the discrete wavelet transform for a symmetric anti symmetric multiwavelet family on the interval the chui lian multiwavelet family of approximation order three is extended to serve as a basis for an orthogonal discrete multiwavelet transform on the interval this requires in addition to the original symmetric and antisymmetric pairs of scaling functions and wavelets four new functions of each type at each edge of the interval i e one more than required just to preserve the approximation order the criteria are given that such edge functions must satisfy in order to allow multiresolution analysis of data sequences without special constraints on endpoint behavior or use of periodic boundary conditions and a specific choice is examined in applications to both noiseless and noisy data these are accomplished through the development of interval basis extensions for both an earlier multiwavelet projection based prefilter postfilter combination and a multiwavelet vector thresholding algorithm

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

high rate girth eight low density parity check codes on rectangular integer lattices this letter introduces a combinatorial construction of girth eight high rate low density parity check codes based on integer lattices the parity check matrix of a code is defined as a point line incidence matrix of a 1 configuration based on a rectangular integer lattice and the girth eight property is achieved by a judicious selection of sets of parallel lines included in a configuration a class of codes with a wide range of lengths and column weights is obtained the resulting matrix of parity checks is an array of circulant matrices

the application of multiwavelet filterbanks to image processing multiwavelets are a new addition to the body of wavelet theory realizable as matrix valued filterbanks leading to wavelet bases multiwavelets offer simultaneous orthogonality symmetry and short support which is not possible with scalar two channel wavelet systems after reviewing this theory we examine the use of multiwavelets in a filterbank setting for discrete time signal and image processing multiwavelets differ from scalar wavelet systems in requiring two or more input streams to the multiwavelet filterbank we describe two methods repeated row and approximation deapproximation for obtaining such a vector input stream from a one dimensional 1 d signal algorithms for symmetric extension of signals at boundaries are then developed and naturally integrated with approximation based preprocessing we describe an additional algorithm for multiwavelet processing of two dimensional 2 d signals two rows at a time and develop a new family of multiwavelets the constrained pairs that is well suited to this approach this suite of novel techniques is then applied to two basic signal processing problems denoising via wavelet shrinkage and data compression after developing the approach via model problems in one dimension we apply multiwavelet processing to images frequently obtaining performance superior to the comparable scalar wavelet transform

results on the propositional μ calculus abstract in this paper we define and study a propositional calculus l which consists essentially of propositional modal logic with a least fixpoint operator l is syntactically simpler yet strictly more expressive than propositional dynamic logic pdl for a restricted version we give an exponential time decision procedure small model property and complete deductive system theory subsuming the corresponding results for pdl

multiwavelet moments and projection prefilters an efficient projection procedure is derived for use of orthogonal multiwavelets in the analysis of discrete data sequences a family of simple prefilters corresponding to numerical quadrature evaluation of the projection integrals provides exact results for locally polynomial data the full approximation order of the multiwavelet basis can thus always be enabled for nonpolynomial signals the prefilters provide approximations to the coefficients of the multiwavelet series whose convergence accelerates quickly with increase in sampling rate comparison is also made with previous time invariant multiwavelet prefilters

bounds on the size of parity check matrices for quasi cyclic low density parity check codes in this paper we investigate the cycle properties of quasi cyclic low density parity check qc ldpc codes using the sequence representation of a parity check matrix for a qc ldpc code we analyze a necessary and sufficient condition for a cycle of a given length to exist we then derive bounds which are necessary conditions for a qc ldpc code to have a given girth in terms of its parameters we also give a bound which is a sufficient condition for a qc ldpc code of a given girth to be constructed by a greedy algorithm the bounds derived here are applicable to any regular or irregular qc ldpc codes as well as they improve the existing bounds in many classes of regular ldpc codes

enhanced frequency analysis using wavelets an enhanced frequency analysis method is introduced using a phase shift wavelet representation to analyse individual frequency components and to reveal features of a given signal the frequency representation provides details at each time location without the loss caused by local orthogonality between the wavelet basis functions and the given signal which occurs when a normal wavelet decomposition is employed movement of the local orthogonality locations corresponding to the phase shift of the wavelet basis functions are studied and it is shown that these movements are important and can be used to reveal features of the signal in the frequency domain such as continuities smoothness and trends several examples are presented to illustrate the method and an application to a permanent magnet synchronous drive system is provided to demonstrate the approach

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

reed solomon based quasi cyclic ldpc codes designs girth cycle structure and reduction of short cycles designs and constructions of quasi cyclic qc ldpc codes for the awgn channel are presented the codes are constructed based on the conventional parity check matrices of reed solomon rs codes and are referred to as rs qc ldpc codes several classes of rs qc ldpc codes are given cycle structural properties of the tanner graphs of codes in these classes are analyzed and specific methods for constructing codes with girth at least eight and reducing their short cycles are presented the designed codes perform well in both waterfall and low error rate regions

the µ calculus alternation hierarchy collapses over structures with restricted connectivity the alternation hierarchy of least and greatest fixpoint operators in the calculus is strict however the strictness of the hierarchy does not necessarily carry over when considering restricted classes of structures for instance over the class of infinite words the alternation free fragment of the calculus is already as expressive as the full logic our current understanding of when and why the calculus alternation hierarchy is and is not strict is limited this article makes progress in answering these questions by showing that the alternation hierarchy of the calculus collapses to the alternation free fragment over some classes of structures including infinite nested words and finite graphs with feedback vertex sets of a bounded size common to these classes is that the connectivity between the components in a structure from such a class is restricted in the sense that the removal of certain vertices from the structure s graph decomposes it into graphs in which all paths are of finite length the collapse results herein are obtained in an automata theoretic setting they subsume generalize and strengthen several prior results on the expressivity of the calculus over restricted classes of structures

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

balanced multiwavelet bases based on symmetric fir filters this paper describes a basic difference between multiwavelets and scalar wavelets that explains without using zero moment properties why certain complications arise in the implementation of discrete multiwavelet transforms assuming we wish to avoid the use of prefilters in implementing the discrete multiwavelet transform it is suggested that the behavior of the iterated filter bank associated with a multiwavelet basis of multiplicity r is more fully revealed by an expanded set of r sup 2 scaling functions spl phi sub i j this paper also introduces new k balanced orthogonal multiwavelet bases based on symmetric fir filters the nonlinear design equations arising in this work are solved using the grobner basis the minimal length k balanced multiwavelet bases based on even length symmetric fir filters are better behaved than those based on odd length symmetric fir filters as illustrated by special relations they satisfy and by examples constructed

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

the μ calculus alternation depth hierarchy is infinite over finite planar graphs abstract we prove that the alternation hierarchy of the modal calculus is infinite over finite planar graphs

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

adaptive multiwavelet initialization the multiwavelet concept uses a set of scaling functions and a set of wavelets to generate an orthogonal multichannel multiresolution pyramid decomposition for finite energy signals the multiple scaling functions extract the lowpass information of the input data set and the multiwavelets extract the bandpass information when the lowpass filters associated with the scaling functions are cascaded a multiresolution pyramid decomposition reconstruction system is formed with each pyramid convolution operator having several inputs and several outputs however there is only one data set available to initialize this process this paper addresses the question of how to best modify the data set using prefilters so that its decomposition contains the most useful information for the chosen application the best prefilters are determined by the minimization of the energy of preselected decomposition components the resulting decomposition is therefore signal adaptive and under appropriate conditions perfect reconstruction of the input data set can be achieved with a proper postfiltering the detailed discussions in the paper are focussed on the two wavelet and two scaling function case the general multiwavelet case is addressed at the end of the paper a compression example is provided to demonstrate the performance of the optimally initialized multiwavelet method and to compare it with a single wavelet method and another multiwavelet initialization method proposed by other authors

design of prefilters for discrete multiwavelet transforms the pyramid algorithm for computing single wavelet transform coefficients is well known the pyramid algorithm can be implemented by using tree structured multirate filter banks the authors propose a general algorithm to compute multiwavelet transform coefficients by adding proper premultirate filter banks before the vector filter banks that generate multiwavelets the proposed algorithm can be thought of as a discrete vector valued wavelet transform for certain discrete time vector valued signals the proposed algorithm can be also thought of as a discrete multiwavelet transform for discrete time signals the authors then present some numerical experiments to illustrate the performance of the algorithm which indicates that the energy compaction for discrete multiwavelet transforms may be better than the one for conventional discrete wavelet transforms

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

quasi cyclic low density parity check codes with girth larger than 12 a quasi cyclic qc low density parity check ldpc code can be viewed as the protograph code with circulant permutation matrices or circulants in this correspondence we find all the subgraph patterns of protographs of qc ldpc codes having inevitable cycles of length 2i i 6 7 8 9 10 i e the cycles that always exist regardless of the shift values of circulants it is also derived that if the girth of the protograph is 2g g 2 its protograph code cannot have the inevitable cycles of length smaller than 6g based on these subgraph patterns we propose new combinatorial construction methods of the protographs whose protograph codes can have girth larger than or equal to 14 or 18 we also propose a couple of shift value assigning rules for circulants of a qc ldpc code guaranteeing the girth 14

on the μ calculus over transitive and finite transitive frames we prove that the modal m calculus collapses to first order logic over the class of finite transitive frames the proof is obtained by using some byproducts of a new proof of the collapse of the m calculus to the alternation free fragment over the class of transitive frames moreover we prove that the modal m calculus is buchi and co buchi definable over the class of all models where in a strongly connected component vertexes are distinguishable by means of the propositions they satisfy

solution of cartesian and curvilinear quantum equations via multiwavelets on the interval abstract it is shown how orthogonal compact support multiwavelets may be used for the solution of quantum mechanical eigenvalue problems subject to specific boundary conditions special scaling functions and wavelets with convenient limiting behaviors at the edges of an interval are constructed in analogy to earlier work on single wavelet families all of the integrals required for hamiltonian matrix elements involving both regular and edge functions are calculated efficiently through use of recursion and quadrature methods it is demonstrated through accurate eigenvalue determination that both cartesian and curvilinear degrees of freedom are readily accommodated with such a basis using as examples the particle in a box and the hydrogen atom in spherical polar coordinates

characterization and efficient search of non elementary trapping sets of ldpc codes with applications to stopping sets in this paper we propose a characterization for non elementary trapping sets netss of low density parity check ldpc codes the characterization is based on viewing an nets as a hierarchy of embedded graphs starting from an ets the characterization corresponds to an efficient search algorithm that under certain conditions is exhaustive as an application of the proposed characterization search we obtain lower and upper bounds on the stopping distance s min of ldpc codes we examine a large number of regular and irregular ldpc codes and demonstrate the efficiency and versatility of our technique in finding lower and upper bounds on and in many cases the exact value of s min finding s min or establishing search based lower or upper bounds for many of the examined codes are out of the reach of any existing algorithm for a constant degree distribution and range of search the worst case computational complexity of the proposed search algorithms for finding netss and stopping sets is linear in the code s block length n the average search complexity for stopping sets however is constant in n if the simple cycles required as input to the search algorithm are available

short wavelets and matrix dilation equations scaling functions and orthogonal wavelets are created from the coefficients of a lowpass and highpass filter in a two band orthogonal filter bank for multifilters those coefficients are matrices this gives a new block structure for the filter bank and leads to multiple scaling functions and wavelets geronimo hardin and massopust see j approx theory vol 78 p 373 401 1994 constructed two scaling functions that have extra properties not previously achieved the functions spl phi sub 1 and spl phi sub 2 are symmetric linear phase and they have short support two intervals or less while their translates form an orthogonal family for any single function spl phi apart from haar s piecewise constants those extra properties are known to be impossible the novelty is to introduce 2 spl times 2 matrix coefficients while retaining orthogonality of the multiwavelets this note derives the properties of spl phi sub 1 and spl phi sub 2 from the matrix dilation equation that they satisfy then our main step is to construct associated wavelets two wavelets for two scaling functions the properties were derived by geronimo hardin and massopust from the iterated interpolation that led to spl phi 1 and spl phi sub 2 one pair of wavelets was found earlier by direct solution of the orthogonality conditions using mathematica our construction is in parallel with recent progress by hardin and geronimo to develop the underlying algebra from the matrix coefficients in the dilation equation in another language to build the 4 spl times 4 paraunitary polyphase matrix in the filter bank the short support opens new possibilities for applications of filters and wavelets near boundaries

a unified approach to the construction of binary and nonbinary quasi cyclic ldpc codes based on finite fields a unified approach for constructing binary and nonbinary quasi cyclic ldpc codes under a single framework is presented six classes of binary and nonbinary quasi cyclic ldpc codes are constructed based on primitive elements additive subgroups and cyclic subgroups of finite fields numerical results show that the codes constructed perform well over the awgn channel with iterative decoding

a new prefilter design for discrete multiwavelet transforms in conventional wavelet transforms prefiltering is not necessary due to the lowpass property of a scaling function this is no longer true for multiwavelet transforms a few research papers on the design of prefilters have appeared but the existing prefilters are usually not orthogonal which often causes problems in coding moreover the condition on the prefilters was imposed based on the first step discrete multiwavelet decomposition we propose a new prefilter design that combines the ideas of the conventional wavelet transforms and multiwavelet transforms the prefilters are orthogonal but nonmaximally decimated they are derived from a very natural calculation of multiwavelet transform coefficients in this new prefilter design multiple step discrete multiwavelet decomposition is taken into account our numerical examples by taking care of the redundant prefiltering indicate that the energy compaction ratio with the geronimo hardin massopust 1994 2 wavelet transform and our new prefiltering is better than the one with daubechies d sub 4 wavelet transform

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed