linear time nearest point algorithms for coxeter lattices the coxeter lattices are a family of lattices containing many of the important lattices in low dimensions this includes an e 7 e 8 and their duals an e 7 and e 8 we consider the problem of finding a nearest point in a coxeter lattice we describe two new algorithms one with worst case arithmetic complexity o nlogn and the other with worst case complexity o n where n is the dimension of the lattice we show that for the particular lattices an and an the algorithms are equivalent to nearest point algorithms that already exist in the literature

type 2 fuzzistics for nonsymmetric interval type 2 fuzzy sets forward problems interval type 2 fuzzy sets it2 fs play a central role in fuzzy sets as models for words and in engineering applications of t2 fss these fuzzy sets are characterized by their footprints of uncertainty fou which in turn are characterized by their boundaries upper and lower membership functions mf the centroid of an it2 fs which is an it1 fs provides a measure of the uncertainty in the it2 fs the main purpose of this paper is to quantify the centroid of a non symmetric it2 fs with respect to geometric properties of its fou this is very important because interval data collected from subjects about words suggests that the fous of most words are non symmetrical using the results in this paper it is possible to formulate and solve forward problems i e to go from parametric non symmetric it2 fs models to data with associated uncertainty bounds we provide some solutions to such problems for non symmetrical triangular trapezoidal gaussian and shoulder fous

a lower bound on the minimum euclidean distance of trellis coded modulation schemes a lower bound on the minimum free euclidean distance of trellis coded modulation tcm is derived that guarantees the existence of good tcm codes of any complexity the bound is used to compare trellis codes combined with phase shift keying pulse amplitude modulation and quadratic amplitude shift keying modulation this random coding bound is the first lower bound on the free distance of trellis codes is tighter than any upper bound for large constraint lengths and predicts the asymptotic performance of tcm when the complexity of the code becomes large the bound can be used with any code rate and any modulation scheme and shows that the free distance increases linearly with the constraint length for large values of the constraint length

mpeg vbr video traffic modeling and classification using fuzzy technique we present an approach for mpeg variable bit rate vbr video modeling and classification using fuzzy techniques we demonstrate that a type 2 fuzzy membership function i e a gaussian mf with uncertain variance is most appropriate to model the log value of i p b frame sizes in mpeg vbr video the fuzzy c means fcm method is used to obtain the mean and standard deviation std of t p b frame sizes when the frame category is unknown we propose to use type 2 fuzzy logic classifiers flcs to classify video traffic using compressed data five fuzzy classifiers and a bayesian classifier are designed for video traffic classification and the fuzzy classifiers are compared against the bayesian classifier simulation results show that a type 2 fuzzy classifier in which the input is modeled as a type 2 fuzzy set and antecedent membership functions are modeled as type 2 fuzzy sets performs the best of the five classifiers when the testing video product is not included in the training products and a steepest descent algorithm is used to tune its parameters

on type 2 fuzzy relations and interval valued type 2 fuzzy sets this paper introduces new operations on the algebra of fuzzy truth values extended supremum and extended infimum which are generalizations of the extended operations of maximum and minimum between fuzzy truth values for type 2 fuzzy sets respectively using these new operations the properties of type 2 fuzzy relations are discussed especially the compositions of type 2 fuzzy relations on this basis this paper introduces interval valued type 2 fuzzy sets and interval valued type 2 fuzzy relations and discusses their properties

a fast encoding method for lattice codes and quantizers in an earlier paper the authors described a very fast method which for the root lattices a n d n e n their duals and certain other lattices finds the closest lattice point to an arbitrary point of the underlying space if the lattices are used as codes for a gaussian channel the algorithm provides a fast decoding procedure or if they are used as vector quantizers the algorithm performs the analog to digital conversion efficiently the present paper offers a solution to the inverse problem for the same lattices the encoding problem for channel codes or the digital to analog part of quantizing namely given an integer k to find the kth code vector and to the closely related problem of finding the index k of a given code vector

probabilistic construction of large constraint length trellis codes for sequential decoding probabilistic algorithms are given for constructing good large constraint length trellis codes for use with sequential decoding that can achieve the channel cutoff rate bound at a bit error rate ber of 10 sup 5 10 sup 6 the algorithms are motivated by the random coding principle that an arbitrary selection of code symbols will produce a good code with high probability one algorithm begins by choosing a relatively small set of codes randomly the error performance of each of these codes is evaluated using sequential decoding and the code with the best performance among the chosen set is retained another algorithm treats the code construction as a combinatorial optimization problem and uses simulated annealing to direct the code search trellis codes for 8 psk and 16 qam constellations with constraint lengths v up to 20 are obtained simulation results with sequential decoding show that these codes reach the channel cutoff rate bound at a ber of 10 sup 5 10 sup 6 and achieve 5 0 6 35 db real coding gains over uncoded systems with the same spectral efficiency and up to 2 0 db real coding gains over 64 state trellis codes using viterbi decoding

on km algorithms for solving type 2 fuzzy set problems computing the centroid and performing type reduction for type 2 fuzzy sets and systems are operations that must be taken into consideration karnik mendel km algorithms are the standard ways to do these operations however because these algorithms are iterative much research has been conducted during the past decade about centroid and type reduction computations this tutorial paper focuses on the research that has been conducted to 1 improve the km algorithms 2 understand the km algorithms leading to further improved algorithms 3 eliminate the need for km algorithms 4 use the km algorithms to solve other nonfuzzy logic system problems and 5 use or not use km algorithms for general type 2 fuzzy sets and fuzzy logic systems

finite element theory for curved and twisted beams based on exact solutions for three dimensional solids part 2 anisotropic and advanced beam models consistent finite element formulations for beams made of anisotropic materials and taking into account non classic inhomogeneous torsion have been developed the formulations are based on a kinematical hypothesis that includes exact solutions for three dimensional solids under terminal loading they describe warping of the cross sections in and out of their planes as well as their rigid displacements and rotations their large deformation and geometrically exact description by finite rotations are considered for the cases of monoclinic orthotropic and transversely isotropic materials exact solutions for the solid made from a monoclinic material have been deduced c 1998 elsevier science s a all rights reserved

shaping of multidimensional signal constellations using a lookup table this paper describes a lookup table for the addressing of an optimally shaped constellation the method is based on partitioning the subconstellations into shaping macro shells of integer bit rate and increasing average energy the macro shells do not need to have an equal number of points a lookup table is used to select a subset of the partitions in the cartesian product space by devising appropriate partitioning merging rules we obtain suboptimum schemes of very low addressing complexity and small performance degradation the performance is computed using the weight distribution of an optimally shaped constellation

a corotational based geometrically nonlinear generalized beam theory buckling fe analysis abstract a geometrically nonlinear generalized beam theory is formulated and the results in the framework of buckling analyses are discussed the geometrically nonlinear model is recovered reusing the model available in the linear context this generalization to the nonlinear context is obtained exploiting the corotational based method called implicit corotational method starting from a mixed energy description of the continuum in terms of nonlinear biot stress strain tensors and using convenient assumptions for the linear stress strain tensors once obtained the nonlinear model has been implemented using a flexibility based finite element the results of buckling analyses for different and complex beam cross section geometries emphasizing distortional and local buckling behaviors are presented and comparisons with finite element shell models made

nested linear lattice codes for structured multiterminal binning network information theory promises high gains over simple point to point communication techniques at the cost of higher complexity however lack of structured coding schemes limited the practical application of these concepts so far one of the basic elements of a network code is the binning scheme wyner 1974 1978 and other researchers proposed various forms of coset codes for efficient binning yet these schemes were applicable only for lossless source or noiseless channel network coding to extend the algebraic binning approach to lossy source or noisy channel network coding previous work proposed the idea of nested codes or more specifically nested parity check codes for the binary case and nested lattices in the continuous case these ideas connect network information theory with the rich areas of linear codes and lattice codes and have strong potential for practical applications we review these developments and explore their tight relation to concepts such as combined shaping and precoding coding for memories with defects and digital watermarking we also propose a few novel applications adhering to a unified approach

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

tables of sphere packings and spherical codes the theta function of a sphere packing gives the number of centers at each distance from the origin the theta functions of a number of important packings a n d n e n the leech lattice and others and tables of the first fifty or so of their coefficients are given in this paper

generalized concatenation of encoded tamed frequency modulation a novel construction for encoded tamed frequency modulation tfm is introduced which is based on the principles of generalized concatenation the inner tfm is partitioned into nested subsystems which increases the free euclidean distances in order to obtain a large distance among the nested tfm subsystems the scrambler matrices have to be computed which transfer the original tfm into the equivalent tfm with better partitioning properties then outer convolutional codes with different error correcting capabilities are used to protect the partitioning the new concatenated and generalized concatenated constructions were simulated in an additive white gaussian noise channel a multistep decoding algorithm based on soft output demodulation was used we present various simulation results which show a significant coding gain in comparison with the best known trellis codes having the same trellis state complexity

ensuring the centroid of an interval type 2 fuzzy set in computing with words establishing a fuzzy set fs model for a word to capture its uncertainties is an important issue an interval type 2 it2 fs can be used to model a word how to establish an it2 fs from the collected data about a word has been a challenging problem it has been reported that one way is to extract the centroid of an it2 fs from the collected data and to obtain geometric parameters of its footprint of uncertainty fou such that its centroid matches the extracted one how to extract the centroid of an it2 fs from the collected data about a word has thoroughly been studied however there exists no method for obtaining fou parameters for an it2 fs such that its centroid matches the desired one to fill this gap this paper presents an approach to obtaining fou parameters for an it2 fs by establishing equations using the centroid requirement to propose this approach a sufficient and necessary condition for ensuring the centroid of an it2 fs is developed using this sufficient and necessary condition two equations about all of the fou parameters are established to obtain the fou parameters all of them except two are predetermined so that the established equations can be simplified to two single variable equations the other two fou parameters can then be determined by solving these two single variable equations using existing root finding algorithms among existing root finding algorithms the false position algorithm is recommended the overall merits of the proposed approach are its simplicity in implementation and its applicability to it2 fss with arbitrary fou shapes in addition numerical examples are provided to further illustrate how to apply the proposed approach to obtain fou parameters for an it2 fs

rotational invariance of trellis codes i encoders and precoders we present a theoretical framework for rotational invariance of trellis codes the distinction between codes and encoders plays a pivotal role necessary and sufficient conditions for rotational invariance are derived under general assumptions and a construction is presented that obtains a rotationally invariant encoder for almost any rotationally invariant code independent of the code s algebraic structure encoders that use a differential precoder are considered as a separate case where a system theoretic characterization of precoding is used to find two alternative and slightly less general encoder constructions

channel coding with quadrature quadrature phase shift keying q sup 2 psk signals channel coding due to trellis modulation has been proved to be useful for bandlimited channels however these modulations mostly designed with n use of 2d signals are primarily aimed at coding gain only it is pointed out that utilization of all available signal dimensions which is limited by the time bandwidth product may improve the bandwidth efficiency and simultaneously bring an additional coding gain trellis coding with a spectrally efficient 4d signal set based on q sup 2 psk is addressed without any expansion of the q sup 2 psk signal set a simple hand designed 16 state trellis code provides a coding gain of 6 02 db with the same number of states in g ungergoeck s 1982 8 psk trellis the gain is 4 1 db in low intersymbol interference situations the bandwidth efficiency of this coded q sup 2 psk is twice that of coded 8 psk if both operate at 2 b s hz and p sub b e 10 sup 5 the coded q sup 2 psk provides a saving of about 4 db over the coded 8 psk some fully connected trellises with an expanded signal due to 2 use of q sup 2 psk signals are also presented two such codes ar rate 7 8 achieve a gain of 5 45 db with only eight states

a fast geometric method for defuzzification of type 2 fuzzy sets generalized type 2 fuzzy logic systems cannot currently be used for practical problems because the amount of computation required to defuzzify a generalized type 2 fuzzy set is too large this paper presents a new method for defuzzifing a type 2 fuzzy set the new much faster technique is based on geometric representations and operations the results of a real world example contained in this paper show this new approach to be over 200 000 times faster than type reduction we present a new method for assessing the accuracy of the membership function of a type 2 fuzzy set this method is used to show that the new representation used by the defuzzifier is not detrimental to the accuracy of the set we also discuss the differences between the new approach and type reduction identifying the origin of this massive improvement in execution speed

equalization of nonlinear time varying channels using type 2 fuzzy adaptive filters presents a kind of adaptive filter type 2 fuzzy adaptive filter faf one that is realized using an unnormalized type 2 takagi sugeno kang tsk fuzzy logic system fls we apply this filter to equalization of a nonlinear time varying channel and demonstrate that it can implement the bayesian equalizer for such a channel has a simple structure and provides fast inference a clustering method is used to adaptively design the parameters of the faf two structures are used for the equalizer transversal equalizer te and decision feedback equalizer dfe a decision tree structure is used to implement the decision feedback equalizer in which each leaf of the tree is a type 2 faf this dfe vastly reduces computational complexity as compared to a te simulation results show that equalizers based on type 2 fafs perform much better than nearest neighbor classifiers nnc or equalizers based on type 1 fafs

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

uncertainty bounds and their use in the design of interval type 2 fuzzy logic systems we derive inner and outer bound sets for the type reduced set of an interval type 2 fuzzy logic system fls based on a new mathematical interpretation of the karnik mendel iterative procedure for computing the type reduced set the bound sets can not only provide estimates about the uncertainty contained in the output of an interval type 2 fls but can also be used to design an interval type 2 fls we demonstrate by means of a simulation experiment that the resulting system can operate without type reduction and can achieve similar performance to one that uses type reduction therefore our new design method based on the bound sets can relieve the computation burden of an interval type 2 fls during its operation which makes an interval type 2 fls useful for real time applications

channel coding with multilevel phase signals a coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth this is achieved by channel coding with expanded sets of multilevel phase signals in a manner which increases free euclidean distance soft maximum likelihood ml decoding using the viterbi algorithm is assumed following a discussion of channel capacity simple hand designed trellis codes are presented for 8 phase shift keying psk and 16 quadrature amplitude shift keying qask modulation these simple codes achieve coding gains in the order of 3 4 db it is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals which we call mapping by set partitioning based on a new distance measure between binary code sequences which efficiently lower bounds the euclidean distance between the corresponding channel signal sequences a search procedure for more powerful codes is developed codes with coding gains up to 6 db are obtained for a variety of multilevel phase modulation schemes simulation results are presented and an example of carrier phase tracking is discussed

graph based iterative decoding algorithms for parity concatenated trellis codes we construct parity concatenated trellis codes in which a trellis code is used as the inner code and a simple parity check code is used as the outer code from the tanner wiberg loeliger 1981 1996 graph representation several iterative decoding algorithms can be derived however since the graph of the parity concatenated code contains many short cycles the conventional min sum and sum product algorithms cannot achieve near optimal decoding after some simple modifications we obtain near optimal iterative decoders the modifications include either a introducing a normalization operation in the min sum and sum product algorithms or b cutting the short cycles which arise in the iterative viterbi algorithm iva after modification all three algorithms can achieve near optimal performance but the iva has the least average complexity we also show that asymptotically maximum likelihood ml decoding and a posteriori probability app decoding can be achieved using iterative decoders with only two iterations unfortunately this asymptotic behavior is only exhibited when the bit energy to noise ratio is above the cutoff rate simulation results show that with trellis shaping iterative decoding can perform within 1 2 db of the shannon limit at a bit error rate ber of 4 spl times 10 sup 5 for a block size of 20000 symbols for a block size of 200 symbols iterative decoding can perform within 2 1 db of the shannon limit

finite element theory for curved and twisted beams based on exact solutions for three dimensional solids part 1 beam concept and geometrically exact nonlinear formulation a geometrically exact and completely consistent finite element theory for curved and twisted beams is proposed it is based on the kinematical hypothesis generally formulated for large deformation and accounts for all kinds of deformation of a three dimensional solid translational and rotational displacements of the cross sections warping of their plane and distortion of their contours the principle of virtual work is applied in a straightforward manner to all non zero six components of the strain and stress tensors expressions are given for tangent matrices of elastic inertia and external forces and specific techniques for discretization and updating are developed for the analysis of beams in inertial and non inertial frames finally the numerical properties of the finite element models are demonstrated through examples c 1998 elsevier science s a all rights reserved

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

the reduction of interval type 2 lr fuzzy sets type reduction of interval type 2 it2 fuzzy sets is essential in conducting the type 2 fuzzy sets expressed with the resolution forms of it2 fuzzy sets several type reduction methods such as km ekm and centroid flow have been proposed these methods are relatively easy to implement but still computation intensive because they need to invoke an iterative switching point finding procedure this study derives a theorem and proposes a heuristic algorithm which can fast and accurately identify the minimum and maximum switching points of a piecewise smooth it2 fuzzy set it also demonstrates that it is easy to derive the close form expressions of the switching points of a piecewise smooth it2 fuzzy set if both of its upper and lower membership functions can be parameterized as lr fuzzy sets which are defined in this paper then the type reduction of piecewise smooth it2 fuzzy sets can be simplified to solve the close form expressions of their switching points in terms of lr parameters experiments on it2 fuzzy sets with various piecewise smooth membership functions including linear gaussian and hybrid shaped ones were made the results showed that the proposed type reduction method can obtain solutions which accurately approximate to the desired switching points with much lower computational overhead than the karnik mondel km and enhanced km ekm methods

multiple trellis coded frequency and phase modulation multiple trellis coded modulation of constant envelope frequency and phase modulated signal sets mtcm fpm is investigated for performance on the additive white gaussian noise awgn channel and on the one sided normal rayleigh and rician fading channels the nakagami m fading model is used as an alternative to the rician fading model to calculate the error probability upper bound for trellis coded schemes on the fading channel the likeliness and the disparity between the upper bounds to the error probability for the two fading models are discussed the design criteria for the one sided normal fading channel modeled by the nakagami m distribution are observed to be the same as those for the rayleigh fading channel for the mtcm fpm schemes it is demonstrated that the set partitioning designed to maximize symbol diversity optimum for fading channels is optimum for performance on the awgn channel as well the mtcm fpm schemes demonstrate improved performance over mtcm mpsk schemes and tcm fpm schemes on the awgn channel and the fading channel

on block coded modulation with interblock memory block coded modulation with interblock memory bcmim is a variation of block coded modulation bcm which is designed for multilevel coding by providing interblock memory between adjacent blocks the coding rate of a bcmim scheme can be increased without decreasing the minimum squared euclidean distance msed as compared to the original bcm in an early version of bcmim interblock coding is provided only between the first two coding levels of adjacent blocks in this paper we design bcmim with a more general form for which interblock coding can be introduced among many coding levels in this way we can further increase the coding rate of bcmim without decreasing the msed we provide many examples to show the advantages of bcmim with the general form most of the examples are designed based on multidimensional signal sets since a multidimensional signal set can provide more coding levels than a two dimensional 2 d signal set

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

an interval type 2 fuzzy model of computing with words as a methodology computing with words allows the use of words instead of numbers or symbols in the process of computing and reasoning and thus conforms more to humans inference when it is used to describe real world problems motivated by zadeh s paradigm of computing with words the literature has proposed a kind of type 1 fuzzy automata as a formal model of computing with words which takes type 1 fuzzy subsets of symbols as input however type 1 fuzzy representation provides a limited platform for approximating the meaning of words since it is not able to capture linguistic uncertainty in this paper we develop a formal interval type 2 fuzzy model of computing with words by generalizing the existing type 1 fuzzy sets based model to an interval type 2 fuzzy environment concretely we take interval type 2 fuzzy automata i e interval type 2 fuzzy finite automata and interval type 2 fuzzy pushdown automata which combine interval type 2 fuzzy set theory and automaton theory as a computational model of computing with words furthermore we develop the extension principles to extend from computing with values to computing with words and show that computing with words can be implemented with computing with values in the interval type 2 fuzzy setting at an extra computational cost

voronoi regions of lattices second moments of polytopes and quantization if a point is picked at random inside a regular simplex octahedron 600 cell or other polytope what is its average squared distance from the centroid in n dimensional space what is the average squared distance of a random point from the closest point of the lattice a n or d n e n a n ast or d n ast the answers are given here together with a description of the voronoi or nearest neighbor regions of these lattices the results have applications to quantization and to the design of signals for the gaussian channel for example a quantizer based on the eight dimensional lattice e8 has a mean squared error per symbol of 0 0717 cdots when applied to uniformly distributed data compared with 0 08333 cdots for the best one dimensional quantizer

on answering the question where do i start in order to solve a new problem involving interval type 2 fuzzy sets this paper which is tutorial in nature demonstrates how the embedded sets representation theorem rt for a general type 2 fuzzy set t2 fs when specialized to an interval i t2 fs can be used as the starting point to solve many diverse problems that involve it2 fss the problems considered are set theoretic operations centroid uncertainty measures similarity inference engine computations for mamdani it2 fuzzy logic systems linguistic weighted average person membership function approach to type 2 fuzzistics and interval approach to type 2 fuzzistics each solution obtained from the rt is a structural solution but is not a practical computational solution however the latter are always found from the former it is this author s recommendation that one should use the rt as a starting point whenever solving a new problem involving it2 fss because it has had such great success in solving so many such problems in the past and it answers the question where do i start in order to solve a new problem involving it2 fss

an efficient block based addressing scheme for the nearly optimum shaping of multidimensional signal spaces we introduce an efficient addressing scheme for the nearly optimum shaping of a multidimensional signal constellation the 2 d two dimensional subspaces are partitioned into k energy shells of equal cardinality the average energy of a 2 d shell can be closely approximated by a linear function of its index in an n 2n d space we obtain k sup n shaping clusters of equal cardinality shaping is achieved by selecting t spl les k sup n of the n d clusters with the least sum of the 2 d indices this results in a set of t integer n tuples with the components in the range 0 k 1 and the sum of the components being at most a given number l the problem of addressing is to find a one to one mapping between the set of such n tuples and the set of integers 0 t 1 such that the mapping and its inverse can be easily implemented in the proposed scheme the n d clusters are grouped into blocks of identical binary weight vectors this results in a simple rule for the addressing of points within the blocks the addressing of the blocks is based on some recursive relationship which allows us to decompose the problem into simpler parts the overall scheme requires a modest amount of memory and has a small computational complexity

wei type trellis coded modulation with 2n dimensional rectangular constellation for n not a power of two the wei method of constructing trellis codes with 2n dimensional rectangular signal constellations is limited to n being an integer power of two in the present correspondence this limitation is removed and examples of six dimensional 6 d and twelve dimensional 12 d trellis codes are given

alpha plane representation for type 2 fuzzy sets theory and applications this paper 1 reviews the alpha plane representation of a type 2 fuzzy set t2 fs which is a representation that is comparable to the alpha cut representation of a type 1 fs t1 fs and is useful for both theoretical and computational studies of and for t2 fss 2 proves that set theoretic operations for t2 fss can be computed using very simple alpha plane computations that are the set theoretic operations for interval t2 it2 fss 3 reviews how the centroid of a t2 fs can be computed using alpha plane computations that are also very simple because they can be performed using existing karnik mendel algorithms that are applied to each alpha plane 4 shows how many theoretically based geometrical properties can be obtained about the centroid even before the centroid is computed 5 provides examples that show that the mean value defuzzified value of the centroid can often be approximated by using the centroids of only 0 and 1 alpha planes of a t2 fs 6 examines a triangle quasi t2 fuzzy logic system q t2 fls whose secondary membership functions are triangles and for which all calculations use existing t1 or it2 fs mathematics and hence they may be a good next step in the hierarchy of flss from t1 to it2 to t2 and 7 compares t1 it2 and triangle q t2 flss to forecast noise corrupted measurements of a chaotic mackey glass time series

two level coding based on trellis coded modulation and reed solomon codes a number of two level codes based on trellis coded modulation tcm schemes and their matched single or double error correcting reed solomon codes using qam constellations are presented and evaluated compared to the well known 4d 16 state tcm scheme with a comparable bandwidth efficiency these two level codes provide greater immunity from both gaussian and impulse noise at the cost of a reasonably higher complexity and a longer decoding delay on the order of a few hundred symbols the authors also show how to cope with phase ambiguities of the constellation in these two level codes both rotationally invariant coding and rotation detecting techniques are considered

finite state codes a class of codes called finite state fs codes is defined and investigated the codes which generalize both block and convolutional codes are defined by their encoders which are finite state machines with parallel inputs and outputs a family of upper bounds on the free distance of a given fs code is derived a general construction for fs codes is given and it is shown that in many cases the fs codes constructed in this way have a free distance that is the largest possible catastrophic error propagation cep for fs codes is also discussed it is found that to avoid cep one must solve the graph theoretic problem of finding a uniquely decodable edge labeling of the state diagram

centroid of a type 2 fuzzy set abstract in this paper we introduce the centroid and generalized centroid of a type 2 fuzzy set both of which are essential for implementing a type 2 fuzzy logic system and explain how to compute them for practical use we show how to compute the centroid of interval and gaussian type 2 fuzzy sets an exact computation procedure is provided for an interval type 2 set whereas an approximation is provided for both interval and gaussian type 2 sets examples are given that compare the exact computational results with the approximate results

new trellis codes based on lattices and cosets a new technique is proposed for constructing trellis codes which provides an alternative to ungerboeck s method of set partitioning the new codes use a signal constellation consisting of points from an n dimensional lattice lambda with an equal number of points from each coset of a sublattice lambda one part of the input stream drives a generalized convolutional code whose outputs are cosets of lambda while the other part selects points from these cosets several of the new codes are better than those previously known

vector coding for partial response channels a linear technique for combining equalization and coset codes on partial response channels with additive white gaussian noise is developed the technique vector coding uses a set of transmit filters or vectors to partition the channel into an independent set of parallel intersymbol interference isi free channels for any given finite or infinite block length the optimal transmit vectors for such channel partitioning are shown to be the eigenvectors of the channel covariance matrix for the specified block length and the gains of the individual channels are the eigenvalues an optimal bit allocation and energy distribution are derived for the set of parallel channels under an accurate extension of the continuous approximation for power in optimal multidimensional signal sets for constellations with unequal signal spacing in different dimensions examples are presented that demonstrate performance advantages with respect to zero forcing decision feedback methods that use the same coset code on the same partial response channel only resampling the channel at an optimal rate and assuming no errors in the feedback path will bring the performance of the decision feedback methods up to the level of the vector coded system

trellis coded quantization of memoryless and gauss markov sources trellis coded quantization tcq is developed and applied to the encoding of memoryless and gauss markov sources the theoretical justification for the approach is alphabet constrained rate distortion theory which is a dual to the channel capacity argument that motivates trellis coded modulation tcm the authors adopt the notions of signal set expansion set partitioning and branch labeling of tcm but modify the techniques to account for the source distribution to design tcq coders of low complexity with excellent mean squared error mse performance for a memoryless uniform source tcq provides an mse within 0 21 db of the distortion rate bound at all positive integral rates the performance is superior to that promised by the coefficient of quantization for all of the best lattices known in dimensions 24 or less for a memoryless gaussian source the tcq performance at rates of 0 5 1 and 2 b sample is superior to all previous results the authors found in the literature the encoding complexity of tcq is very modest tcq is incorporated into a predictive coding structure for the encoding of gauss markov sources simulation results for first second and third order gauss markov sources are presented

a systematic approach to the construction of bandwidth efficient multidimensional trellis codes we present a unified approach to the construction of multidimensional trellis codes that have small constellation expansion ratio cer and small peak to average power ratio par and hence are of high bandwidth efficiency in the approach we extend wei s 1987 construction to dimensionalities other than a power of two and propose a low complexity encoder we also present a general partition technique that gives geometrically uniform partitions with a good distance profile the proposed method can generate many good codes including previously reported ungerboeck codes gallager calderbank sloane codes and wei codes with quadrature amplitude modulation signals some newly discovered six dimensional trellis codes are compared with these known codes on their cer par effective coding gain and encoder decoder complexities

new results about the centroid of an interval type 2 fuzzy set including the centroid of a fuzzy granule the centroid of an interval type 2 fuzzy set it2 fs provides a measure of the uncertainty of such a fs its calculation is very widely used in interval type 2 fuzzy logic systems in this paper we present properties about the centroid of an it2 fs we also illustrate many of the general results for a t2 fuzzy granule fg in order to develop some understanding about the uncertainty of the fg in terms of its vertical and horizontal dimensions at present the t2 fg is the only it2 fs for which it is possible to obtain closed form formulas for the centroid and those formulas are in this paper

fast quantizing and decoding and algorithms for lattice quantizers and codes for each of the lattices a n n geq 1 d n n geq 2 e 6 e 7 e 8 and their duals a very fast algorithm is given for finding the closest lattice point to an arbitrary point if these lattices are used for vector quantizing of uniformly distributed data the algorithm finds the minimum distortion lattice point if the lattices are used as codes for a gaussian channel the algorithm performs maximum likelihood decoding

answering an open problem on t norms for type 2 fuzzy sets abstract this paper proves that a binary operation on 0 1 ensuring that the binary operation is a t norm or is a t conorm is a t norm where and are special convolution operations defined by f g x sup f y g z y z x f g x sup f y g z y z x for any f g map 0 1 0 1 where and are a continuous t norm and a continuous t conorm on 0 1 answering negatively an open problem posed in 8 besides some characteristics of t norm and t conorm are obtained in terms of the binary operations and

a large rotation finite element analysis of 3d beams by incremental rotation vector and exact strain measure with all the desirable features abstract different strategies based on rotation vector and exact strain measure have been proposed over the years for analyzing flexible bodies undergoing arbitrary large rotations to avoid the singularity of the vector like parametrization the interpolation of the incremental rotation vector is the most popular approach in this context even if this leads to path dependence and numerical instability i e error accumulation it is also non objective although both objectivity and path independence are recovered with h and p refinement corotational approaches do not have these drawbacks even though the geometrically exact model is achieved by mesh refinement in this work we develop a novel strategy which uses the incremental nodal rotation vectors to define corotational nodal rotations which are then interpolated for the evaluation of the nonlinear strains this choice makes the approach singularity free allows for additive updates within each increment and preserves all the features of the theoretical problem for any mesh and interpolation rotational variables objectivity exact strain measure path independence and symmetric stiffness matrix for conservative loads this last property is a consequence of the direct differentiation of the relation between local and global rotations whose compact form also makes a simple and general definition of the internal forces and the tangent stiffness for any order of interpolation possible in addition we show how the common approach of interpolating incremental vectors can be made stable by a simple updating procedure based on local rotations carried out at the end of each increment in order to avoid cumulative errors geometrically exact 3d beams are considered as a demonstrative example an iterative strategy based on mixed integration points is used to solve the nonlinear discrete equations efficiently

multidimensional mappings for iteratively decoded bicm on multiple antenna channels multidimensional binary mappings for bit interleaved coded modulations bicms on ergodic multiple antenna channels with iterative decoding are presented after derivation of a closed form expression for the pairwise error probability under ideal maximum likelihood ml decoding the design criterion for mapping optimization is established from the ml performance of the ideally interleaved channel it coincides with the figure of merit derived from the genie condition when the iterative receiver converges to perfect a priori information multidimensional mapping constructions that exhibit high signal to noise ratio snr gains without increasing the complexity of the a posteriori probability app detection are proposed they allow for a reduced decoding complexity as they achieve near turbo code performance with a single convolutional code