthogonal designs at low signal to noise ratio snr but worse at high snr this is due to the fact that the slope of the performance curve depends on the diversity it is desired to have the quasi orthogonal stbcs with full diversity to ensure good performance at high snr in this paper we achieve this goal by properly choosing the signal constellations specifically we propose that half of the symbols in a quasi orthogonal design are chosen from a signal constellation set a and the other half of them are chosen from a rotated constellation e sup j spl phi a the resulting stbcs can guarantee both full diversity and fast ml decoding moreover we obtain the optimum selections of the rotation angles spl phi for some commonly used signal constellations simulation results show that the proposed codes outperform the codes from orthogonal designs at both low and high snrs

constellations matched to the rayleigh fading channel we introduce a new technique for designing signal sets matched to the rayleigh fading channel in particular we look for n dimensional n spl ges 2 lattices whose structure provides nth order diversity our approach is based on a geometric formulation of the design problem which in turn can be solved by using a number geometric approach specifically a suitable upper bound on the pairwise error probability makes the design problem tantamount to the determination of what is called a critical lattice of the body s x x sub 1 spl middot spl middot spl middot x sub n spl isin r sup n spl pi sub i 1 sup n x sub i spl les 1 the lattices among which we search for an optimal solution are the standard embeddings in r sup n of the number ring of some totally real number field of degree n over q simulation results confirm that this approach yields lattices with considerable coding gains

delay tolerant distributed tast codes for cooperative diversity in cooperative networks using a decode and forward strategy the multiple relays effectively transmit a distributed space time code the performance of which can be severely degraded when timing synchronization among the relays is not assured e g in cases of broadcast to dispersed recipients or in networks without a shared high quality timing reference recent work by xia and hammons have investigated the design of distributed space time codes that are delay tolerant in the sense that full spatial diversity is achieved regardless of timing offsets in general the previously known space time block codes belonging to the class of c linear codes however which are important because they achieve full spatial diversity and admit near optimal lattice decoding algorithms are not delay tolerant in this paper we present a new family of such codes that are fully delay tolerant the new codes generalize the threaded algebraic space time tast codes introduced by el gamal and damen like their brethren the new distributed tast codes are effective and flexible enabling use of different signaling constellations transmission rates numbers of transmit and receive antennas and decoders of varying levels of complexity

high rate full diversity space time frequency codes for broadband mimo block fading channels a systematic design of high rate full diversity space time frequency stf codes is proposed for multiple input multiple output frequency selective block fading channels it is shown that the proposed stf codes can achieve rate mt and full diversity mtmrmbl i e the product of the number of transmit antennas mt receive antennas mr fading blocks mb and channel taps l the proposed stf codes are constructed from a layered algebraic design where each layer of algebraic coded symbols are parsed into different transmit antennas orthogonal frequency division multiplexing tones and fading blocks without rate loss simulation results show that the proposed stf codes achieve higher diversity gain in block fading channels than some typical space frequency codes

positive solutions for semipositone k n k conjugate boundary value problems for 1 k n 1 positive solutions are obtained for the boundary value problem 1 n k y n fx y x 0 1y i 0 0 0 i k 1 y j 1 0 0 j n k 1 where f x y m and m is a positive constant we show the existence of positive solutions by using a fixed point theorem in cones

an algebraic family of complex lattices for fading channels with application to space time codes a new approach is presented for the design of full modulation diversity fmd complex lattices for the rayleigh fading channel the fmd lattice design problem essentially consists of maximizing a parameter called the normalized minimum product distance d sub p sup 2 of the finite signal set carved out of the lattice we approach the problem of maximizing d sub p sup 2 by minimizing the average energy of the signal constellation obtained from a new family of fmd lattices the unnormalized minimum product distance for every lattice in the proposed family is lower bounded by a nonzero constant minimizing the average energy of the signal set translates to minimizing the frobenius norm of the generator matrices within the proposed family the two strategies proposed for the frobenius norm reduction are based on the concepts of successive minima sm and basis reduction of an equivalent real lattice the lattice constructions in this paper provide significantly larger normalized minimum product distances compared to the existing lattices in certain dimensions the proposed construction is general and works for any dimension as long as a list of number fields of the same degree is available

orthogonal decomposition of finite population statistics and its applications to distributional asymptotics we study orthogonal decomposition of symmetric statistics based on samples drawn without replacement from finite populations several applications to finite population statistics are given we establish one term edge worth expansions for general asymptotically normal symmetric statistics prove an efron stein inequality and the consistency of the jackknife estimator of variance our expansions provide second order a s approximations to wu s jackknife histogram

elementary symmetric polynomials of increasing order the asymptotic behaviour of elementary symmetric polynomials sn k of order k based on n independent and identically distributed random variables x1 xn is investigated for the case that both k and n get large if k o n frac 1 2 then the distribution function of a suitably normalised sn k is shown to converge to a standard normal limit the speed of this convergence to normality is of order mathcal o kn frac 1 2 provided k mathcal o log 1 nlog 2 1 nn frac 1 2 and certain natural moment assumptions are imposed this order bound is sharp and cannot be inferred from one of the existing berry esseen bounds for u statistics if k at the rate n1 2 then a non normal weak limit appears provided the xi s are positive and sn k is standardised appropriately on the other hand if k at a rate faster than n1 2 then it is shown that for positive xj sthere exists no linear norming which causes sn k to converge weakly to a nondegenerate weak limit

linear threaded algebraic space time constellations space time st constellations that are linear over the field of complex numbers are considered relevant design criteria for these constellations are summarized and some fundamental limits to their achievable performances are established the fundamental tradeoff between rate and diversity is investigated under different constraints on the peak power receiver complexity and rate scaling with the signal to noise ratio snr a new family of constellations that achieve optimal or near optimal performance with respect to the different criteria is presented the proposed constellations belong to the threaded algebraic st tast signaling framework and achieve the optimal minimum squared euclidean distance and the optimal delay for systems with one receive antenna these constellations also achieve the optimal peak to average power ratio for quadrature amplitude modulation qam and phase shift keying psk input constellations as well as optimal coding gains in certain scenarios the framework is general for any number of transmit and receive antennas and allows for realizing the optimal tradeoff between rate and diversity under different constraints simulation results demonstrate the performance gains offered by the proposed designs in average power and peak power limited systems

diagonal algebraic space time block codes we construct a new family of linear space time st block codes by the combination of rotated constellations and the hadamard transform and we prove them to achieve the full transmit diversity over a quasi static or fast fading channels the proposed codes transmit at a normalized rate of 1 symbol s when the number of transmit antennas n 1 2 or n is a multiple of four we spread a rotated version of the information symbol vector by the hadamard transform and send it over n transmit antennas and n time periods for other values of n we construct the codes by sending the components of a rotated version of the information symbol vector over the diagonal of an n spl times n st code matrix the codes maintain their rate diversity and coding gains for all real and complex constellations carved from the complex integers ring z i and they outperform the codes from orthogonal design when using complex constellations for n 2 the maximum likelihood ml decoding of the proposed codes can be implemented by the sphere decoder at a moderate complexity it is shown that using the proposed codes in a multiantenna system yields good performances with high spectral efficiency and moderate decoding complexity

on normal approximations to u statistics let x1 xn be i i d random observations taking their values in a measurable space consider a u statistic of order k says s x1 xn based on the sample x1 xn assume that s can be represented in the form s l t with el et 0 where l is a linear statistic and t is a stochastically smaller statistic assume further that the linear statistic l is approximately normal and that we have a bound for error of such approximation our objective is to show that a similar bound holds for the distribution ofs provided that we correct it by adding vart i ln 2 3 1 vart where vart et 2 is the variance of t it occurs that one has to correct the normal approximation as well replacing it by some asymptotically as n 1 equivalent distribution furthermore we show that the class of u statistics up to negligible error terms is closed under studentization or self normalization for example using the jackknife estimators of variances this leads to error bounds of normal approximations for non deterministically normalized u statistics our results extend reflne and yield a number of related known results the bounds have a simple and convenient form for applications and analysis since they involve only the variance of the noise termt in the case k 3 of higher order statistics it remains open whether one can improve the bounds removing the logarithmic factor ln 2 3 1 vart we provide as well an optimal bound under a lower moment assumption ejtj fi 1 fi 2