wavelets and filter banks theory and design the wavelet transform is compared with the more classical short time fourier transform approach to signal analysis then the relations between wavelets filter banks and multiresolution signal processing are explored a brief review is given of perfect reconstruction filter banks which can be used both for computing the discrete wavelet transform and for deriving continuous wavelet bases provided that the filters meet a constraint known as regularity given a low pass filter necessary and sufficient conditions for the existence of a complementary high pass filter that will permit perfect reconstruction are derived the perfect reconstruction condition is posed as a bezout identity and it is shown how it is possible to find all higher degree complementary filters based on an analogy with the theory of diophantine equations an alternative approach based on the theory of continued fractions is also given these results are used to design highly regular filter banks which generate biorthogonal continuous wavelet bases with symmetries

multi d wavelet filter bank design using quillen suslin theorem for laurent polynomials in this paper we present a new approach for constructing the wavelet filter bank our approach enables constructing nonseparable multidimensional non redundant wavelet filter banks with fir filters using the quillen suslin theorem for laurent polynomials our construction method presents some advantages over the traditional methods of multidimensional wavelet filter bank design first it works for any spatial dimension and for any sampling matrix second it does not require the initial lowpass filters to satisfy any additional assumption such as interpolatory condition third it provides an algorithm for constructing a wavelet filter bank from a single lowpass filter so that its vanishing moments are at least as many as the accuracy number of the lowpass filter

on the realizability of biorthogonal m dimensional two band filter banks we show an algebraic approach for the design of ladder structures for causal biorthogonal filter banks the key ingredient of the approach is known in literature as euclid s algorithm using this algorithm we derive some strong result on the design freedom for ladder structures in particular we show that the dimensionality of the problem plays an important role we end by with some conjectures concerning the extensions to multichannel and noncausal filter banks

on factorization of m channel paraunitary filterbanks we systematically investigate the factorization of causal finite impulse response fir paraunitary filterbanks with given filter length based on the singular value decomposition of the coefficient matrices of the polyphase representation a fundamental order one factorization form is first proposed for general paraunitary systems then we develop a new structure for the design and implementation of paraunitary system based on the decomposition of hermitian unitary matrices within this framework the linear phase filterbank and pairwise mirror image symmetry filterbank are revisited their structures are special cases of the proposed general structures compared with the existing structures more efficient ones that only use approximately half the number of free parameters are derived the proposed structures are complete and minimal although the factorization theory with or without constraints is discussed in the framework of m channel filterbanks the results can be applied to wavelets and multiwavelet systems and could serve as a general theory for paraunitary systems

invertible deinterlacing with sampling density preservation theory and design a novel class of deinterlacing for intra frame field based motion picture coding such as motion jpeg2000 as well as an inter frame based coding without the support of interlaced scanning video such as mpeg 1 is developed this technique has two features sampling density preservation and invertibility these features mean that the amount of deinterlaced pictures is not increased and the original pictures can be perfectly reconstructed this deinterlacing technique is a kind of sampling lattice alteration and is regarded as a generalization of conventional field interleaving and field separation with the help of multidimensional m d multirate theory it is shown that the design problem of such a system that is invertible deinterlacer with sampling density preservation can be replaced to finding a 2 spl times 2 multivariable polynomial matrix with a monomial determinant this problem resembles the design of two channel m d maximally decimated perfect reconstruction finite impulse response fir filterbanks the inverse system which is referred to as a reinterlacer is given as fir when the dual deinterlacing system is fir a practical design procedure is provided by suggesting three constraints considered to be preferable normalization regularity and vertical symmetry the significance of the procedure is verified by showing some design examples of deinterlacing and reinterlacing filters simulation results show that the developed method causes fewer comb shaped artifacts than conventional field interleaving

stable tameness of two dimensional polynomial automorphisms over a regular ring abstract in this paper it is established that all two dimensional polynomial automorphisms over a regular ring r are stably tame this results from the main theorem of this paper which asserts that an automorphism in any dimension n is stably tame if said condition holds point wise over spec r a key element in the proof is a theorem which yields the following corollary over an artinian ring a all two dimensional polynomial automorphisms having jacobian determinant one are stably tame and are tame if a is a q algebra another crucial ingredient of interest in itself is that stable tameness is a local property if an automorphism is locally tame then it is stably tame

perfect reconstruction fir filter banks some properties and factorizations perfect reconstruction finite impulse response fir filter banks are analyzed both in the z transform and time domains showing the alternatives between designs in the two domains various classes of perfect reconstruction schemes are indicated and relations between previously known systems are given windowed modulated filter banks with low computational complexity and perfect reconstruction are shown new factorizations of polyphase filter matrices leading in particular to linear phase filters are given the computational complexity and the architecture of the new structures are indicated

the role of linear semi infinite programming in signal adapted qmf bank design we consider the problem of designing a perfect reconstruction fir quadrature mirror filter qmf bank h g adapted to input signal statistics with the coding gain as the adaptation criterion maximization of the coding gain has so far been viewed as a difficult nonlinear constrained optimization problem it is shown that the coding gain depends only on the product filter p z h z h z sup 1 and this transformation leads to a stable class of linear optimization problems having finitely many variables and infinitely many constraints termed linear semi infinite programming sip problems the sought for original filter h z is obtained by deflation and spectral factorization of p z with the sip formulation every locally optimal solution is also globally optimal and can be computed using reliable numerical algorithms the natural regularity properties inherent in the sip formulation enhance the performance of these algorithms we present a comprehensive theoretical analysis of the sip problem and its dual characterize the optimal filters and analyze uniqueness and sensitivity issues all these properties are intimately related to those of the input signal and bring considerable insight into the nature of the adaptation process we present discretization and cutting plane algorithms and apply both methods to several examples

nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for r sup n new results on multidimensional filter banks and their connection to multidimensional nonseparable wavelets are presented among the topics discussed are sampling in multiple dimensions multidimensional perfect reconstruction filter banks the two channel case in multiple dimensions the synthesis of multidimensional filter banks and the design of compactly supported wavelets

rational coefficient dual tree complex wavelet transform design and implementation the dual tree complex wavelet transform cwt has recently received significant interest in the wavelet community owing primarily to its directional selective and near shift invariant properties it has been shown that with two separate maximally decimated and dyadic decompositions where filters are offset by a half sample the resulting cwt wavelet bases form an approximate hilbert transform pair in this paper we present the design implementation and applications of several families of orthogonal as well as biorthogonal rational coefficient wavelet filters that satisfy the hilbert transform pair condition and meet other desirable properties such as high coding gain good directional sensitivity and sufficient degree of regularity the wavelet filters presented here which confirm to selesnick s and kingsbury s design schemes are designed and implemented directly in the lattice and lifting domain using vlsi friendly dyadic coefficients we confirm the fact that rational coefficient constraint does not impose a significant loss in terms of energy compaction wavelet smoothness time invariance or directionality we also propose the time reversal relationships between the two cwt filter pairs in lattice and lifting domain thereby facilitating both the design and implementation process in the end we present several applications and evaluations to illustrate the performance of the proposed designs

the strong anick conjecture recently umirbaev proved the long standing anick conjecture that is there exist wild automorphisms of the free associative algebra kx y z over a field k of characteristic 0 in particular the well known anick automorphism is wild in this work we obtain a stronger result the strong anick conjecture that implies the anick conjecture namely we prove that there exist wild coordinates of kx y z in particular the two nontrivial coordinates in the anick automorphism are both wild we establish a similar result for several large classes of automorphisms of kx y z we also discover a large previously undescribed class of wild automorphisms of kx y z that is not covered by the results of umirbaev

on the optimal choice of a wavelet for signal representation two techniques for finding the discrete orthogonal wavelet of support less than or equal to some given integer that leads to the best approximation to a given finite support signal up to a desired scale are presented the techniques are based on optimizing certain cost functions the first technique consists of minimizing an upper bound that is derived on the l sub 2 norm of error in approximating the signal up to the desired scale it is shown that a solution to the problem of minimizing that bound does exist and it is explained how the constrained minimization over the parameters that define discrete finite support orthogonal wavelets can be turned into an unconstrained one the second technique is based on maximizing an approximation to the norm of the projection of the signal on the space spanned by translates and dilates of the analyzing discrete orthogonal wavelet up to the desired scale both techniques can be implemented much faster than the optimization of the l sub 2 norm of either the approximation to the given signal up to the desired scale or that of the error in that approximation

general analysis of two band qmf banks the two channel perfect reconstruction quadrature mirror filter banks pr qmf banks are analyzed in detail by assuming arbitrary analysis and synthesis filters solutions where the filters are fir or iir correspond to the fact that a certain function is monomial or nonmonomial respectively for the monomial case the design problem is formulated as a nonlinear constrained optimization problem the formulation is quite robust and is able to design various two channel filter banks such as orthogonal and biorthogonal arbitrary delay linear phase filter banks to name a few same formulation is used for causal and stable pr iir filter bank solutions

from frame like wavelets to wavelet frames keeping approximation properties and symmetry abstract for a given symmetric refinable mask obeying the sum rule of order n an explicit method is suggested for the construction of mutually symmetric almost frame like wavelet system providing approximation order n a transformation based on the lifting scheme is described that allows to improve almost frame like wavelets up to dual wavelet frames and preserve other properties a direct method for the construction of dual wavelet frames providing approximation order n and mutual symmetry properties is also discussed for an abelian symmetry group h a technique providing the h symmetry property for each wavelet function is given for the above three methods

filters for distortion free two band multirate filter banks in this paper conditions are given for a two band multi rate filter bank to be alias free and to have a unity frequency response it is shown that the class of quadrature mirror filters qmf s that satisfies these conditions is quite limited a class of filters which does satisfy these conditions is given and a simple procedure for designing filters from this class is presented with an example

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

an algorithmic proof of suslin s stability theorem for polynomial rings abstract let k be a field then gaussian elimination over k and the euclidean division algorithm for the univariate polynomial ring k x allow us to write any matrix in sln k or sln k x n 2 as a product of elementary matrices suslin s stability theorem states that the same is true for sln k xl xm with n 3 and m 1 in this paper we present an algorithmic proof of suslin s stability theorem thus providing a method for finding an explicit factorization of a given polynomial matrix into elementary matrices grobner basis techniques may be used in the implementation of the algorithm

sampled data consensus for multi agent systems with quantised communication this paper studies the sampled data consensus problem of first order and second order multi agent systems mass with logarithmic quantised information transmission both undirected and directed communication topologies are considered under the undirected connected communication topology we adopt for each agent the standard consensus protocols based on the logarithmic quantised relative states between neighbouring agents under the directed communication topology so as to preserve the weighted average of agents states we utilise the encoding decoding strategy to transmit quantised information and estimate neighbouring agents states and then distributed control laws based on the state estimates are employed for each agent under both kinds of communication topologies we prove that for any given sampling period by designing appropriate control gains first order mass can reach consensus for any given quantiser accuracy while second order mass can reach consensus if the quantiser accuracy is smal

on coefficient quantization and computational roundoff effects in lossless multirate filter banks it is shown that fir lossless transfer matrices in multirate analysis synthesis systems having the perfect reconstruction property can be implemented such that the lossless property and hence perfect reconstruction can be retained regardless of coefficient quantization such a result was previously shown to be true only for two channel filter banks a noise gain result is presented for lossless systems which finds application in analyzing roundoff noise in multirate filter banks

near perfect reconstruction pseudo qmf banks a novel approach to the design of m channel pseudo quadrature mirror filter qmf banks is presented in this approach the prototype filter is constrained to be a linear phase spectral factor of a 2mth band filter as a result the overall transfer function of the analysis synthesis system is a delay moreover the aliasing cancellation ac constraint is derived such that all the significant aliasing terms are canceled consequently the aliasing level at the output is comparable to the stopband attenuation of the prototype filter in other words the only error at the output of the analysis synthesis system is the aliasing error which is at the level of stopband attenuation using this approach it is possible to design a pseudo qmf bank where the stopband attenuation of the analysis and thus synthesis filters is on the order of spl minus 100 db moreover the resulting reconstruction error is also on the order of spl minus 100 db several examples are included

lattice structure for regular paraunitary linear phase filterbanks and m band orthogonal symmetric wavelets parauninary linear phase pulp m channel uniform filterbanks which are also known as the generalized lapped orthogonal transforms genlots can be designed and implemented using lattice structures this paper discusses how to impose regularity constraints onto the lattice structure of pulp filterbanks these conditions are expressed in term of the rotation angles of the lattice components by which the resulting filterbanks are guaranteed to have one or two degrees of regularity iterating these new regular filterbanks on the lowpass subband generates a large family of symmetric m band orthonormal wavelets design procedures with many design examples are presented smooth interpolation using regular pulp filterbanks is illustrated through image coding experiments where the novel m band wavelets consistently yield smoother reconstructed images and better perceptual quality

algorithms for the quillen suslin theorem abstract we give an algorithm for computing a free basis of a projective c x1 xn module which is presented as the image kernel or cokernel of a polynomial matrix our method can be implemented using grobner bases and it provides an elementary constructive proof of the quillen suslin theorem

applications of grobner bases to signal and image processing a survey this paper is a tutorial and survey paper on the grobner bases method and some of its applications in signal and image processing although all the results presented in the paper are available in the literature we give an elementary treatment here in the hope that it will further bring awareness of and stimulate interest in grobner bases among researchers in signal and image processing we give first a tutorial on grobner bases and then a survey of the design of multidimensional wavelets and filter banks applications of grobner bases to other areas of signal and image processing are also briefly reviewed

maximally decimated perfect reconstruction fir filter banks with pairwise mirror image analysis and synthesis frequency responses structures are presented for the perfect reconstruction quadrature mirror filter bank that are based on lossless building blocks these structures ensure that the frequency responses of the analysis and synthesis filters have pairwise symmetry with respect to pi 2 and require fewer parameters than recently reported structures also based on lossless building blocks the design time for the proposed structures is correspondingly much less than for the earlier methods which did not incorporate such symmetry

on the admissibility and consistency of tests for homogeneity of variances 1 introduction and summary the problem of testing homogeneity of variances in a one way fixed analysis of variance model has been studied for the past 50 years if we let s2 denote the sample variance for the ith population i 1 2 p then the s2 s are independently distributed and each estimates its corresponding variance a under the assumption of normality of each of the p populations and the further assumption that each s2 is based on n 1 the same number of observations laue 1965 studied a two parameter family of tests t x q which reject the homogeneity hypothesis when r x rq d where

hybrid wavelet support vector classification of waveforms the support vector machine svm represents a new and very promising technique for machine learning tasks involving classification regression or novelty detection improvements of its generalization ability can be achieved by incorporating prior knowledge of the task at hand we propose a new hybrid algorithm consisting of signal adapted wavelet decompositions and hard margin svms for waveform classification the adaptation of the wavelet decompositions is tailored for hard margin sv classifiers with radial basis functions as kernels it allows the optimization of the representation of the data before training the svm and does not suffer from computationally expensive validation techniques we assess the performance of our algorithm against the background of current concerns in medical diagnostics namely the classification of endocardial electrograms and the detection of otoacoustic emissions here the performance of hard margin svms can significantly be improved by our adapted preprocessing step

an algorithm for unimodular completion over laurent polynomial rings abstract we present a new and simple algorithm for completion of unimodular vectors with entries in a multivariate laurent polynomial ring r k x 1 x k over an infinite field k more precisely given n 3 and a unimodular vector v t v 1 v n r n that is such that v 1 v n r the algorithm computes a matrix m in m n r whose determinant is a monomial such that m v t 1 0 0 and thus m 1 is a completion of v to an invertible matrix

an algorithm for unimodular completion over noetherian rings abstract we give an algorithm for the well known result asserting that if r is a polynomial ring in a finite number of variables over a noetherian ring a of krull dimension d then for n max 3 d 2 sl n r acts transitively on um n r for technical reasons we demand that the noetherian ring a has a theory of grobner bases and contains an infinite set e y 1 y 2 such that y i y j a for each i j the most important guiding examples are affine rings k x 1 x m i and localizations of polynomial rings s 1 k x 1 x m with k an infinite field moreover we give an algorithmic proof of suslin s stability theorem over these rings for the purpose to prepare the ground for this algorithmic generalizations of the quillen suslin theorem corresponding to the particular case a is a field we will give in the first section a constructive proof of an important lemma of suslin which is the only nonconstructive step in suslin s second elementary solution of serre s conjecture this lemma says that for a commutative ring a if v 1 x v n x a x where v 1 is monic and n 3 then there exist 1 l e n 1 a x such that res v 1 e 1 1 v 2 v n t res v 1 e 1 l v 2 v n t a thanks to this constructive proof suslin s second proof of serre s conjecture becomes fully constructive

perfect reconstruction filter banks having linear phase fir filters with equiripple response the design of equiripple linear phase analysis and synthesis fir filters of two channel perfect reconstruction pr filter banks is formulated as the minimization of a weighted peak error under both linear inequality arising from the desired responses of the analysis filters and nonlinear equality pr constraints the effectiveness of a proposed method to solve the design problem a modified dual affine scaling variant of karmarkar s 1989 algorithm and an approximation scheme is illustrated through several design examples

automorphisms fixing a variable of k x y z we study automorphisms of the free associative algebra k x y z over a field k such that x y are linear with respect to x y and z z we establish a sufficient and necessary condition for the tameness of these automorphisms in the class of all automorphisms fixing z which gives an algorithm to recognize the wild ones in particular we prove that the well known anick automorphism is wild in this sense this class of automorphisms induces tame automorphisms of the polynomial algebra k x y z for n 2 the automorphisms of k x1 xn z which fix z and are linear in the xis are tame

a generalization of serre s conjecture and some related issues abstract several topics concerned with multivariate polynomial matrices like unimodular matrix completion matrix determinantal or primitive factorization matrix greatest common factor existence and subsequent extraction along with relevant primeness and coprimeness issues are related to a conjecture which may be viewed as a type of generalization of the original serre problem conjecture solved nonconstructively in 1976 and constructively more recently this generalized serre conjecture is proved to be equivalent to several other unsettled conjectures and therfore all these conjectures constitute a complete set in the sense that solution to any one also solves all the remaining

an efficient algorithm to design weighted minimax perfect reconstruction quadrature mirror filter banks an efficient algorithm is presented to design lattice structure two channel perfect reconstruction quadrature mirror filter pr qmf banks we formulate the filter bank design problem as an unconstrained weighted least squares problem with respect to the coefficients of the lattice structure the proposed iterative algorithm optimizes the lattice coefficients and provides flexible control of the filters stopband ripple profiles typically only a few iterations of the algorithm are needed to obtain an optimal solution in the weighted minimax sense we include a set of practical design rules for use with our algorithm these rules allow very good estimation of important filter bank characteristics such as the filter length and the number of signed digits for quantization of the lattice coefficients into canonic signed digit representation to meet a given set of pr qmf bank specifications

signal adapted multiresolution transform for image coding the authors consider the problem of designing multiresolution transforms that are adapted to the given image signal in the sense that they maximize the coding gain at each resolution level a simple alternating optimization algorithm is derived for solving this problem in the framework of the lattice realization of para unitary quadrature mirror filters the resulting large coding scheme is discussed in some detail and its performance is compared with that of the discrete cosine transform jpeg technique and with that of some nonadapted multiresolution transforms

decentralised minimum time consensus we consider the discrete time dynamics of a network of agents that exchange information according to a nearest neighbour protocol under which all agents are guaranteed to reach consensus asymptotically we present a fully decentralised algorithm that allows any agent to compute the final consensus value of the whole network in finite time using the minimum number of successive values of its own state history we show that the minimum number of steps is related to a jordan block decomposition of the network dynamics and present an algorithm to compute the final consensus value in the minimum number of steps by checking a rank condition of a hankel matrix of local observations furthermore we prove that the minimum number of steps is related to graph theoretical notions that can be directly computed from the laplacian matrix of the graph and from the minimum external equitable partition

graph diameter eigenvalues and minimum time consensus we consider the problem of achieving average consensus in the minimum number of linear iterations on a fixed undirected graph we are motivated by the task of deriving lower bounds for consensus protocols and by the so called definitive consensus conjecture which states that for an undirected connected graph g with diameter d there exist d matrices whose nonzero pattern complies with the edges in g and whose product equals the all ones matrix our first result is a counterexample to the definitive consensus conjecture which is the first improvement of the diameter lower bound for linear consensus protocols we then provide some algebraic conditions under which this conjecture holds which we use to establish that all distance regular graphs satisfy the definitive consensus conjecture

filters for anisotropic wavelet decompositions abstract like the continuous shearlet transform and their relatives discrete transformations based on the interplay between several filterbanks with anisotropic dilations provide a high potential to recover directed features in two and more dimensions due to simplicity most of the directional systems constructed so far were using prediction correction methods based on interpolatory subdivision schemes in this paper we give a simple but effective construction for qmf quadrature mirror filter filterbanks which are the usual discrete tools in orthogonal wavelet analysis we characterize when the filterbank gives rise to the existence of refinable functions and hence wavelets and give a generalized shearlet construction for arbitrary dimensions and arbitrary scalings for which the filterbank construction ensures the existence of an orthogonal wavelet analysis we also show that under some restriction on the dilation factors this multiple filterbank system satisfies the slope resolution property which is a key feature in all types of directional transforms

wavelet families of increasing order in arbitrary dimensions we build discrete time compactly supported biorthogonal wavelets and perfect reconstruction filter banks for any lattice in any dimension with any number of primal and dual vanishing moments the associated scaling functions are interpolating our construction relies on the lifting scheme and inherits all of its advantages fast transform in place calculation and integer to integer transforms we show that two lifting steps suffice predict and update the predict step can be built using multivariate polynomial interpolation while update is a multiple of the adjoint of predict while we concentrate on the discrete time case some discussion of convergence and stability issues together with examples is given

theory and design of m channel maximally decimated quadrature mirror filters with arbitrary m having the perfect reconstruction property based on the concept of losslessness in digital filter structures this paper derives a general class of maximally decimated m channel quadrature mirror filter banks that lead to perfect reconstruction the perfect reconstruction property guarantees that the reconstructed signal hat x n is a delayed version of the input signal x n i e hat x n x n n 0 it is shown that such a property can be satisfied if the alias component matrix ac matrix for short is unitary on the unit circle of the z plane the number of channels m is arbitrary and when m is two the results reduce to certain recently reported 2 channel perfect reconstruction qmf structures a procedure based on recently reported fir cascaded lattice structures is presented for optimal design of such fir m channel filter banks design examples are included

an analysis of means type test for variances from normal populations after a brief review of the literature a test for homogeneity of variances is presented the test which is the variances analog of the analysis of means anom can be presented in a graphical form this graphical presentation makes it easy for practitioners to assess both practical and statistical significance tables of critical values are presented for balanced designs a large sample version of the test which uses anom critical values is also presented as well as a version of the test for unbalanced designs for which approximate critical values can be calculated monte carlo results indicate that this test has power comparable to well known normal based tests for homogeneity of variances

new quadrature mirror filter structures this paper introduces new quadrature mirror filter qmf structures for the frequency domain analysis and synthesis of digital signals the conventional qmf technique is first extended to cover complex quadrature mirror filters cqmf in which a digital signal is split into n adjacent complex subbands where the real and imaginary parts are subsampled by 1 2n with respect to the original signal the computational complexity of qmf banks is then analyzed and a new scheme which reduces the computational complexity by about a factor of two over conventional qmf implementations is proposed finally the filter design tradeoffs are discussed and the microprogramed implementation of qmf banks is evaluated

decentralized estimation of laplacian eigenvalues in multi agent systems in this paper we present a decentralized algorithm to estimate the eigenvalues of the laplacian matrix that encodes the network topology of a multi agent system we consider network topologies modeled by undirected graphs the basic idea is to provide a local interaction rule among agents so that their state trajectory is a linear combination of sinusoids oscillating only at frequencies function of the eigenvalues of the laplacian matrix in this way the problem of decentralized estimation of the eigenvalues is mapped into a standard signal processing problem in which the unknowns are the finite number of frequencies at which the signal oscillates

on the statistics of fixed point roundoff error roundoff error after fixed point multiplication is commonly modeled as uniformly distributed white noise that is uncorrelated with the signal this paper presents a statistical analysis of fixed point roundoff error that identifies the conditions under which this model is valid and examines the statistical behavior of roundoff error when these conditions are not satisfied the results show that if the multiplier coefficient is expressed as a n m where m is a positive integral power of two and n is an odd integer then the errors generated by roundoff after multiplication can generally be modeled as uniformly distributed white and uncorrelated with the signal if the signal has sufficiently wide bandwidth and has a dynamic range that extends over approximately m quantum steps for narrow band low level signals the roundoff error statistics can differ significantly from the uniform white uncorrelated model in addition these results show that statistical behavior of roundoff error can differ significantly from that of the quantization error that is generated when a continuous random variable is quantized

new stably tame automorphisms of polynomial algebras abstract let k x y k x1 xn y1 ym be the polynomial algebra in m n variables over a field k of characteristic 0 let be a locally nilpotent derivation of k x y such that yi 0 i 1 m and let act as a k y affine transformation over the free k y module freely generated by x1 xn we prove that the automorphism exp w of k x y is stably tame for every polynomial w from the kernel ker of this result is applied to the automorphisms of the polynomial algebra in five variables introduced recently by drensky and gupta and arising from wild automorphisms of generic matrix algebras we also give an algorithm for constructing new stably tame automorphisms in any number of variables and hence new potential candidates for wild automorphisms

on orthonormal wavelets and paraunitary filter banks the known result that a binary tree structured filter bank with the same paraunitary polyphase matrix on all levels generates an orthonormal basis is generalized to binary trees having different paraunitary matrices on each level a converse result that every orthonormal wavelet basis can be generated by a tree structured filter bank having paraunitary polyphase matrices is then proved the concept of orthonormal bases is extended to generalized nonbinary tree structures and it is seen that a close relationship exists between orthonormality and paraunitariness it is proved that a generalized tree structure with paraunitary polyphase matrices produces an orthonormal basis since not all phases can be generated by tree structured filter banks it is proved that if an orthonormal basis can be generated using a tree structure it can be generated specifically by a paraunitary tree

a new generalized consensus problem and its crt based solution this technical note introduces a new generalized consensus problem where nodes in a network are guaranteed to reach the final consensus on a common vector whose elements are exactly the initial values arbitrarily chosen by nodes we propose a fully decentralized algorithm able to solve the above problem and derive conditions to guarantee that consensus is reached in a finite number of steps in particular we prove that in a network composed by n nodes the proposed consensus problem is solvable in at most 2n steps moreover we introduce a finite field solution based on the chinese remainder theorem crt able to reduce the complexity of the proposed approach in the case of capacity constraints and we discuss an illustrative case study

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

the steinberg group of a monoid ring nilpotence and algorithms for a regular ring r and an affine monoid m the homotheties of m act nilpotently on the milnor unstable groups of r m this strengthens the k2 part of the main result of j gubeladze the nilpotence conjecture in k theory of toric varieties invent math 160 2005 173 216 in two ways the coefficient field of characteristic 0 is extended to any regular ring and the stable k2 group is substituted by the unstable ones the proof is based on a polyhedral combinatorial technique computations in steinberg groups and a substantially corrected version of an old result on elementary matrices by mushkudiani z mushkudiani k2 groups of monoid algebras over regular rings proc a razmadze math inst 113 1995 120 137 a similar stronger nilpotence result for k1 and algorithmic consequences for factorization of high frobenius powers of invertible matrices are also derived

analysis design and implementation of two channel linear phase filter banks a new approach the problem of designing two channel perfect reconstruction fir filter banks with linear phase analysis and synthesis filters is revisited based on a new algebraic formulation all the possible factorized forms for this two band filter bank are derived we thus obtain complete and canonical solutions for the filter banks composed of odd order symmetric and antisymmetric filters type a systems and for those built with symmetric even order filters type b systems a strong characteristic of these new cascade structures which until now had not been identified is related to a defectivity property taking this into account is the key issue to cover all the fir solutions and to design cascade structures being robust to the quantization of their parameters design examples are provided that illustrate our method

structures for m channel perfect reconstruction fir qmf banks which yield linear phase analysis filters the authors develop structures for fir finite impulse response perfect reconstruction qmf quadrature mirror filter banks which cover a subclass of systems that yield linear phase analysis filters for arbitrary m the parameters of these structures can be optimized to design analysis filters with minimum stopband energy which at the same time have linear phase and satisfy the perfect reconstruction property if there are m subbands then depending on whether the coefficients h sub k n of each analysis filter are symmetric or antisymmetric several combinations of filter banks are possible some of these permit perfect reconstruction and some do not for a given m the authors develop a formula for the number of combinations for a subclass of linear phase perfect reconstruction structures as an example they elaborate on a perfect reconstruction linear phase lattice structure for three channels the lattice structure is such that regardless of the parameter values the qmf bank enjoys the perfect reconstruction property while at the same time the analysis filters have linear phase a design example is presented along with tables of impulse response coefficients

one and two level filter bank convolvers in a recent paper it was shown in detail that in the case of orthonormal and biorthogonal filter banks we can convolve two signals by directly convolving the subband signals and combining the results in this paper we further generalize the result we also derive the statistical coding gain for the generalized subband convolver as an application we derive a novel low sensitivity structure for fir filters from the convolution theorem we define and derive a deterministic coding gain of the subband convolver over direct convolution for a fixed wordlength implementation this gain serves as a figure of merit for the low sensitivity structure several numerical examples are included to demonstrate the usefulness of these ideas by using the generalized polyphase representation we show that the subband convolvers linear periodically time varying systems and digital block filtering can be viewed in a unified manner furthermore the scheme called ifir filtering is shown to be a special case of the convolver

lattice structures for optimal design and robust implementation of two channel perfect reconstruction qmf banks a lattice structure and an algorithm are presented for the design of two channel qmf quadrature mirror filter banks satisfying a sufficient condition for perfect reconstruction the structure inherently has the perfect reconstruction property while the algorithm ensures a good stopband attenuation for each of the analysis filters implementations of such lattice structures are robust in the sense that the perfect reconstruction property is preserved in spite of coefficient quantization the lattice structure has the hierarchical property that a higher order perfect reconstruction qmf bank can be obtained from a lower order perfect reconstruction qmf bank simply by adding more lattice sections several numerical examples are provided in the form of design tables

fast multiplierless approximations of the dct with the lifting scheme we present the design implementation and application of several families of fast multiplierless approximations of the discrete cosine transform dct with the lifting scheme called the bindct these bindct families are derived from chen s 1977 and loeffler s 1989 plane rotation based factorizations of the dct matrix respectively and the design approach can also be applied to a dct of arbitrary size two design approaches are presented in the first method an optimization program is defined and the multiplierless transform is obtained by approximating its solution with dyadic values in the second method a general lifting based scaled dct structure is obtained and the analytical values of all lifting parameters are derived enabling dyadic approximations with different accuracies therefore the bindct can be tuned to cover the gap between the walsh hadamard transform and the dct the corresponding two dimensional 2 d bindct allows a 16 bit implementation enables lossless compression and maintains satisfactory compatibility with the floating point dct the performance of the bindct in jpeg h 263 and lossless compression is also demonstrated

digital filter bank design quadratic constrained formulation formulate the filter bank design problem as an quadratic constrained least squares minimization problem the solution of the minimization problem converges very quickly since the cost function as well as the constraints are quadratic functions with respect to the unknown parameters the formulations of the perfect reconstruction cosine modulated filter bank of the near perfect reconstruction pseudo qmf bank and of the two channel biorthogonal linear phase filter bank are derived using the proposed approach compared with other design methods the proposed technique yields pr filter banks with much higher stopband attenuation the proposed technique can also be extended to design multidimensional filter banks

the theory and design of arbitrary length cosine modulated filter banks and wavelets satisfying perfect reconstruction it is well known that fir filter banks that satisfy the perfect reconstruction pr property can be obtained by cosine modulation of a linear phase prototype filter of length n 2mm where m is the number of channels in this paper we present a pr cosine modulated filter bank where the length of the prototype filter is arbitrary the design is formulated as a quadratic constrained least squares optimization problem where the optimized parameters are the prototype filter coefficients additional regularity conditions are imposed on the filter bank to obtain the cosine modulated orthonormal bases of compactly supported wavelets design examples are given

partial spectrum reconstruction using digital filter banks the problem of reconstructing a part of the spectrum is reduced to designing the filter bank to satisfy a set of conditions for the case considered here these conditions cannot be satisfied simultaneously so perfect reconstruction is not possible the necessary and sufficient conditions on the filters so that the resulting filter bank cancels most alias components are found such filter banks are called partial alias cancellation filter banks the product of the polyphase transfer matrices of these filter banks must be a block pseudocirculant matrix an algorithm design procedure is discussed and examples are given to demonstrate the theory

quadrature mirror filter design in the time domain a new technique for designing quadrature mirror filters is described the formulation carried out in the time domain is shown to result in an optimization problem requiring minimization of a quartic multinomial an iterative solution is suggested which involves computation of the eigenvector of a matrix with a dimensionality equal to one half the number of filter taps our experiments show that convergence to the optimum tap weights is stable and the accuracy of the final solution is limited only by the accuracy of the eigenvalue eigenvector routine as in an earlier technique the user can specify the stop band frequency the relative weights of the passband ripple energy and the stopband residual energy and of course the number of filter taps

a new design algorithm for two band orthonormal rational filter banks and orthonormal rational wavelets we present a new algorithm for the design of orthonormal two band rational filter banks owing to the connection between iterated rational filter banks and rational wavelets this is also a design algorithm for orthonormal rational wavelets it is basically a simple iterative procedure which explains its exponential convergence and adaptability under various linear constraints e g regularity although the filters obtained from this algorithm are suboptimally designed they show excellent frequency selectivity after an in depth account of the algorithm we discuss the properties of the rational wavelets generated by some designed filters in particular we stress the possibility to design almost shift error free wavelets which allows the implementation of a rational wavelet transform

making the use of maximal ideals constructive the purpose of this paper is to decipher constructively a lemma of suslin which played a central role in his second solution of serre s problem on projective modules over polynomial rings this lemma says that for a commutative ring a if a x where v 1 is monic and n 3 then there exist c 1 c e n 1 a x such that denoting by w i the first coordinate of c it v 2 v n we have a by the constructive proof we give suslin s proof of serre s problem becomes fully constructive moreover the new method with which we treat this academic example may be a model for miming constructively abstract proofs in which one works modulo a generic maximal ideal in order to prove that an ideal contains 1

subband coder design incorporating recursive quadrature filters and optimum adpcm coders this paper presents the results of an analytical and experimental study of the use of iir quadrature mirror filters and forward adapting adpcm with optimum quantizers for octave band subband coders for speech it is shown that the underlying analysis synthesis systems can be designed such that there is no interband aliasing distortion in the coded speech and such that the analysis synthesis transfer function has no frequency distortion or no phase distortion but not both the experimental study showed that subband coders based on a mixture of infinite impulse response iir and finite duration impulse response fir filters resulted in higher quality coding systems using fewer multiplies than for systems based on fir filters alone it was also found that a signal to noise ratio snr gain of about 4 db could be obtained using optimum quantizers for forward adapting adpcm coders over linear quantizers in backward adapting apcm codes but no perceptual quality gain was observed

a new approach to the realization of low sensitivity iir digital filters a new implementation of an iir digital filter transfer function is presented that is structurally passive and hence has extremely low pass band sensitivity the structure is based on a simple parallel interconnection of two all pass sections with each section implemented in a structurally lossless manner the structure shares a number of properties in common with wave lattice digital filters computer simulation results verifying the low sensitivity feature are included along with results on roundoff noise dynamic range interaction a large number of alternatives is available for the implementation of the all pass sections giving rise to the well known wave lattice digital filters as a specific instance of the implementation

quadrature mirror filter design for an arbitrary number of equal bandwidth channels the two channel quadrature mirror filter structure of croisier and esteban may be extended to an arbitrary number of equal bandwidth channels given certain restrictions on the bandpass filters the most serious restriction is that the stopband attenuation of eacli band pass filter must be high for all frequencies outside twice the nominal 3 db bandwidth of the filter this restriction is not really a limiting factor for speech subband waveform coding since high adjacent channel attenuation is a necessity for the confinement of quantization noise a key feature of our filter structure is that the number of multiplies adds and stored coefficients required for implementation is significantly less than those needed for the conventional qmf structure given the same number of channels fortran code for a 16 channel filter structure is listed as an example of efficient implementation

factorability of lossless time varying filters and filter banks we study the factorability of linear time varying ltv lossless filters and filter banks we give a complete characterization of all degree one lossless ltv systems and show that all degree one lossless systems can be decomposed into a time dependent unitary matrix followed by a lossless dyadic based ltv system the lossless dyadic based system has several properties that make it useful in the factorization of lossless ltv systems the traditional lapped orthogonal transform lot is also generalized to the ltv case we identify two classes of tvlots namely the invertible inverse lossless iil and noninvertible inverse lossless nil tvlots the minimum number of delays required to implement a tvlot is shown to be a nondecreasing function of time and it is a constant if and only if the tvlot is iil we also show that all iil tvlots can be factorized uniquely into the proposed degree one lossless building block the factorization is minimal in terms of the delay elements for nil tvlots there are factorable and unfactorable examples both necessary and sufficient conditions for the factorability of lossless ltv systems are given we also introduce the concept of strong eternal reachability ser and strong eternal observability seo of ltv systems the ser and seo of an implementation of ltv systems imply the minimality of the structure using these concepts we are able to show that the cascade structure for a factorable iil ltv system is minimal that implies that if a iil ltv system is factorable in terms of the lossless dyadic based building blocks the factorization is minimal in terms of delays as well as the number of building blocks we also prove the bibo stability of the ltv normalized iir lattice

step size sequence design for finite time average consensus in secure wireless sensor networks this paper concerns the study of average consensus in wireless sensor networks with aim of providing a way to reach consensus in a finite number of steps in particular we investigate the design of consensus protocols when for security reasons for instance the underlying graph is constrained to be strongly regular or distance regular the proposed design method is based on parameters of the intersection array characterizing the underlying graph with this protocol at execution time average consensus is achieved in a number of steps equal to the diameter of the graph i e the smallest possible number of steps to achieve consensus

a study of two channel complex valued filterbanks and wavelets with orthogonality and symmetry properties we investigate two channel complex valued filterbanks and wavelets that simultaneously have orthogonality and symmetry properties first the conditions for the filterbank to be orthogonal symmetric and regular for generating smooth wavelets are presented then a complete and minimal lattice structure is developed which enables a general design approach for filterbanks and wavelets with arbitrary length and arbitrary order of regularity finally two integer implementation methods that preserve the perfect reconstruction property of the filterbank are proposed their performances are evaluated via experimental results