an architecture for addition and subtraction of long word length numbers in the logarithmic number system an architecture is described for performing addition and subtraction of numbers in the logarithmic number system using small lookup tables previous implementations require approximately 4 2 sup f words for f bits of precision in the fraction the author shows how to reduce the size of the lookup table to fewer than 18 f 2 sup f 2 words the key to this is the use of a linear approximation of the nonlinear functions stored in the lookup tables the functions involved are highly nonlinear in some regions so variable size regions are used for the approximation the replacement of the f exponential dependence on the number of words by f 2 allows roughly 50 more bits of fractional precision to be obtained for a given amount of rom the architecture is mathematically analyzed yielding explicit expressions for all design parameters the approach is illustrated with an example logarithmic addition and subtraction unit using 32 b words with 30 b exponents containing 22 fractional bits a factor of 118 reduction in table size compared to previous techniques is achieved for this example

minimality and observability of group systems abstract group systems are a generalization of willems type linear systems that are useful in error control coding it is shown that the basic ideas of willems s treatment of linear systems are easily generalized to linear systems over arbitrary rings and to group systems the interplay between systems behaviors and trellises evolution laws is discussed with respect to completeness minimality controllability and observability it is pointed out that for trellises of group systems and willems type linear systems minimality is essentially the same as observability the development is universal algebraic in nature and holds unconditionally for linear systems over the real numbers

implementation of four common functions on an lns co processor we propose a scheme for evaluating four commonly used functions namely 1 inverse trigonometric functions 2 trigonometric functions 3 the exponential function and 4 the logarithmic function with the help of a logarithmic number system lns processor a novel idea of series folding has been introduced for computing the above functions expressed in the form of infinite series we also show that with a suitable choice of the radix for the lns we can evaluate exponential and logarithmic functions without using any extra hardware

loop calculus in statistical physics and information science considering a discrete and finite statistical model of a general position we introduce an exact expression for the partition function in terms of a finite series the leading term in the series is the bethe peierls belief propagation bp contribution the rest are expressed as loop contributions on the factor graph and calculated directly using the bp solution the series unveils a small parameter that often makes the bp approximation so successful applications of the loop calculus in statistical physics and information science are discussed

statistical measures of dna sequence dissimilarity under markov chain models of base composition in molecular biology the issue of quantifying the similarity between two biological sequences is very important past research has shown that word based search tools are computationally efficient and can find some new functional similarities or dissimilarities invisible to other algorithms like fasta recently under the independent model of base composition wu burke and davison 1997 biometrics 53 1431 1439 characterized a family of word based dissimilarity measures that defined distance between two sequences by simultaneously comparing the frequencies of all subsequences of n adjacent letters i e n words in the two sequences specifically they introduced the use of mahalanobis distance and standardized euclidean distance into the study of dna sequence dissimilarity they showed that both distances had better sensitivity and selectivity than the commonly used euclidean distance the purpose of this article is to extend mahalanobis and standardized euclidean distances to markov chain models of base composition in addition a new dissimilarity measure based on kullback leibler discrepancy between frequencies of all n words in the two sequences is introduced applications to real data demonstrate that kullback leibler discrepancy gives a better performance than euclidean distance moreover under a markov chain model of order kq for base composition where kq is the estimated order based on the query sequence standardized euclidean distance performs very well under such a model it performs as well as mahalanobis distance and better than kullback leibler discrepancy and euclidean distance since standardized euclidean distance is drastically faster to compute than mahalanobis distance in a usual workstation pc computing environment the use of standardized euclidean distance under the markov chain model of order kq of base composition is generally recommended however if the user is very concerned with computational efficiency then the use of kullback leibler discrepancy which can be computed as fast as euclidean distance is recommended this can significantly enhance the current technology in comparing large datasets of dna sequences

redundant logarithmic arithmetic a number system that offers advantages in some situations over conventional floating point and sign logarithmic number systems is described redundant logarithmic arithmetic like conventional logarithmic arithmetic relies on table lookups to make the arithmetic unit simpler than an equivalent floating point unit the cost of 32 bit subtraction in a redundant logarithmic number system is lower than previously published logarithmic subtraction methods the total memory requirement for a 29 bit redundant logarithmic unit is 16 k words compared to 22 k words by the best previously published conventional sign logarithm unit assuming similar addition techniques are employed a redundant logarithmic number system can be implemented with online arithmetic which would be impractical for a conventional sign logarithm number system the disadvantages of redundant arithmetic are typical of redundant number systems first the redundancy doubles the storage requirements for data values second the representation can become ill conditioned especially as a result of iterated multiplications third division and square root operations are more difficult to implement in redundant logarithmic arithmetic

the use of logistic models for the analysis of codon frequencies of dna sequences in terms of explanatory variables the development of the regressive logistic model applicable to the analysis of codon frequencies of dna sequences in terms of explanatory variables is presented a codon is a triplet of nucleotides that code for an amino acid and may be considered as a trivariate response b1 b2 b3 where bi i 1 2 3 is a categorical random variable with values a c g t the linear order of bases in the dna and possible statistical dependence of the bases in a given codon make the regressive logistic model a suitable tool for the analysis of codon frequencies a problem of structural zeros arises from the fact that the stopping codons terminators do not code for amino acids this is solved by normalizing the likelihood function codon frequencies may also depend on the function of the gene and they are known to differ between genes of the same genome differences also occur between synonymous codons for the same amino acid thus the use of covariates that differ between synonymous codons as well as covariates that are constant within codons of the same amino acid may be useful in explaining the frequencies as an illustration the method is applied to the human mitochondrial genome using the following as explanatory variables 1 tscore a measure of the number of single base mutations required for a given codon to become a terminator 2 aarisk an indicator of a codon s ability of changing by a single base substitution to triplets coding for amino acids with very different characteristics 3 avdist a measure of the typicality of the amino acid coded for by the triplets the results indicate that models that incorporate dependency structure and covariates are to be preferred to either the models comprising covariates alone or dependency structure alone

a recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

the dynamics of group codes dual abelian group codes and systems fundamental results concerning the dynamics of abelian group codes behaviors and their duals are developed duals of sequence spaces over locally compact abelian lca groups may be defined via pontryagin duality dual group codes are orthogonal subgroups of dual sequence spaces the dual of a complete code or system is finite and the dual of a laurent code or system is anti laurent if c and c sup spl perp are dual codes then the state spaces of c act as the character groups of the state spaces of c sup spl perp the controllability properties of c are the observability properties of c sup spl perp in particular c is strongly controllable if and only if c sup spl perp is strongly observable and the controller memory of c is the observer memory of c sup spl perp the controller granules of c act as the character groups of the observer granules of c sup spl perp examples of minimal observer form encoder and syndrome former constructions are given finally every observer granule of c is an end around controller granule of c

a hybrid number system processor with geometric and complex arithmetic capabilities the architecture design and performance of a hybrid number system processor are described the processor performs multiplication division square root and square in the logarithmic number system lns domain however the input output addition and subtraction are all executed in the 32 b ieee standard floating point number system with the lns multiplier and pipelined architecture the processor is able to perform the geometric and complex arithmetic very effectively the processor is also shown to compare well to an existing 32 b floating point dsp digital signal processor chip for the same level of cmos technology the performance ratios between the hybrid number system and the floating point processor are shown to be 6 4 1 and 8 1 for division and square root respectively for the complex fft fast fourier transform algorithm the ratio is around 2 1

normal factor graphs and holographic transformations this paper stands at the intersection of two distinct lines of research one line is holographic algorithms a powerful approach introduced by valiant for solving various counting problems in computer science the other is normal factor graphs an elegant framework proposed by forney for representing codes defined on graphs we introduce the notion of holographic transformations for normal factor graphs and establish a very general theorem called the generalized holant theorem which relates a normal factor graph to its holographic transformation we show that the generalized holant theorem on the one hand underlies the principle of holographic algorithms and on the other hand reduces to a general duality theorem for normal factor graphs a special case of which was first proved by forney in the course of our development we formalize a new semantics for normal factor graphs which highlights various linear algebraic properties that potentially enable the use of normal factor graphs as a linear algebraic tool

on minimal tree realizations of linear codes a tree decomposition of the coordinates of a code is a mapping from the coordinate set to the set of vertices of a tree a tree decomposition can be extended to a tree realization i e a cycle free realization of the code on the underlying tree by specifying a state space at each edge of the tree and a local constraint code at each vertex of the tree the constraint complexity of a tree realization is the maximum dimension of any of its local constraint codes a measure of the complexity of maximum likelihood ml decoding for a code is its treewidth which is the least constraint complexity of any of its tree realizations it is known that among all tree realizations of a linear code that extends a given tree decomposition there exists a unique minimal realization that minimizes the state space dimension at each vertex of the underlying tree in this paper we give two new constructions of these minimal realizations as a by product of the first construction a generalization of the state merging procedure for trellis realizations we obtain the fact that the minimal tree realization also minimizes the local constraint code dimension at each vertex of the underlying tree the second construction relies on certain code decomposition techniques that we develop we further observe that the treewidth of a code is related to a measure of graph complexity also called treewidth we exploit this connection to resolve a conjecture of forney s regarding the gap between the minimum trellis constraint complexity and the treewidth of a code we present a family of codes for which this gap can be arbitrarily large

minimal trellises for block codes basic concepts in the study of trellises of block codes are defined it is shown that minimal proper trellises exists for all block codes bounds on the size of such trellises are established these bounds are shown to be exact for maximum distance separable codes and nearly so for perfect codes

characteristic generators and dualization for tail biting trellises this paper focuses on dualizing tail biting trellises particularly kv trellises these trellises are based on characteristic generators as introduced by koetter vardy 2003 and may be regarded as a natural generalization of minimal conventional trellises even though they are not necessarily minimal two dualization techniques will be investigated the local dualization introduced by forney 2001 for general normal graphs and a linear algebra based dualization tailored to the specific class of tail biting bahl cocke jelinek raviv bcjr trellises introduced by nori shankar 2006 it turns out that in general the bcjr dual is a subtrellis of the local dual while for kv trellises these two coincide furthermore making use of both the bcjr construction and the local dualization it will be shown that for each complete set of characteristic generators of a code there exists a complete set of characteristic generators of the dual code such that their resulting kv trellises are dual to each other if paired suitably this proves a stronger version of a conjecture formulated by koetter vardy

pseudocodewords of tanner graphs this paper presents a detailed analysis of pseudocodewords of tanner graphs pseudocodewords arising on the iterative decoder s computation tree are distinguished from pseudocodewords arising on finite degree lifts lower bounds on the minimum pseudocodeword weight are presented for the bec bsc and awgn channel some structural properties of pseudocodewords are examined and pseudocodewords and graph properties that are potentially problematic with min sum iterative decoding are identified an upper bound on the minimum degree lift needed to realize a particular irreducible lift realizable pseudocodeword is given in terms of its maximal component and it is shown that all irreducible lift realizable pseudocodewords have components upper bounded by a finite value t that is dependent on the graph structure examples and different tanner graph representations of individual codes are examined and the resulting pseudocodeword distributions and iterative decoding performances are analyzed the results obtained provide some insights in relating the structure of the tanner graph to the pseudocodeword distribution and suggest ways of designing tanner graphs with good minimum pseudocodeword weight

semi logarithmic number systems we present a new class of number systems called semi logarithmic number systems that constitute a family of various compromises between floating point and logarithmic number systems this allows trade between the speed of the arithmetic operations and the size of the required tables we give arithmetic algorithms addition subtraction multiplication division for the semi logarithmic number systems and we compare these number systems to the classical floating point or logarithmic number systems

on the algebraic structure of linear tail biting trellises tail biting trellises are important graphical representations of codes which can be simpler to decode than conventional trellises while conventional trellises are well understood the general theory of tail biting trellises is still under development in this paper we first develop a new algebraic framework for a systematic analysis of linear trellises which enables us to address open foundational questions in particular we present a useful and powerful characterization of linear trellis isomorphy we also obtain a new proof of the linear factorization theorem of koetter vardy and point out unnoticed problems for the group case next we apply our framework to 1 describe all the factorizations of linear trellises 2 determine and classify all the minimal linear trellises for a given linear code and 3 prove that nonmergeable one to one linear trellises are determined by the edge label sequences of certain closed paths we also provide new insight into mergeability and state how results on reduced linear trellises can be extended to nonreduced ones our classification results are relevant to iterative linear programming decoding as we show that minimal linear trellises can yield different pseudocodewords even if they have the same graph structure

local irreducibility of tail biting trellises this paper investigates tail biting trellis realizations for linear block codes intrinsic trellis properties are used to characterize irreducibility on given intervals of the time axis it proves beneficial to always consider the trellis and its dual simultaneously a major role is played by trellis properties that amount to observability and controllability of trellis fragments of various lengths for fragments of length less than the minimum span length of the code it is shown that fragment observability and fragment controllability are equivalent to irreducibility for reducible trellises a constructive reduction procedure is presented the considerations also lead to a characterization for when the dual of a trellis allows a product factorization into elementary atomic trellises

generation of a precise binary logarithm with difference grouping programmable logic array the design algorithm of a differential group programmable logic array dgpla to generate the precise binary logarithm function is suggested it can reach an optimal condition such that the number of bits in a pla is minimized while the error is still kept as small as possible thus the space in the pla is saved estimated at only 15 94 percent of the space for a readonly memory rom counterpart

the distribution of the frequency of occurrence of nucleotide subsequences based on their overlap capability dna s genetic code can be represented as an alphabetic sequence composed of the four letters a c g and t which represent the four types of nucleotides adenylic cytidylic guanylic and thymidylic acid of which dna is composed now that these sequences have been identified for many genes and are available in computer readable form scientists can analyze these data and search for patterns in an attempt to learn more about the regulatory functions of the gene one area of study is that of the frequency of occurrence of specific nucleotide subsequences e g acac within part or all of a nucleotide sequence this paper derives the probability distribution of the frequency of occurrence of a subsequence within a nucleotide sequence under the hypothesis that the four nucleotides occur at random and with equal probability this distribution is nontrivial because different subsequences have different overlap capability for example the subsequence aaaa can occur up to 17 times in a sequence of length 20 which would happen if the sequence were composed solely of a s but the subsequence acgt cannot occur more than 5 times in a sequence of length 20 thus the frequency distributions are different for each type of overlap capability it is of interest to assess and compare the degree of nonrandomness for different subsequences or among different portions of a sequence the existence and degree of nonrandomness may be related to the type and degree of functionality of a nucleotide sub sequence the frequency distributions provided here can be used to perform exact significance tests of the hypothesis of randomness an approximate test is also described for use with long sequences this can be used to test a more general null hypothesis of nucleotides occurring with unequal probabilities

on the pseudocodeword redundancy of binary linear codes for a binary linear code the pseudocodeword redundancy with respect to the additive white gaussian noise channel the binary symmetric channel or the max fractional weight is defined to be the smallest number of rows in a parity check matrix such that the corresponding minimum pseudoweight is equal to the minimum hamming distance of the code it is shown that most codes do not have a finite pseudocodeword redundancy also upper bounds on the pseudocodeword redundancy for some families of codes including codes based on designs are provided the pseudocodeword redundancies for all codes of small length at most 9 are computed furthermore comprehensive results are provided on the cases of cyclic codes of length at most 250 for which the eigenvalue bound of vontobel and koetter is sharp

a measure of dna sequence dissimilarity based on mahalanobis distance between frequencies of words summary a number of algorithms exist for searching genetic databases for biologically significant similarities in dna sequences past research has shown that word based search tools are computationally efficient and can find similarities or dissimilarities invisible to other algorithms like fasta we characterize a family of word based dissimilarity measures that define distance between two sequences by simultaneously comparing the frequencies of all subsequences of n adjacent letters i e n words in the two sequences applications to real data demonstrate that currently used word based methods that rely on euclidean distance can be significantly improved by using mahalanobis distance which accounts for both variances and covariances between frequencies of n words furthermore in those cases where mahalanobis distance may be too difficult to compute using standardized euclidean distance which only corrects for the variances of frequencies of n words still gives better performance than the euclidean distance also a simple way of combining distances obtained at different n words is considered the goal is to obtain a single measure of dissimilarity between two dna sequences the performance ranking of the preceding three distances still holds for their combined counterparts all results obtained in this paper are applicable to amino acid sequences with minor modifications

minimal realizations of linear systems the shortest basis approach given a discrete time linear system c a shortest basis for c is a set of linearly independent generators for c with the least possible lengths a basis b is a shortest basis if and only if it has the predictable span property i e has the predictable delay and degree properties and is non catastrophic or alternatively if and only if it has the subsystem basis property for any interval j the generators in b whose span is in j is a basis for the subsystem cj the dimensions of the minimal state spaces and minimal transition spaces of c are simply the numbers of generators in a shortest basis b that are active at any given state or symbol time respectively a minimal linear realization for c in controller canonical form follows directly from a shortest basis for c and a minimal linear realization for c in observer canonical form follows directly from a shortest basis for the orthogonal system c this approach seems conceptually simpler than that of classical minimal realization theory

low power logarithmic number system addition subtraction and their impact on digital filters this paper presents techniques for low power addition subtraction in the logarithmic number system lns and quantifies their impact on digital filter vlsi implementation the impact of partitioning the look up tables required for lns addition subtraction on complexity performance and power dissipation of the corresponding circuits is quantified two design parameters are exploited to minimize complexity namely the lns base and the organization of the lns word a roundoff noise model is used to demonstrate the impact of base and word length on the signal to noise ratio of the output of finite impulse response fir filters in addition techniques for the low power implementation of an lns multiply accumulate mac units are investigated furthermore it is shown that the proposed techniques can be extended to cotransformation based circuits that employ interpolators the results are demonstrated by evaluating the power dissipation complexity and performance of several fir filter configurations comprising one two or four mac units simulations of placed and routed vlsi lns based digital filters using a 90 nm 1 0 v cmos standard cell library reveal that significant power dissipation savings are possible by using optimized lns circuits at no performance penalty when compared to linear fixed point two s complement equivalents

codes on graphs constraint complexity of cycle free realizations of linear codes cycle free graphical realizations of linear codes generalize trellis realizations given a linear code c and a cycle free graph topology there exists a well defined minimal realization for c on that graph in which each constraint is a linear code with a well defined length and dimension the constraint complexity of the realization is defined as maximum dimension of any constraint code there exists a graph that minimizes constraint complexity in which all internal nodes have degree 3 and all interface nodes have degree 2 and which moreover can be put in the form of a tree structured trellis realization the constraint complexity of a general cycle free graph realization can be less than that of any conventional trellis realization but not by very much such realizations can yield reductions in decoding complexity even when they do not reduce constraint complexity

pipelined computation of very large word length lns addition subtraction with polynomial hardware cost a novel pipelined method is proposed to compute the addition subtraction in very large word length logarithmic number system lns arithmetic digit parallel additive normalization and digit on line multiplicative normalization methods are adopted to compute the exponential and logarithmic functions respectively in lns addition subtraction these two methods can both be implemented in a pipelined and regular architecture the size of the required lookup tables is now proportional to a third order polynomial function instead of an exponential function of the word length the total size of the tables in a 32 bit lns unit is estimated to be less than 53 5 kbits and the total size of the table s in a 64 bit lns unit will be less than 471 kbits furthermore the hardware cost of the other circuits in the proposed lns unit is only proportional to the square of the word length this study also develops a simple leading zero bits prediction technique that can significantly enhance the precision in lns addition subtraction computation when compared to floating point arithmetic design our approach still suffers from large hardware cost and pipeline latency however the proposed approach has provided a theoretical advancement in the development of very large word length lns arithmetic

codes on graphs duality and macwilliams identities a conceptual framework involving partition functions of normal factor graphs is introduced paralleling a similar recent development by al bashabsheh and mao the partition functions of dual normal factor graphs are shown to be a fourier transform pair whether or not the graphs have cycles the original normal graph duality theorem follows as a corollary within this framework macwilliams identities are found for various local and global weight generating functions of general group or linear codes on graphs this generalizes and provides a concise proof of the macwilliams identity for linear time invariant convolutional codes that was recently found by gluesing luerssen and schneider further macwilliams identities are developed for terminated convolutional codes particularly for tail biting codes similar to those studied recently by bocharova hug johannesson and kudryashov

counting in graph covers a combinatorial characterization of the bethe entropy function we present a combinatorial characterization of the bethe entropy function of a factor graph such a characterization being in contrast to the original analytical definition of this function we achieve this combinatorial characterization by counting valid configurations in finite graph covers of the factor graph analogously we give a combinatorial characterization of the bethe partition function whose original definition was also of an analytical nature as we point out our approach has similarities to the replica method but also stark differences the above findings are a natural backdrop for introducing a decoder for graph based codes that we will call symbolwise graph cover decoding a decoder that extends our earlier work on blockwise graph cover decoding both graph cover decoders are theoretical tools that help toward a better understanding of message passing iterative decoding namely blockwise graph cover decoding links max product min sum algorithm decoding with linear programming decoding and symbolwise graph cover decoding links sum product algorithm decoding with bethe free energy function minimization at temperature one in contrast to the gibbs entropy function which is a concave function the bethe entropy function is in general not concave everywhere in particular we show that every code picked from an ensemble of regular low density parity check codes with minimum hamming distance growing with high probability linearly with the block length has a bethe entropy function that is convex in certain regions of its domain

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

applying features of ieee 754 to sign logarithm arithmetic various features found in standard floating point arithmetic ieee 754 are examined in light of their appropriateness for sign logarithm arithmetic the emphasis is on a 32 b word size comparable to ieee 754 single precision although other word sizes are possible a multilayer sign logarithm format is considered the lowest layer similar to previous implementations would provide only normalized representations but would not provide representations for zero denormalized values infinities and nans the highest layer would provide most of the features found in ieee 754 including zeros denormalized values infinities and nans novel algorithms for implementing logarithmic denormalized arithmetic are presented simulation results show that the error characteristics of the proposed logarithmic denormalized arithmetic algorithms are similar to those of the denormalized floating point arithmetic in ieee 754

arithmetic co transformations in the real and complex logarithmic number systems the real logarithmic number system which represents a value with a sign bit and a quantized logarithm can be generalized to create the complex logarithmic number system which replaces the sign bit with a quantized angle in a log polar coordinate system although multiplication and related operations are easy in both real and complex systems addition and subtraction are hard especially when interpolation is used to implement the system both real and complex logarithmic arithmetic benefit from the use of co transformation which converts an addition or subtraction from a region where interpolation is expensive to a region where it is easier two co transformations that accomplish this goal are introduced the first is an approximation based on real analysis of the subtraction logarithm the second is based on simple algebra that applies for both real and complex values and that works for both addition and subtraction

sign logarithm arithmetic for fft implementation sign logarithm arithmetic is applicable to a variety of numerical applications where wide dynamic range and small wordsize are required in this paper the basic sign logarithm arithmetic operations required for signal processing i e addition subtraction and multiplication are reviewed the computational errors are analyzed for fft realization and simulation results are presented which serve to verify the analysis it is shown that the sign logarithm approach provides improved arithmetic quantization error performance for a given word size over fft s implemented with conventional fixed or floating point arithmetic and that the sign logarithm implementation is faster and less complex than conventional approaches

interleaved memory function interpolators with application to an accurate lns arithmetic unit this paper describes a new method for polynomial interpolation in hardware with advantages demonstrated by its application to an accurate logarithmic number system lns arithmetic unit the use of an interleaved memory reduces storage requirements by allowing each stored function value to be used in interpolation across several segments this strategy can be shown to always use fewer words of memory than an optimized polynomial with stored polynomial coefficients interleaved memory function interpolators are then applied to the specific goal of an accurate logarithmic number system arithmetic unit many accuracy requirements for the lns arithmetic unit are possible although a round to nearest would be desirable it cannot be easily achieved the goal suggested is to insure that the worst case lns relative error is smaller than the worst case floating point fp relative error using the interleaved memory interpolator the detailed design of an lns arithmetic unit is performed using a second order polynomial interpolator including approximately 91k bits of rom this arithmetic unit has better accuracy and less complexity than previous lns units

linear tail biting trellises characteristic generators and the bcjr construction this paper investigates the constructions of tail biting trellises for linear block codes as introduced by koetter and vardy 2003 and nori and shankar 2006 for a given code the sets of characteristic generators are defined slightly more generally than by koetter and vardy in particular they are not uniquely determined by the code the effect of the choice of characteristic generators on the resulting product trellises called kv trellises is discussed in detail it is shown that each kv trellis is a span based bcjr trellis and that the latter are always nonmergeable finally a duality conjecture posed by koetter and vardy is addressed by making use of a dualization technique of bcjr trellises the conjecture is proven for minimal trellises

constraint complexity of realizations of linear codes on arbitrary graphs a graphical realization of a linear code c consists of an assignment of the coordinates of c to the vertices of a graph along with a specification of linear state spaces and linear local constraint codes to be associated with the edges and vertices respectively of the graph the kappa complexity of a graphical realization is defined to be the largest dimension of any of its local constraint codes kappa complexity is a reasonable measure of the computational complexity of a sum product decoding algorithm specified by a graphical realization the main focus of this paper is on the following problem given a linear code c and a graph g how small can the kappa complexity of a realization of c on g be as useful tools for attacking this problem we introduce the vertex cut bound and the notion of vc treewidth for a graph which is closely related to the well known graph theoretic notion of treewidth using these tools we derive tight lower bounds on the kappa complexity of any realization of c on g our bounds enable us to conclude that good error correcting codes can have low complexity realizations only on graphs with large vc treewidth along the way we also prove the interesting result that the ratio of the kappa complexity of the best conventional trellis realization of a length n code c to the kappa complexity of the best cycle free realization of c grows at most logarithmically with code length n such a logarithmic growth rate is in fact achievable

codes on graphs observability controllability and local reducibility this paper investigates properties of realizations of linear or group codes on general graphs that lead to local reducibility trimness and properness are dual properties of constraint codes a linear or group realization with a constraint code that is not both trim and proper is locally reducible a linear or group realization on a finite cycle free graph is minimal if and only if every local constraint code is trim and proper a realization is called observable if there is a one to one correspondence between codewords and configurations and controllable if it has independent constraints a linear or group realization is observable if and only if its dual is controllable a simple counting test for controllability is given an unobservable or uncontrollable realization is locally reducible parity check realizations are controllable if and only if they have independent parity checks in an uncontrollable tail biting trellis realization the behavior partitions into disconnected sub behaviors but this property does not hold for nontrellis realizations on a general graph the support of an unobservable configuration is a generalized cycle

integrated circuit logarithmic arithmetic units this correspondence examines integrated circuit logarithmic arithmetic units which include adders subtracters multipliers and dividers the design of these arithmetic units is reviewed and an example arithmetic unit which performs multiplication followed by addition is designed in detail the design results are used to develop a size and speed comparison of integrated circuit logarithmic and fixed point arithmetic units this comparison is exercised through a video signal processing example it is concluded from this comparison that logarithmic arithmetic units are smaller than and as fast as fixed point arithmetic units with comparable capabilities in digital signal processing applications characterized by large dynamic range and moderate computational accuracy requirements further this comparison quantitatively illustrates the interaction of digital signal processing and integrated circuit issues in the design of special purpose digital signal processors