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

algebraic construction of sparse matrices with large girth in this correspondence we present a method for constructing sparse matrices that have a compact description and whose associated bipartite graphs have large girth based on an arbitrary seed matrix of nonnegative integers a new matrix is constructed which replaces each entry of the seed matrix with a sum of permutation matrices algebraic conditions that lead to short cycles in the associated bipartite graph are analyzed and methods to achieve large girth in two special cases are presented in one all the permutation matrices are circulants in the other they are all affine permutation matrices when used to define a low density parity check ldpc code the compact description should lead to efficient implementation and the large girth to good error correction performance the method is adaptable to a variety of rates and a variety of row and column degrees

low density parity check codes based on finite geometries a rediscovery and new results this paper presents a geometric approach to the construction of low density parity check ldpc codes four classes of ldpc codes are constructed based on the lines and points of euclidean and projective geometries over finite fields codes of these four classes have good minimum distances and their tanner 1981 graphs have girth 6 finite geometry ldpc codes can be decoded in various ways ranging from low to high decoding complexity and from reasonably good to very good performance they perform very well with iterative decoding furthermore they can be put in either cyclic or quasi cyclic form consequently their encoding can be achieved in linear time and implemented with simple feedback shift registers this advantage is not shared by other ldpc codes in general and is important in practice finite geometry ldpc codes can be extended and shortened in various ways to obtain other good ldpc codes several techniques of extension and shortening are presented long extended finite geometry ldpc codes have been constructed and they achieve a performance only a few tenths of a decibel away from the shannon theoretical limit with iterative decoding

spectrum estimation by wavelet thresholding of multitaper estimators current methods for power spectrum estimation by wavelet thresholding use the empirical wavelet coefficients derived from the log periodogram unfortunately the periodogram is a very poor estimate when the true spectrum has a high dynamic range and or is rapidly varying in addition because the distribution of the log periodogram is markedly non gaussian special wavelet dependent thresholding schemes are needed these difficulties can be bypassed by starting with a multitaper spectrum estimator the logarithm of this estimator is close to gaussian distributed if a moderate number spl ges 5 of tapers are used in contrast to the log periodogram log multitaper estimates are not approximately pairwise uncorrelated at the fourier frequencies but the form of the correlation can be accurately and simply approximated for scale independent thresholding the correlation acts in accordance with the wavelet shrinkage paradigm to suppress small scale noise spikes while leaving informative coarse scale coefficients relatively unattenuated this simple approach to spectrum estimation is demonstrated to work very well in practice additionally the progression of the variance of wavelet coefficients with scale can be accurately calculated allowing the use of scale dependent thresholds this more involved approach also works well in practice but is not uniformly preferable to the scale independent approach

long term analysis of the stormer verlet method for hamiltonian systems with a solution dependent high frequency the long time behaviour of the stormer verlet leapfrog method is studied when this method is applied to highly oscillatory hamiltonian systems with a slowly varying solution dependent high frequency using the technique of modulated fourier expansions with state dependent frequencies which is newly developed here the following results are proved the considered hamiltonian systems have the action as an adiabatic invariant over long times that cover arbitrary negative powers of the small parameter the stormer verlet method approximately conserves a modified action and a modified total energy over a long time interval that covers a negative integer power of the small parameter this power depends on the size of the product of the stepsize with the high frequency

the averaged lagrangian method in this paper we present a new class of symplectic methods namely variational integrators employing the same integral averaging found in the averaged vector field method extensions to higher order are obtained using a collocation approach the obtained methods are time symmetric p series methods the proposed methods are compared with existing integrators the tests focus on highly oscillatory problems like the fermi pasta ulam problem

multitaper spectral estimation of power law processes in many branches of science particularly astronomy and geophysics power spectra of the form f sup spl beta where spl beta is a positive power law exponent are common this form of spectrum is characterized by a sharp increase in the spectral density as the frequency f decreases toward zero a power spectrum analysis method that has proven very powerful wherever the spectrum of interest is detailed and or varies rapidly with a large dynamic range is the multitaper method with multitaper spectral estimation a set of orthogonal tapers are applied to the time series and the resulting direct spectral estimators eigenspectra are averaged thus reducing the variance one class of processes with spectra of the power law type are fractionally differenced gaussian processes that are stationary and can model certain types of long range persistence spectral decay f sup spl beta can be modeled for 0 spl beta 1 estimation of the spectral slope parameter by regression on multitaper spectral ordinates is examined for this class of processes it is shown that multitapering or using sine or slepian tapers produces much better results than using the periodogram and is attractive compared with other competing methods the technique is applied to a geophysical estimation problem

on the construction of local parities for r t availability in distributed storage motivated by applications involving hot data the notion of r t availability was introduced where a symbol is said to have r t availability if it can be reconstructed respectively from t disjoint repair alternatives of other symbols each of size at most r the key to constructing a family of locally repairable codes with r t availability is the design of a membership matrix for dividing global parities into local ones such that each repair group contains only one parity symbol although explicit designs of membership matrices are available it remains unclear whether membership matrices with significantly better parameters can be constructed which was a question left open to tackle the open problem this paper first provides a connection between the design of membership matrices and the design of a combinatorial object in the well known combinatorial design theory called resolvable configurations this paper then proposes several direct designs of membership matrices based on euclidean geometry finite fields circulant permutation matrices affine permutation matrices and reed solomon codes by leveraging the notion of resolvable configurations and the kronecker product this paper further proposes a two step design of membership matrices the proposed direct and two step designs of membership matrices are shown to have significantly better parameters than those recently available

should structure functions be used to estimate power laws in turbulence a comparative study second order structure functions are widely used to characterize turbulence in the inertial range because they are simple to estimate particularly in comparison to spectral density functions and wavelet variances structure function estimators however are highly autocorrelated and as a result no suitable theory has been established to provide confidence intervals for turbulence parameters when determined via regression fits in log log space monte carlo simulations were performed to compare the performance of structure function estimators of turbulence parameters with corresponding multitaper spectral and wavelet variance estimators the simulations indicate that these latter estimators have smaller variances than estimators based upon the structure function in contrast to structure function estimators the statistical properties of the multitaper spectral and wavelet variance estimators allow for the construction of confidence intervals for turbulence parameters the monte carlo simulations also confirm the validity of the statistical theory behind the multitaper spectral and wavelet variance estimators the strengths and weaknesses of the various estimators are further illustrated by analyzing an atmospheric temperature time series

near shannon limit quasi cyclic low density parity check codes this letter presents two classes of quasi cyclic low density parity check codes that perform close to the shannon limit

on finding bipartite graphs with a small number of short cycles and large girth the problem of finding bipartite tanner graphs with given degree sequences that have large girth and few short cycles is of great interest in many applications including construction of good low density parity check ldpc codes in this paper we prove that for given integers alpha beta and gamma and degree sequences pi and pi the problem of determining whether there exists a simple bipartite graph with degree sequences pi pi that has at most alpha beta and gamma cycles of length four six and eight respectively is np complete this is proved by a two step polynomial time reduction from the 3 partition problem on the other hand using connections to linear hypergraphs we prove that given the degree sequence pi a polynomial time algorithm can be devised to determine whether there exists a bipartite graph whose degree sequence on one side of the bipartition is pi and has a girth of at least six in addition to these complexity results we devise a quasi polynomial time algorithm that can construct a bipartite graph with given irregular degree sequences and a girth that increases logarithmically with the size of the graph compared to the well known progressive edge growth peg algorithm the proposed method in some cases results in tanner graphs with larger girth particularly for scenarios where the degree sequences of the graph are strictly enforced

hierarchical and high girth qc ldpc codes we present an approach to designing capacity approaching high girth low density parity check ldpc codes that are friendly to hardware implementation and compatible with some desired input code structure defined using a protograph the approach is based on a mapping of any class of codes defined using a protograph into a family of hierarchical quasi cyclic hqc ldpc codes whereas the parity check matrices of standard quasi cyclic qc ldpc codes are composed of circulant submatrices those of hqc ldpc codes are composed of a hierarchy of circulant submatrices that are in turn constructed from circulant submatrices and so on through some number of levels next we present a girth maximizing algorithm that optimizes the degrees of freedom within the family of codes to yield a high girth hqc ldpc code subject to bounds imposed by the fact that hqc codes are still quasi cyclic finally we discuss how certain characteristics of a code protograph will lead to inevitable short cycles and show that these short cycles can be eliminated using a squashing procedure that results in a high girth qc ldpc code although not a hierarchical one we illustrate our approach with three design examples of qc ldpc codes two girth 10 codes of rates 1 3 and 0 45 and one girth 8 code of rate 0 7 all of which are obtained from protographs of one sided spatially coupled codes

assessing neuronal coherence with single unit multi unit and local field potentials the purpose of this study was to obtain a better understanding of neuronal responses to correlated input in particular focusing on the aspect of synchronization of neuronal activity the first aim was to obtain an analytical expression for the coherence between the output spike train and correlated input and for the coherence between output spike trains of neurons with correlated input for poisson neurons we could derive that the peak of the coherence between the correlated input and multi unit activity increases proportionally with the square root of the number of neurons in the multi unit recording the coherence between two typical multi unit recordings 2 to 10 single units with partially correlated input increases proportionally with the number of units in the multi unit recordings the second aim of this study was to investigate to what extent the amplitude and signal to noise ratio of the coherence between input and output varied for single unit versus multi unit activity and how they are affected by the duration of the recording the same problem was addressed for the coherence between two single unit spike series and between two multi unit spike series the analytical results for the poisson neuron and numerical simulations for the conductance based leaky integrate and fire neuron and for the conductance based hodgkin huxley neuron show that the expectation value of the coherence function does not increase for a longer duration of the recording the only effect of a longer duration of the spike recording is a reduction of the noise in the coherence function the results of analytical derivations and computer simulations for model neurons show that the coherence for multi unit activity is larger than that for single unit activity this is in agreement with the results of experimental data obtained from monkey visual cortex v4 finally we show that multitaper techniques greatly contribute to a more accurate estimate of the coherence by reducing the bias and variance in the coherence estimate

ldpc block and convolutional codes based on circulant matrices a class of algebraically structured quasi cyclic qc low density parity check ldpc codes and their convolutional counterparts is presented the qc codes are described by sparse parity check matrices comprised of blocks of circulant matrices the sparse parity check representation allows for practical graph based iterative message passing decoding based on the algebraic structure bounds on the girth and minimum distance of the codes are found and several possible encoding techniques are described the performance of the qc ldpc block codes compares favorably with that of randomly constructed ldpc codes for short to moderate block lengths the performance of the ldpc convolutional codes is superior to that of the qc codes on which they are based this performance is the limiting performance obtained by increasing the circulant size of the base qc code finally a continuous decoding procedure for the ldpc convolutional codes is described

quasi cyclic ldpc codes based on pre lifted protographs quasi cyclic low density parity check qc ldpc codes based on protographs are of great interest to code designers because analysis and implementation are facilitated by the protograph structure and the use of circulant permutation matrices for protograph lifting however these restrictions impose undesirable fixed upper limits on important code parameters such as minimum distance and girth in this paper we consider an approach to constructing qc ldpc codes that uses a two step lifting procedure based on a protograph and by following this method instead of the usual one step procedure we obtain improved minimum distance and girth properties we also present two new design rules for constructing good qc ldpc codes using this two step lifting procedure and in each case we obtain a significant increase in minimum distance and achieve a certain guaranteed girth compared with one step circulant based liftings the expected performance improvement is verified by simulation results

on the girth of quasi cyclic protograph ldpc codes in this paper we study the relationships between the girth of the tanner graph of a quasi cyclic qc protograph low density parity check ldpc code the lifting degree and the size and the structure of the base graph as a result for a given base graph we derive a lower bound on the lifting degree as a necessary condition for the lifted graph to have a certain girth this also provides an upper bound on the girth of the family of graphs lifted from a given base graph with a given lifting degree the upper bounds derived here which are applicable to both regular and irregular base graphs with no parallel edges are in some cases more general and in some other cases tighter than the existing bounds the results presented in this work can be used to design cyclic liftings with relatively small degree and relatively large girth as an example we present new qc protograph ldpc code constructions with girth 8 using fully connected base graphs these constructions provide upper bounds on the lifting degree required for achieving girth 8 using fully connected base graphs

reduced mean square error quadratic inverse spectrum estimator a new spectrum estimator is introduced the new estimator exploits quadratic inverse theory to attain improved mean square error performance over the standard multitaper spectrum estimators the standard non adaptive eigenvalue weighted multitaper estimator is obtained by averaging a high resolution inconsistent spectrum estimator over the estimator bandwidth the improved performance of the proposed estimator results from replacing this average with a weighted average computed in the space spanned by the quadratic inverse basis the weighting determined analytically is chosen such that the resulting estimator minimizes the sum of the variance and the square of the in band bias neglecting bias due to spectral leakage and potential bias due to the possible incompleteness of the quadratic inverse basis for a white spectrum the neglected bias is found to be as small as that of a standard non adaptive multitaper spectrum estimator the relative reduction of the mean square error of the proposed spectrum estimator is validated by simulation for an arma 4 2 process and results in a typical mean square error reduction of 5 for large time bandwidth parameters and 20 for a time bandwidth parameter of four when compared to the non adaptive non eigenvalue weighted multitaper estimator when compared to the adaptive multitaper spectrum estimator larger mean square error improvements are attainable an expression for the theoretical probability density function for the proposed estimator is given it is found to be as accurate as the asymptotic probability density function for the standard multitaper estimator

a new multi edge metric constrained peg algorithm for designing binary ldpc code with improved cycle structure the progressive edge growth peg algorithm constructs an edge in each stage to maximize the variable node vn of interest s local girth in real time thus this vn s local girths after more than one edge is added to the current tanner graph tg setting may not be maximized relative to that tg setting to address this problem we define the multi edge local girth and edge trial and based on these definitions propose a new multi edge metric constrained peg algorithm mm pega to improve the design at each vn the mm pega constructs an edge in each stage that relative to the current tg setting can potentially maximize the vn of interest s local girth after a certain number up to the edge trial of edges are added to the tg setting we first analyze the properties of the multi edge local girth and then propose an algorithm for calculating the multi edge local girth we also propose a method for accelerating the mm pega moreover we generalize the mm pega for improving different peg like designs according to the theoretical analysis increasing the edge trial of the mm pega is expected to positively affect the cycle structure and the error performance of resulting low density parity check ldpc code this expectation is verified by simulations

on algebraic construction of gallager and circulant low density parity check codes this correspondence presents three algebraic methods for constructing low density parity check ldpc codes these methods are based on the structural properties of finite geometries the first method gives a class of gallager codes and a class of complementary gallager codes the second method results in two classes of circulant ldpc codes one in cyclic form and the other in quasi cyclic form the third method is a two step hybrid method codes in these classes have a wide range of rates and minimum distances and they perform well with iterative decoding

finite length analysis of low density parity check codes on the binary erasure channel in this paper we are concerned with the finite length analysis of low density parity check ldpc codes when used over the binary erasure channel bec the main result is an expression for the exact average bit and block erasure probability for a given regular ensemble of ldpc codes when decoded iteratively we also give expressions for upper bounds on the average bit and block erasure probability for regular ldpc ensembles and the standard random ensemble under maximum likelihood ml decoding finally we present what we consider to be the most important open problems in this area

joseph geology and seismic deconvolution common practice in seismic deconvolution is to assume that the reflection sequence is uncorrelated that is that the sequence has a white power spectrum and a delta function autocorrelation a white spectrum implies that the acoustic impedance function has a power spectrum proportional to 1 f2 which is characteristic of a nonstationary brownian process f is frequency however the maximum power spectrum permissible for the acoustic impedance function is 1 f we call a spectrum of this kind a joseph spectrum a joseph spectrum corresponds to a reflection sequence with a power spectrum proportional to f and a negative autocorrelation at small lags joseph spectrum behavior for reflection sequences has been seen before and we show it again in a well off newfoundland and in two wells from quebec if the power spectrum is proportional to f then the first term of the discretized autocorrelation function is 0 405 of the zero lag term and higher terms are negligible we construct a joseph filter analogous t

searching for voltage graph based ldpc tailbiting codes with large girth the relation between parity check matrices of quasi cyclic qc low density parity check ldpc codes and biadjacency matrices of bipartite graphs supports searching for powerful ldpc block codes using the principle of tailbiting compact representations of bipartite graphs based on convolutional codes can be found bounds on the girth and the minimum distance of ldpc block codes constructed in such a way are discussed algorithms for searching iteratively for ldpc block codes with large girth and for determining their minimum distance are presented constructions based on all one matrices steiner triple systems and qc block codes are introduced finally new qc regular ldpc block codes with girth up to 24 are given

symmetrical constructions for regular girth 8 qc ldpc codes in this paper we propose new constructions for regular girth 8 quasi cyclic low density parity check qc ldpc codes based on circulant permutation matrices cpm the constructions assume symmetries in the structure of the parity check matrix and employ a greedy exhaustive search algorithm to find the permutation shifts of the cpms as a result of symmetries the new codes have a more compact representation compared with their counterparts in majority of cases also they achieve the girth 8 at a shorter block length for the same degree distribution code rate deterministic explicit constructions are also presented to expand the proposed parity check matrices to larger block lengths and higher rates the proposed long high rate codes are often substantially shorter than regular girth 8 qc ldpc codes of similar rate in the literature simulation results demonstrate that the proposed symmetric codes have competitive performance in comparison with similar existing qc ldpc codes that lack symmetry

decomposition methods for large scale lp decoding when binary linear error correcting codes are used over symmetric channels a relaxed version of the maximum likelihood decoding problem can be stated as a linear program lp this lp decoder can be used to decode error correcting codes at bit error rates comparable to state of the art belief propagation bp decoders but with significantly stronger theoretical guarantees however lp decoding when implemented with standard lp solvers does not easily scale to the block lengths of modern error correcting codes in this paper we draw on decomposition methods from optimization theory specifically the alternating direction method of multipliers admm to develop efficient distributed algorithms for lp decoding the key enabling technical result is a two slice characterization of the parity polytope the polytope formed by taking the convex hull of all codewords of the single parity check code this new characterization simplifies the representation of points in the polytope using this simplification we develop an efficient algorithm for euclidean norm projection onto the parity polytope this projection is required by the admm decoder and its solution allows us to use lp decoding with all its theoretical guarantees to decode large scale error correcting codes efficiently we present numerical results for ldpc codes of lengths more than 1000 the waterfall region of lp decoding is seen to initiate at a slightly higher snr than for sum product bp however an error floor is not observed for lp decoding which is not the case for bp our implementation of lp decoding using the admm executes as fast as our baseline sum product bp decoder is fully parallelizable and can be seen to implement a type of message passing with a particularly simple schedule

efficient search of girth optimal qc ldpc codes in this paper we study the cycle structure of quasi cyclic qc low density parity check ldpc codes with the goal of obtaining the shortest code with a given degree distribution and girth we focus on qc ldpc codes whose tanner graphs are cyclic liftings of fully connected base graphs of size 3 times n n geq 4 and obtain minimal lifting degrees that result in girths 6 and 8 this is performed through an efficient exhaustive search and as a result we also find all the possible non isomorphic codes with the same minimum block length girth and degree distribution the exhaustive search which is ordinarily a formidable task is made possible by pruning the search space of many codes that are isomorphic to those previously examined in the search process many of the pruning techniques proposed in this paper are also applicable to qc ldpc codes with base graphs other than the 3 times n fully connected ones discussed here as well as to codes with a larger girth to further demonstrate the effectiveness of the pruning techniques we use them to search for qc ldpc codes with girths 10 and 12 and find a number of such codes that have a shorter block length compared with the best known similar codes in the literature in addition motivated by the exhaustive search results we tighten the lower bound on the block length of qc ldpc codes of girth 6 constructed from fully connected 3 times n base graphs and construct codes that achieve the lower bound for an arbitrary value of n geq 4

modeling persistence in hydrological time series using fractional differencing the class of autoregressive integrated moving average arima time series models may be generalized by permitting the degree of differencing d to take fractional values models including fractional differencing are capable of representing persistent series d 0 or short memory series d 0 the class of fractionally differenced arima processes provides a more flexible way than has hitherto been available of simultaneously modeling the long term and short term behavior of a time series in this paper some fundamental properties of fractionally differenced arima processes are presented methods of simulating these processes are described estimation of the parameters of fractionally differenced arima models is discussed and an approximate maximum likelihood method is proposed the methodology is illustrated by fitting fractionally differenced models to time series of streamflows and annual temperatures

maximum likelihood estimation of the parameters of discrete fractionally differenced gaussian noise process a maximum likelihood estimation procedure is constructed for estimating the parameters of discrete fractionally differenced gaussian noise from an observation set of finite size n the procedure does not involve the computation of any matrix inverse or determinant it requires n sup 2 2 o n operations the expected value of the loglikelihood function for estimating the parameter d of fractionally differenced gaussian noise which corresponds to a parameter of the equivalent continuous time fractional brownian motion related to its fractal dimension is shown to have a unique maximum that occurs at the true value of d a cramer rao bound on the variance of any unbiased estimate of d obtained from a finite sized observation set is derived it is shown experimentally that the maximum likelihood estimate of d is unbiased and efficient when finite size data sets are used in the estimation procedure the proposed procedure is extended to deal with noisy observations of discrete fractionally differenced gaussian noise

modified trigonometric integrators we study modified trigonometric integrators which generalize the popular class of trigonometric integrators for highly oscillatory hamiltonian systems by allowing the fast frequencies to be modifi

adaptive smoothing of the log spectrum with multiple tapering a hybrid estimator of the log spectral density of a stationary time series is proposed first a multiple taper estimated is performed followed by kernel smoothing the log multiple taper estimate this procedure reduces the expected mean square error by spl pi sup 2 4 sup 4 5 over simply smoothing the log tapered periodogram a data adaptive implementation of a variable bandwidth kernel smoother is given

from cages to trapping sets and codewords a technique to derive tight upper bounds on the minimum size of trapping sets and minimum distance of ldpc codes cages defined as regular graphs with minimum number of nodes for a given girth are well studied in graph theory trapping sets are graphical structures responsible for error floor of low density parity check ldpc codes and are well investigated in coding theory in this paper we make connections between cages and trapping sets in particular starting from a cage or a modified cage we construct a trapping set in multiple steps based on the connection between cages and trapping sets we then use the available results in graph theory on cages and derive tight upper bounds on the size of the smallest trapping sets for variable regular ldpc codes with a given variable degree and girth the derived upper bounds in many cases meet the best known lower bounds and thus provide the actual size of the smallest trapping sets considering that non zero codewords are a special case of trapping sets we also derive tight upper bounds on the minimum weight of such codewords i e the minimum distance of variable regular ldpc codes as a function of variable degree and girth

optimized weighted averaging of peak matched multiple window spectrum estimators periodogram averaging with multiple windows can be used in spectrum analysis of nonstationary data usually however the windows for the subspectra are equally weighted in the estimate in this correspondence a criterion for the optimization of weighting factors is formulated as the average of normalized bias variance or mean square error in a certain frequency interval around a predefined peaked spectrum the weighting factors are optimized using the peak matched multiple windows the sinusoid multiple windows and the discrete prolate spheroidal sequences

the effective bandwidth of a multitaper spectral estimator summary the bandwidth of a spectral estimator is a measure of the minimum separation in frequency between approximately uncorrelated spectral estimates we determine an effective bandwidth measure for multitaper spectral estimators a relatively new and very powerful class of spectral estimators proving to be very valuable whenever the spectrum of interest is detailed and or varies rapidly with a large dynamic range the multitaper spectral estimator is the average of several direct spectral estimators each of which uses one of a set of orthogonal tapers we show that the equivalent width of the autocorrelation of the overall spectral window is a suitable measure of the effective bandwidth of a multitaper spectral estimator and illustrate its use in the case of both slepian and sinusoidal orthogonal tapers this measure allows a unified treatment of bandwidth for the class of quadratic spectral estimators hence for example it is now possible properly to compare multitaper spectral estimators with traditional lag window spectral estimators by assigning a fixed and equal effective bandwidth to both methods an application is given to the spectral analysis of ocean wave data

on girth conditioning for low density parity check codes low density parity check ldpc codes are gaining interest for high data rate applications in both terrestrial and spatial communications they can be designed and studied through a bipartite graph whose characteristics affect the performance this paper proposes a low complexity method to improve the performance of ldpc codes by selectively removing some cycles from the associated bipartite graph the method is based on a modified version of the breadth first search bfs algorithm that we call modified bfs mbfs which is applied to find cycles and a greedy procedure to eliminate them throughout the paper we will give a detailed description of the algorithm proposed and analytically study its complexity simulation results show that this girth conditioning method applied to some classes of codes whose structure allows further optimization can lead to a significative complexity reduction and a performance improvements with respect to other methods

numerical integrators based on the magnus expansion for nonlinear dynamical systems abstract explicit numerical integration algorithms up to order four based on the magnus expansion for nonlinear non autonomous ordinary differential equations are presented and tested on problems possessing qualitative very often geometric features that is convenient to preserve under numerical discretization the range of applications covers augmented dynamical systems highly oscillatory problems and nonlinear non autonomous partial differential equations of evolution previously discretized in space

quasicyclic low density parity check codes from circulant permutation matrices in this correspondence the construction of low density parity check ldpc codes from circulant permutation matrices is investigated it is shown that such codes cannot have a tanner graph representation with girth larger than 12 and a relatively mild necessary and sufficient condition for the code to have a girth of 6 8 10 or 12 is derived these results suggest that families of ldpc codes with such girth values are relatively easy to obtain and consequently additional parameters such as the minimum distance or the number of redundant check sums should be considered to this end a necessary condition for the codes investigated to reach their maximum possible minimum hamming distance is proposed

searching for binary and nonbinary block and convolutional ldpc codes a unified approach to search for and optimize codes determined by their sparse parity check matrices is presented replacing the nonzero elements of a binary parity check matrix the base or parent matrix either by circulants or by companion matrices of elements from a finite field gf 2 m we obtain quasi cyclic low density parity check ldpc block codes and binary images of nonbinary ldpc block codes respectively by substituting monomials of a formal variable d we obtain the polynomial description of an ldpc convolutional code a set of performance measures applicable to different classes of ldpc codes is considered and a greedy algorithm for code performance optimization is presented the heart of the new optimization algorithm is a fast procedure for searching for ldpc codes with large girth of their tanner graphs for a few classes of ldpc codes examples of codes combining good error correcting performance with compact representation are obtained in particular we present optimized convolutional ldpc codes and conclude that the ldpc block codes are still superior to their convolutional counterparts if both decoding complexity and coding delay are considered moreover a specific channel model can easily be embedded into the optimization loop thereby the code can be optimized for a specific channel the efficiency of such an optimization is demonstrated via an example of faster than nyquist ftn signaling using ldpc codes the ftn strategy combined with a rate r 1 2 ldpc code of length 64 800 optimized for effective data rate r 3 4 gains more than 0 5 db compared with the standard ldpc codes of the same rate and length the obtained gain corresponds to transmission at the capacity of the binary input additive white gaussian noise channel in most numerical examples we consider codes with bidiagonal structure of the parity check matrix this restriction preserves low encoding complexity and allows fair comparison with codes selected for communication standards

additive structural and multiplicative transformations for the construction of quasi cyclic ldpc matrices the construction of a quasi cyclic low density parity check qc ldpc matrix is usually carried out in two steps in the first step a prototype matrix is defined according to certain criteria size girth check and variable node degrees and so on the second step involves the expansion of the prototype matrix during this last phase an integer value is assigned to each non null position in the prototype matrix corresponding to the right rotation of the identity matrix the problem of determining these integer values is complex the state of the art solutions use either some mathematical constructions to guarantee a given girth of the final qc ldpc code or a random search of values until the target girth is satisfied in this paper we propose an alternative complementary method that reduces the search space by defining large equivalence classes of topologically identical matrices through row and column permutations using additive structural and multiplicative transformations selecting only a single element per equivalence class can reduce the search space by a few orders of magnitude then we use the formalism of constraint programming to list the exhaustive sets of solutions for a given girth and a given expansion factor an example is presented in all sections of the paper to illustrate the methodology

on the encoding complexity of quasi cyclic ldpc codes in this paper we propose a parity check matrix pcm construction technique that reduces the encoding complexity of quasi cyclic qc ldpc codes the proposed construction method is based on a constraint selection of shifting factors shown here to reduce the density of an inverted matrix used in several encoding algorithms furthermore it demonstrates that the complexity of encoding schemes involving inverted matrices can be defined by the density of the small inverted binary base matrix and not by the extended qc pcm comparisons of the proposed codes with codes employed in international standards and with randomly shifted qc ldpc codes of comparable characteristics show the low complexity of the corresponding hardware implementations and a ber performance equivalent to that of previously reported codes without increasing the decoding complexity furthermore adoption of the proposed method can decrease the complexity of several encoding procedures in particular an area reduction of 40 55 is reported for qc ldpc encoders while a reduction of 86 is reported for multi level qc ldpc encoders

quasi cyclic ldpc codes influence of proto and tanner graph structure on minimum hamming distance upper bounds quasi cyclic qc low density parity check ldpc codes are an important instance of proto graph based ldpc codes in this paper we present upper bounds on the minimum hamming distance of qc ldpc codes and study how these upper bounds depend on graph structure parameters like variable degrees check node degrees girth of the tanner graph and of the underlying proto graph moreover for several classes of proto graphs we present explicit qc ldpc code constructions that achieve or come close to the respective minimum hamming distance upper bounds because of the tight algebraic connection between qc codes and convolutional codes we can state similar results for the free hamming distance of convolutional codes in fact some qc code statements are established by first proving the corresponding convolutional code statements and then using a result by tanner that says that the minimum hamming distance of a qc code is upper bounded by the free hamming distance of the convolutional code that is obtained by unwrapping the qc code

design of multiple edge protographs for qc ldpc codes avoiding short inevitable cycles there have been lots of efforts on the construction of quasi cyclic qc low density parity check ldpc codes with large girth however most of them focus on protographs with single edges and little research has been done for the construction of qc ldpc codes lifted from protographs with multiple i e parallel edges compared to single edge protographs multiple edge protographs have benefits such that qc ldpc codes lifted from them can potentially have larger minimum hamming distance in this paper all subgraph patterns of multiple edge protographs which prevent qc ldpc codes from having large girth by inducing inevitable cycles are fully investigated based on a graph theoretic approach by using combinatorial designs a systematic construction method of multiple edge protographs is proposed for regular qc ldpc codes with girth at least 12 and another method is proposed for regular qc ldpc codes with girth at least 14 moreover a construction algorithm of qc ldpc codes based on certain liftings of multiple edge protographs is proposed and it is shown that the resulting qc ldpc codes have larger upper bounds on the minimum hamming distance than those lifted from single edge protographs simulation results are provided to compare the performance of the proposed qc ldpc codes with progressive edge growth peg ldpc codes and with peg qc ldpc codes

fitting long memory models by generalized linear regression summary there is an increasing awareness of the importance of long memory models in statistical applications if long memory is present it has to be taken into account in order to obtain reliable tests and confidence intervals one obstacle to using models with long memory in routine statistical analysis has been the lack of easily available and sufficiently versatile statistical software here we propose a simple but flexible class of parametric models which can be used to model such behaviour we demonstrate that these models can be fitted by generalized linear regression standard statistical software packages can be used a data example is discussed

minimum bias multiple taper spectral estimation two families of orthonormal tapers are proposed for multitaper spectral analysis minimum bias tapers and sinusoidal tapers spl upsisup k where spl upsisub nsup k spl radic 2 n 1 sin spl pi kn n 1 and n is the number of points the resulting sinusoidal multitaper spectral estimate is s spl circ f 1 2k n 1 spl sigmasub j 1sup k y f j 2n 2 y f j 2n 2 sup 2 where y f is the fourier transform of the stationary time series s f is the spectral density and k is the number of tapers for fixed j the sinusoidal tapers converge to the minimum bias tapers like 1 n since the sinusoidal tapers have analytic expressions no numerical eigenvalue decomposition is necessary both the minimum bias and sinusoidal tapers have no additional parameter for the spectral bandwidth the bandwidth of the jth taper is simply 1 n centered about the frequencies spl plusmn j 2n 2 thus the bandwidth of the multitaper spectral estimate can be adjusted locally by simply adding or deleting tapers the band limited spectral concentration spl intsub wsup w v f sup 2 df of both the minimum bias and sinusoidal tapers is very close to the optimal concentration achieved by the slepian 1978 tapers in contrast the slepian tapers can have the local bias spl intsub 1 2 sup 1 2 f sup 2 v f sup 2 df much larger than of the minimum bias tapers and the sinusoidal tapers

on the minimum distance of array codes as ldpc codes for a prime q and an integer j spl les q the code c q j is a class of low density parity check ldpc codes from array codes which has a nice algebraic structure in this correspondence we investigate the minimum distance d q j of the code in an algebraic way we first prove that the code is invariant under a doubly transitive group of affine permutations then we show that d 5 4 8 d 7 4 8 and d q 4 spl ges 10 for any prime q 7 in addition we also analyze the codewords of weight 6 in the case of j 3 and the codewords of weight 8 in c 5 4 and c 7 4

the variance of multitaper spectrum estimates for real gaussian processes multitaper spectral estimation has proven very powerful as a spectral analysis method wherever the spectrum of interest is detailed and or varies rapidly with a large dynamic range in his original paper d j thomson 1982 gave a simple approximation for the variance of a multitaper spectral estimate which is generally adequate when the spectrum is slowly varying over the taper bandwidth the authors show that near zero or nyquist frequency this approximation is poor even for white noise and derive the exact expression of the variance in the general case of a stationary real valued time series this expression is illustrated on an autoregressive time series and a convenient computational approach outlined it is shown that this multitaper variance expression for real valued processes is not derivable as a special case of the multitaper variance for complex valued circularly symmetric processes as previously suggested in the literature