noncoherent coded modulation trellis coded modulation with two or multidimensional signal constellations together with coherent maximum likelihood detection is considered an attractive solution for communications over the additive white gaussian noise awgn channel in this paper a new noncoherent communication system is introduced called noncoherent coded modulation ncm as an alternative to coherent coded modulation ncm achieves almost the same power efficiency without bandwidth expansion or an extensive increase in complexity as a noncoherent system the method does not need carrier phase estimation nonetheless differential encoding is not required high performance noncoherent detection is achieved by using multiple symbol observations unlike previous approaches a sliding window for the observations is used with each observation covering several branches of the trellis such that the observations are time overlapped we define a new type of noncoherent maximum likelihood sequence estimator mlse and analyze its performance over the awgn channel by numerical calculation of the union bound we perform a computerized search and present new codes for noncoherent detection with their performance the new codes cover many useful rates and complexities and achieve higher performance than existing codes for noncoherent detection the method can also be used for multiple symbol demodulation of mdpsk with better results than existing methods

new 16 psk group trellis codes the theory of group codes simplifies the construction of good rotationally invariant trellis codes for the additive white gaussian noise channel new geometrically uniform trellis codes with spectral efficiency 3 bits 2d based on multidimensional l spl times 16 psk constellations in 4 6 and 8 dimensions obtained by group trellis codes over nonbinary groups of the kind z sub 16 sup l are presented most of the codes improve in terms of asymptotic gain and rotational invariance the best codes known so far in the literature

on computing the edge connectivity of an uncertain graph uncertain graphs are employed to describe graph models with imprecise expert data in uncertain graphs due to the existence of uncertain edges edge connectivity is essentially an uncertain variable different from that in a deterministic graph it is more suitable to investigate the possibility uncertain measure that an uncertain graph is k edge connected which is the main aim of this paper we first deduce a theoretical formula to calculate the uncertain measure and on this basis we then propose an algorithm which is derived from maximum flow algorithms to numerically calculate the uncertain measure the proposed algorithm is also proved to be a polynomial time algorithm and its effectiveness and efficiency are illustrated by numerical examples

a new decoding algorithm for geometrically uniform trellis codes a new decoding algorithm for geometrically uniform trellis codes is presented the group structure of the codes is exploited in order to improve the decoding process analytical bounds to the algorithm performance and to its computational complexity are derived the algorithm complexity does not depend on the number of states of the trellis describing the code extensive simulations yield results on the algorithm performance and complexity and permit a comparison with the viterbi algorithm and the sequential fano algorithm

codes based on a trellis cut set transformation ii codes for noncoherent detection for pt i see ibid vol 46 p 610 20 1998 we continue the study of cut set trellis section t sub cb codes begun in part i and show that such codes are well suited for use with multiple symbol noncoherent detection and conventional two symbol differential detection the information bit assignment for a cut set trellis section can be modified to make the resulting code noncatastrophic for noncoherent detection by this procedure many existing two dimensional 2 d and multidimensional mpsk and qam trellis codes designed for coherent detection can be easily converted into codes for both coherent and noncoherent detection compared to m dpsk trellis codes it is shown that 2 d mpsk cut set trellis section codes have improved performance for multiple symbol noncoherent detection and comparable performance for conventional two symbol differential detection thus cut set trellis section codes are a class of universal channel codes since a single encoder performs well for coherent detection multiple symbol noncoherent detection and conventional two symbol differential detection

geometrically uniform tcm codes over groups based on l spl times mpsk constellations the theory of geometrically uniform signal sets and codes over groups is applied to the case of l spl times mpsk constellations conditions for rotational invariance of group codes are discussed the tables of geometrically uniform partitions found in benedetto et al 1993 are used to construct good geometrically uniform trellis codes over nonbinary abelian groups the present authors consider l spl times 4psk and l spl times 8psk constellations used to transmit information rates of 1 and 2 bit two dimensions respectively and present tables of good codes over generating groups z4 sup l and z8 sup l for l ranging from 1 to 4 in most cases they improve the tables of codes known so far moreover the geometrical uniformity of codes allows a very easy performance evaluation so that the authors also present a complete set of curves of error event probability for the obtained codes

uncertain calculus with yao process uncertain calculus is a branch of mathematics that deals with the differentiation and integration of functions of uncertain processes this paper investigates the yao process defined by the yao integral and extends the uncertain calculus on the yao process some important results are developed for finite variation and linearity of integral moreover this paper proposes a concept of the multifactor yao process and also gives a fundamental theorem for it

signal sets matched to groups recently linear codes over z sub m the ring of integers mod m have been presented that are matched to m ary phase modulation the general problem of matching signal sets to generalized linear algebraic codes is addressed based on these codes a definition is given for the notion of matching it is shown that any signal set in n dimensional euclidean space that is matched to an abstract group is essentially what d slepian 1968 called a group code for the gaussian channel if the group is commutative this further implies that any such signal set is equivalent to coded phase modulation with linear codes over z sub m some further results on such signal sets are presented and the signal sets matched to noncommutative groups and the linear codes over such groups are discussed

the performance of trellis coded mdpsk with multiple symbol detection the idea of using a multiple more than two symbol observation interval to improve error probability performance is applied to differential detection of trellis coded multiple phase shift keying mpsk over an additive white gaussian noise awgn channels an equivalent euclidean distance measure per trellis branch is determined for this detection scheme this is used to define an augmented larger multiplicity trellis code whose distance measure is the conventional squared euclidean distance typical of conventional trellis coded modulation on the awgn such an augmented multiple trellis code is a convenient mathematical tool for simplifying the analysis results are obtained by a combination of analysis and computer simulation it is shown that only a slight increase e g one symbol in the length of the observation interval will provide a significant improvement in bit error probability performance

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

rotational invariance of trellis codes ii group codes and decoders for pt i see ibid vol 42 no 3 p 751 65 1996 in part i general results on rotationally invariant codes and encoders were derived assuming no algebraic structure in part ii trellis codes based on group systems are considered as a special case for which code and encoder constructions are particularly simple rotational invariance is expressed as an algebraic constraint on a group code and algebraic constructions are found for both absorbed precoder encoders and for encoders with separate differential precoders finally the various encoder forms used to achieve rotational invariance are compared based on their performance on an awgn channel

a transformation invariant matching algorithm for handwritten chinese character recognition abstract all the chinese characters are composed of some fundamental characters called radicals the algorithms we present here are used to match the radicals stored in the data base with the input chinese character from the radicals in the input chinese character we can know what this input chinese character really is the matching process is equivalent to checking the validity of the relation that the radical is a sub character of the input chinese character the least square error estimation and some properties of the chinese characters are also used here to check whether the relation mentioned above is satisfied and if this relation is satisfied the sub character matched with the radical is extracted this matching algorithm has some superior merits which are 1 transformation scaling rotation and translation invariant and furthermore 2 invulnerable to the inherent defects of the thinning algorithms it overcomes these defects at the expense of more computation time but it can be run in parallel so large computation time is no longer a very serious problem experiments with 114 characters and 62 radicals have been conducted and we find that the success rate of our matching algorithms is above 99 thus we conclude that our algorithms are very reliable

on multidimensional coded modulations having uniform error property for generalized decoding and flat fading channels we consider the problem of uniform error property uep for a coded modulation with constant energy multidimensional symbols transmitted over the additive white gaussian noise awgn or fading channels and received by a broad class of decoders this class includes coherent partially coherent double differential and noncoherent decoders decoders designed for fading channels decoders using one or multiple symbol observations and many more these decoders are described as special cases of a general decoder model this decoder operates by maximizing an arbitrary likelihood function that its arguments are front end correlator matched filter outputs a group code structure that guarantees uep is developed by using the theory of geometrically uniform codes and applying it to the general decoder these codes are defined over groups commonly nonbinary with isometric mapping to channel symbols we show the code construction for the specific case of lth dimensional m ary phase shift keying mpsk an additional interesting property of these general uniform error codes is related to the case of noncoherent decoding we show that when using codes of this family if a code is noncoherently catastrophic then it is also rotationally invariant then the use of preceding of the input such that the code becomes rotationally transparent will also make it noncatastrophic

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

convolutional codes over groups the basic algebraic structure theory of convolutional codes and their trellises is developed simultaneously for codes over groups rings and fields the first part which covers fundamental notions such as minimality and observability is semi tutorial in that most definitions are already standard within the modern behavioral theory as are some of the formally stated results however some of the pivotal results emphasizing the role of observability as the basic well behavedness condition for codes are new and several previous results are given simplified proofs the usefulness of the behavioral approach even for convolutional codes over fields is demonstrated by a new minimality test for encoders as well as by the straightforward derivation of some known minimality criteria for generator matrices from the basic minimality criteria for group trellises the second part of the paper deals with issues that are specific to codes over rings and groups the main result is a concise characterization the first such of those groups that can appear as the branch group of any group trellis it is further shown how such groups are presented by shift registers a new large class of noncommutative convolutional codes is also given

multilevel construction of block and trellis group codes the theory of group codes has been shown to be a useful starting point for the construction of good geometrically uniform codes in this paper we study the problem of building multilevel group codes i e codes obtained combining separate coding at different levels in such a way that the resulting code is a group code a construction leading to multilevel group codes for semi direct and direct products is illustrated the codes that can be obtained in this way are identified new geometrically uniform euclidean space codes obtained from multilevel codes over abelian and nonabelian groups are presented

least squares inner product shaping we develop methods that construct an optimal set of vectors with a specified inner product structure from a given set of vectors in a complex hilbert space the optimal vectors are chosen to minimize the sum of the squared norms of the errors between the constructed vectors and the given vectors four special cases are considered in the first the constructed vectors are orthonormal in the second they are orthogonal in the third the gram matrix of inner products of the constructed vectors is a circulant matrix as we show the vectors form a cyclic set in the fourth the gram matrix has the property that the rows are all permutations of each other the constructed vectors are shown to be geometrically uniform

two 16 state rate r 2 4 trellis codes whose free distances meet the heller bound for rate r 1 2 convolutional codes with 16 states there exists a gap between heller s 1968 upper bound on the free distance and its optimal value this article reports on the construction of 16 state binary rate r 2 4 nonlinear trellis and convolutional codes having d sub free 8 a free distance that meets the heller upper bound the nonlinear trellis code is constructed from a 16 state rate r 1 2 convolutional code over z sub 4 using the gray map to obtain a binary code both convolutional codes are obtained by computer search systematic feedback encoders for both codes are potential candidates for use in combination with iterative decoding regarded as modulation codes for 4 psk these codes have free squared euclidean distance d sub e free sup 2 16

simplified viterbi decoding of geometrically uniform tcm codes we present a procedure to design maximum likelihood ml decoders for the new class of geometrically uniform gu trellis coded modulation tcm codes exploiting the algebraic properties of such codes the proposed design has a very efficient vlsi implementation the design of the decoders for the gutcm codes is more complicated in comparison to the standard convolutional codes because between any pair of states in the trellis diagram of a gutcm code there is usually a large number of parallel transitions and the trellis diagram of the code has a much higher degree of connectivity in comparison to binary convolutional codes we present a novel technique for solving the parallel transitions using the algebraic structure of the gutcm codes which represents a significant reduction in complexity in comparison to the direct approach the proposed technique is applied to the design of a simplified viterbi decoder vd for a 64 state nonbinary gutcm code defined over z sub 8 sup 4 for this example we obtain a 58 fold reduction in complexity for the parallel transition solver in comparison to a direct implementation

handwritten chinese radical recognition using nonlinear active shape models handwritten chinese characters can be recognized by first extracting the basic shapes radicals of which they are composed radicals are described by nonlinear active shape models and optimal parameters found using the chamfer distance transform and a dynamic tunneling algorithm the radical recognition rate is 96 5 percent correct writer independent on 280 000 characters containing 98 radical classes

the structure of the state representation of shift invariant controllable and observable group codes in this paper an investigation on the structure of the canonical trellis section of shift invariant l controllable and m observable group codes is carried out necessary and sufficient conditions for a set of group homomorphisms in order that they represent the trellis section of this class of codes are established

group trellis codes over multidimensional unbalanced psk and their optimality new geometrically uniform trellis codes over multidimensional unbalanced 4 8 and 16 psk constellations obtained with the group code approach are presented they improve by up to 1 25 db the asymptotic performance of the best known trellis codes over multidimensional balanced psk to check their optimality an upper bound on the free distance of group trellis codes is derived some of the new codes achieve the largest obtainable free distance and several others are close to the bound exploiting the symmetry properties of the codes curves of tight upper bounds to the error probability are also presented

multiple symbol differential detection of mpsk a differential detection technique for mpsk multiple phase shift keying which uses a multiple symbol observation interval is presented and its performance is analyzed and simulated the technique makes use of maximum likelihood sequence estimation of the transmitted phases rather than symbol by symbol detection as in the conventional differential detection thus the performance of this multiple symbol detection scheme fills the gap between conventional two symbol observation differentially coherent detection of mpsk and ideal coherent of mpsk with differential encoding the amount of improvement gained over conventional differential detection depends on the number of phases m and the number of additional symbol intervals added to the observation what is particularly interesting is that substantial performance improvement can be obtained for only one or two additional symbol intervals of observation the analysis and simulation results presented are for uncoded mpsk

geometrically uniform partitions of l mpsk constellations and related binary trellis codes the theory of geometrically uniform trellis codes is applied to the case of multidimensional psk phase shift keying constellations the symmetry group of an l mpsk m ary psk constellation is completely characterized conditions for rotational invariance of geometrically uniform partitions of a signal constellation are given through suitable algorithms geometrically uniform partitions of l mpsk m 4 8 16 and l 1 2 3 4 constellations are found which present good characteristics in terms of the set of distances at a given partition level the maximum obtainable rotational invariance and the isomorphism of the quotient group associated with the partition these partitions are used as starting points in a search for good geometrically uniform trellis codes based on binary convolutional codes

combined coding and modulation theory and applications the theoretical aspects of the encoding process are investigated resulting in a precise definition of linear codes together with theorems that clarify how they can be obtained a particular subset of linear codes called superlinear codes for which the performance analysis is highly simplified is identified the most relevant performance measures for the analysis of this class of codes are discussed the minimum euclidean distance and the event and bit error probabilities are found analytically using the uniform error property when applicable or variations on it this yields accurate upper and lower bounds to the error rate at the price of reasonable computational complexity the theory is then applied to the search for good codes and to their performance evaluation the cases of 16 and 32 psk codes which are good candidates for use in digital satellite transmission are considered several new results in terms of error event and bit error probabilities are presented showing considerable gains in terms of snr with respect to the uncoded case

the dynamics of group codes state spaces trellis diagrams and canonical encoders a group code c over a group g is a set of sequences of group elements that itself forms a group under a component wise group operation a group code has a well defined state space sigma sub k at each time k each code sequence passes through a well defined state sequence the set of all state sequences is also a group code the state code of c the state code defines an essentially unique minimal realization of c the trellis diagram of c is defined by the state code of c and by labels associated with each state transition the set of all label sequences forms a group code the label code of c which is isomorphic to the state code of c if c is complete and strongly controllable then a minimal encoder in controller canonical feedbackfree form may be constructed from certain sets of shortest possible code sequences called granules the size of the state space sigma sub k is equal to the size of the state space of this canonical encoder which is given by a decomposition of the input groups of c at each time k if c is time invariant and nu controllable then mod sigma sub k mod pi sub 1

theoretical analysis and performance limits of noncoherent sequence detection of coded psk a theoretical performance analysis of noncoherent sequence detection schemes previously proposed by the authors for combined detection and decoding of coded m ary phase shift keying m psk is presented a method for the numerical evaluation of the pairwise error probability for which no closed form expressions exist is described the classical union bound is computed and results are compared with computer simulations an upper bound on this pairwise error probability is also presented this upper bound may be effectively used for the definition of an equivalent distance which may be useful in exhaustive searches for optimal codes using this bound it is proven that in the general coded case the considered noncoherent decoding schemes perform as close as desired to an optimal coherent receiver when a phase memory parameter is sufficiently large in the case of differentially encoded m psk a simple expression of the asymptotic bit error probability is derived which is in agreement with simulations for high as well as low signal to noise ratio snr

optimal decoding of coded psk and qam signals in correlated fast fading channels and awgn a combined envelope multiple differential and coherent detection approach the maximum likelihood sequence estimator for the reception of coded digital phase modulated signals with single or multiamplitude constellations transmitted over a multiplicative frequency nonselective i e flat correlated fast fading rayleigh or rician channel and corrupted by additive white gaussian noise awgn is derived due to this correlation the errors caused by fading tend to occur in bursts in the analysis no assumption simplifying the problem is made for fast fading the authors consider the most general case where both phase and amplitude distortion resulting from the fading process could change significantly and thus cannot be assumed to be constant over a number of transmitted symbols it is shown that the estimator s hardware structure consists of a combination of envelope multiple differential and coherent detectors with multiple differential detectors they define a receiver structure consisting of a combination of more than one distinct differential detectors each of them employing a progressively increasing by the symbol duration time delay element the outputs of these detectors are jointly processed by means of an algorithm which is presented in a recursive form the derivation of this new receiver is general enough to accommodate trellis coded phase shift keying psk and quadrature amplitude modulated qam systems differentially encoded signals such as the spl pi 4 shift differential quadrature phase shift keying dqpsk scheme can also be incorporated in order to reduce the overall receiver implementation complexity several reduced complexity near optimal versions of the algorithm are presented these reduced complexity receivers are based on the use of only a few multiple differential detectors performance evaluation results for reduced complexity trellis coded spl pi 4 shift dqpsk spl pi 4 shift 8 dqam differential quadrature amplitude modulation and 8 dpsk differential phase shift keying systems have demonstrated that the proposed receivers significantly reduce the error floors caused by fading

system theoretic properties of convolutional codes over rings convolutional codes over rings are particularly suitable for representing codes over phase modulation signals in order to develop a complete structural analysis of this class of codes it is necessary to study rational matrices over rings which constitutes the generator matrices encoders for such convolutional codes noncatastrophic minimal systematic and basic generator matrices are introduced and characterized by using a canonical form for polynomial matrices over rings finally some classes of convolutional codes defined according to the generator matrix they admit are introduced and analyzed from a system theoretic point of view

labelings and encoders with the uniform bit error property with applications to serially concatenated trellis codes the well known uniform error property for signal constellations and codes is extended to encompass information bits we introduce a class of binary labelings for signal constellations called bit geometrically uniform bgu labelings for which the uniform bit error property holds i e the bit error probability does not depend on the transmitted signal strong connections between the symmetries of constellations and binary hamming spaces are involved for block coded modulation bcm and trellis coded modulation tcm euclidean space codes bgu encoders are introduced and studied the properties of bgu encoders prove quite useful for the analysis and design of codes aimed at minimizing the bit rather than symbol error probability applications to the analysis and the design of serially concatenated trellis codes are presented together with a case study which realizes a spectral efficiency of 2 b s hz

analysis of optimum detector of trellis coded mpsk in phase noise channels this paper presents a novel analytical expression which approximates the bit error rate ber of the joint phase and symbol maximum a posteriori jmap sequence estimator for trellis code modulation tcm with m ary phase shift keying mpsk modulation any arbitrary phase noise model i e not limited to the wiener process and either a matched or mismatched decoder first we derived convenient closed form expressions for approximating the pairwise error probability of two code sequences for both matched and mismatched decoders since the expressions are formulated either in the time or in the frequency domain it is possible to indicate the contribution of every frequency in the phase noise spectrum we then applied the union bound on the code sequence pairwise errors the analytical expression was tight usually 1db for mpsk constellations with m 4 and code rates 0 5 once developed the expressions will assist designers to consider the influence on receiver performance of the code characteristics decoder implementation and rf synthesizer phase noise it further enables joint optimization of the rf synthesizer the code and the decoder for achieving the lowest error rate or other design targets

on the hamming distance properties of group codes under certain mild conditions the minimum hamming distance d of an n k d group code c over a non abelian group g is bounded by d n 2 consequently there exists no n k n k 1 group code c over an non abelian group g if 1

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

hierarchical attributed graph representation and recognition of handwritten chinese characters abstract this paper presents a new method of recognizing handwritten chinese characters a structural representation called hierarchical attributed graph representation hagr is introduced to describe handwritten chinese characters the hagr provides a simple and direct representation of handwritten chinese characters with hagr the recognition process becomes a simple task of graph matching a cost function mapping a candidate to a model graph is introduced this approach can tolerate the variations of hagr which reflect the instabilities or variabilities of handwritten chinese characters resulting from different writing styles several rules have been introduced to rearrange the order of the vertices of the graphs in order to avoid the combinatorial explosion in graph matching in addition the database of the character models is organized in a search tree structure for a candidate character the search process to find a corresponding model character has been divided into a number of simple and local decisions at different levels of the tree this considerably improves the efficiency and accuracy of the matching process

asymptotic upper bounds on the minimum distance of trellis codes a trellis code is a sliding window method of encoding a binary data stream as a sequence of signal points in rn when a trellis code is used to encode data at the rate of k bits channel symbol each channel input depends not only on the most recent block of k bits to enter the encoder but will also depend on a set of bits preceding this block the bits determine the state of the encoder and the most recent block of k bits generates the channel symbol conditional on the encoder state the performance of a trellis code depends on a suitably defined minimum distance property of that code this paper obtains upper bounds on this minimum distance that are better than any previously known

minimal and systematic convolutional codes over finite abelian groups abstract convolutional codes over abelian groups provide an effective theoretical framework for the analysis of some classes of tcm codes the encoder synthesis for this class of codes is not as simple as in the binary case since minimal encoders in the group case might be necessarily nonlinear in this contribution an algorithmic method testing whether a convolutional code over an abelian group admits a systematic or a minimal homomorphic encoder is provided this test consists in verifying whether a subgroup splits in a group through this method the class of codes admitting systematic encoders and the class of codes admitting minimal encoders can be compared finally this test is applied to some examples of practical relevance

optical recognition of handwritten chinese characters by hierarchical radical matching method abstract in this paper a radical based ocr system for the recognition of handwritten chinese characters is proposed in our approach a recursive hierarchical scheme is developed to perform radical extraction first character features and radical features are then extracted for matching last a hierarchical radical matching scheme is devised to identify the radicals embedded in an input chinese character and recognize the input character accordingly experiments for radical extraction are conducted on 1856 characters the successful rate of radical extraction is 92 5 the average time for radical extraction is 0 65 second per character experiments for matching process are conducted on two sets training set and testing set each set includes 900 characters the overall recognition rate in our experiments is 98 2 and 80 9 for training set and testing set respectively the average recognition time of our hierarchical radical matching scheme is 0 274 s there are totally 4716 radicals in 1800 characters in average one character consists of 2 62 radicals each character will match 7 28 radical templates in average thus each radical will match 2 77 radical temples the experimental results reveal that our proposed method is feasible flexible and effective

trellis coded multidimensional phase modulation a 2l dimensional multiple phase shift keyed l mpsk signal set is obtained by forming the cartesian product of l two dimensional mpsk signal sets a systematic approach to partitioning l mpsk signal sets that is based on block coding is used an encoder system approach is developed it incorporates the design of a differential precoder a systematic convolutional encoder and a signal set mapper trellis coded l 4psk l 8psk and l 16psk modulation schemes are found for 1

the capacity of finite abelian group codes over symmetric memoryless channels the capacity of finite abelian group codes over symmetric memoryless channels is determined for certain important examples such as m psk constellations over additive white gaussian noise awgn channels with m a prime power it is shown that this capacity coincides with the shannon capacity i e there is no loss in capacity using group codes this had previously been known for binary linear codes used over binary input output symmetric memoryless channels on the other hand a counterexample involving a three dimensional geometrically uniform constellation is presented in which the use of abelian group codes leads to a loss in capacity the error exponent of the average group code is determined and it is shown to be bounded away from the random coding error exponent at low rates for finite abelian groups not admitting galois field structure

variation analysis of uncertain stationary independent increment processes a stationary independent increment process is an uncertain process with stationary and independent increments this paper aims to calculate the variance of stationary independent increment processes and gains that for each fixed time the variance is a constant multiplying the square of time based on this result it is proved that the total variation of stationary independent increment process with finite variance is bounded almost surely besides the quadratic variation of stationary independent increment process with finite variance is 0 almost surely and in mean

geometrically uniform frames we introduce a new class of finite dimensional frames with strong symmetry properties called geometrically uniform gu frames that are defined over a finite abelian group of unitary matrices and are generated by a single generating vector the notion of gu frames is then extended to compound gu cgu frames which are generated by a finite abelian group of unitary matrices using multiple generating vectors the dual frame vectors and canonical tight frame vectors associated with gu frames are shown to be gu and therefore also generated by a single generating vector which can be computed very efficiently using a fourier transform ft defined over the generating group of the frame similarly the dual frame vectors and canonical tight frame vectors associated with cgu frames are shown to be cgu the impact of removing single or multiple elements from a gu frame is considered a systematic method for constructing optimal gu frames from a given set of frame vectors that are not gu is also developed finally the euclidean distance properties of gu frames are discussed and conditions are derived on the abelian group of unitary matrices to yield gu frames with strictly positive distance spectrum irrespective of the generating vector