geometric measures of convex sets and bounds on problem sensitivity and robustness for conic linear optimization the effect of data perturbation and uncertainty has always been an important consideration in optimization it is important to know whether a given problem is very sensible to perturbations on the data or on the contrary is more robust problem geometry does have an impact on the sensitivity of the problem and in this paper we analyze this connection by developing bounds to the change in the optimal value of a conic linear problem in terms of some geometric measures related to the radius of inscribed and circumscribed balls to the feasible region of the problem we also present a parametric analysis for linear programming which allows us to construct an estimate of safety limits for perturbations of the data these results are developed in relation to questions in robust optimization

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

lapped transforms for efficient transform subband coding two lapped transforms for subband transform coding of signals are introduced a version of the lapped orthogonal transform lot which can be efficiently computed for any transform length and the modulated lapped transform mlt which is based on a modulated quadrature mirror qmf bank the mlt can also be efficiently computed by means of a type iv discrete sine transform dst iv the lot and the mlt are both asymptotically optimal lapped transforms for coding an ar 1 signal with a high intersample correlation coefficient the coding gains of the lot and mlt of length m are higher than that of the discrete cosine transform dct of the same length they are actually close to the coding gains obtained with a dct of length 2m an mlt based adaptive transform coder act for speech signals is simulated the code is essentially free from frame rate noise and has a better spectral resolution that dct based atc systems

global stability of an hiv 1 infection model with saturation infection and intracellular delay in this paper an hiv 1 infection model with a saturation infection rate and an intracellular delay accounting for the time between viral entry into a target cell and the production of new virus particles is investigated by analyzing the characteristic equations the local stability of an infection free equilibrium and a chronic infection equilibrium of the model is established by using suitable lyapunov functionals and the lasalle invariant principle it is proved that if the basic reproduction ratio is less than unity the infection free equilibrium is globally asymptotically stable if the basic reproduction ratio is greater than unity the chronic infection equilibrium is globally asymptotically stable

lyapunov functionals for delay differential equations model of viral infections we study global properties of a class of delay differential equations model for virus infections with nonlinear transmissions compared with the typical virus infection dynamical model this model has two important and novel features to give a more complex and general infection process a general nonlinear contact rate between target cells and viruses and the removal rate of infected cells are considered and two constant delays are incorporated into the model which describe i the time needed for a newly infected cell to start producing viruses and ii the time needed for a newly produced virus to become infectious mature respectively by the lyapunov direct method and using the technology of constructing lyapunov functionals we establish global asymptotic stability of the infection free equilibrium and the infected equilibrium we also discuss the effects of two delays on global dynamical properties by comparing the results with the stability conditions for the model without delays further we

orthonormal and biorthonormal filter banks as convolvers and convolutional coding gain convolution theorems for filter bank transformers are introduced both uniform and nonuniform decimation ratios are considered and orthonormal as well as biorthonormal cases are addressed all the theorems are such that the original convolution reduces to a sum of shorter decoupled convolutions in the subbands that is there is no need to have cross convolution between subbands for the orthonormal case expressions for optimal bit allocation and the optimized coding gain are derived the contribution to coding gain comes partly from the nonuniformity of the signal spectrum and partly from nonuniformity of the filter spectrum with one of the convolved sequences taken to be the unit pulse function e coding gain expressions reduce to those for traditional subband and transform coding the filter bank convolver has about the same computational complexity as a traditional convolver if the analysis bank has small complexity compared to the convolution itself

computing the radius of pointedness of a convex cone let h denote the set of all nonzero closed convex cones in a finite dimensional hilbert space h consider this set equipped with the bounded pompeiu hausdorff metric the collection of all pointed cones forms an open set in the metric space h one possible way of measuring the degree of pointedness of a cone k is by evaluating the distance from k to the set of all nonpointed cones the number k obtained in this way is called the radius of pointedness of the cone k the evaluation of this number is in general a very cumbersome task in this note we derive a simple formula for computing k and we propose also a method for constructing a nonpointed cone at minimal distance from k our results apply to any cone k whose maximal angle does not exceed 120

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

the role of synaptic transmission in a hiv model with memory a fractional order model for hiv with two transmission modes and treatment is proposed global stability of the disease free equilibrium is analyzed cell to cell transmission has a strong impact in the value of the reproduction number increasing drug efficacy decreases the infectiousness of the disease we propose a mathematical model with memory for the dynamics of hiv epidemics where two transmission modes cell to cell and virus to cell and drug resistance are considered systems with memory or fractional order systems have largely been applied to the modeling of several real life phenomena here we consider a fractional model where the order of the non integer derivative takes values in the interval 0 5 1 0 we prove the local and global stability of the disease free equilibrium we study the role of the cell to cell transmission probability on the dynamics of the model and on the value of the reproduction number r0 for distinct values of the fractional order derivative moreover we show evidence of an improvement of hiv infected patients quality of life due to the increase of the drug efficacy in the end important inferences are drawn

bifrequency and bispectrum maps a new look at multirate systems with stochastic inputs in multirate digital signal processing we often encounter decimators interpolators and complicated interconnections of these with lti filters we also encounter cyclo wide sense stationary cwss processes and linear periodically time varying ptv systems it is often necessary to understand the effects of multirate systems on the statistical properties of their input signals some of these issues have been addressed earlier for example it has been shown that a necessary and sufficient addition for the output of an l fold interpolation filter to be wide sense stationary wss for all wss inputs is that the filter was an alias free l support however several questions of this nature remain unanswered for example what is the necessary and sufficient condition on a pair or more generally a bank of interpolation filters so that their outputs are jointly wss jwss for all jointly wss inputs what is the condition if only the sum of their outputs is required to be wss when is the output of an lptv system for example a uniform filter bank wss for all wss inputs some of these questions may appear to be simple generalizations of the above mentioned result for a single interpolation filter however the frequency domain approaches that proved this result are quite difficult to generalize to answer these questions the purpose of this paper is to provide these answers using analysis based on bifrequency maps and bispectra these tools are two dimensional 2 d fourier transforms that characterize linear time varying ltv systems and nonstationary random processes respectively we show that the questions raised above are addressed elegantly and in a geometrically insightful way using these tools we also derive a bifrequency characterization lossless ltv systems this may potentially lead to an increased understanding of these systems

the price of robustness a robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended under this approach we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes a concern with such an approach is that it might be too conservative in this paper we propose an approach that attempts to make this trade off more attractive that is we investigate ways to decrease what we call the price of robustness in particular we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations an attractive aspect of our method is that the new robust formulation is also a linear optimization problem thus we naturally extend our methods to discrete optimization problems in a tractable way we report numerical results for a portfolio optimization problem a knapsack problem and a problem from the net lib library

linear phase paraunitary filter banks theory factorizations and designs m channel maximally decimated filter banks have been used in the past to decompose signals into subbands the theory of perfect reconstruction filter banks has also been studied extensively nonparaunitary systems with linear phase filters have also been designed the authors study paraunitary systems in which each individual filter in the analysis synthesis banks has linear phase specific instances of this problem have been addressed by other authors and linear phase paraunitary systems have been shown to exist this property is often desirable for several applications particularly in image processing they begin by answering several theoretical questions pertaining to linear phase paraunitary systems next they develop a minimal factorization for a large class of such systems this factorization will be proved to be complete for even m further they structurally impose the additional condition that the filters satisfy pairwise mirror image symmetry in the frequency domain this significantly reduces the number of parameters to be optimized in the design process they then demonstrate the use of these filter banks in the generation of m band orthonormal wavelets several design examples are also given to validate the theory

vector space framework for unification of one and multidimensional filter bank theory a number of results in filter bank theory can be viewed using vector space notations this simplifies the proofs of many important results in this paper we first introduce the framework of vector space and then use this framework to derive some known and some new filter bank results as well for example the relation among the hermitian image property orthonormality and the perfect reconstruction pr property is well known for the case of one dimensional 1 d analysis synthesis filter banks we can prove the same result in a more general vector space setting this vector space framework has the advantage that even the most general filter banks namely multidimensional nonuniform filter banks with rational decimation matrices become a special case many results in 1 d filter bank theory are hence extended to the multidimensional case with some algebraic manipulations of integer matrices some examples are the equivalence of biorthonormality and the pr property the interchangeability of analysis and synthesis filters the connection between analysis synthesis filter banks and synthesis analysis transmultiplexers etc furthermore we obtain the subband convolution scheme by starting from the generalized parseval s relation in vector space several theoretical results of wavelet transform can also be derived using this framework in particular we derive the wavelet convolution theorem

on spectral theory of cyclostationary signals in multirate systems this paper studies two problems in the spectral theory of discrete time cyclostationary signals the cyclospectrum representation and the cyclospectrum transformation by linear multirate systems four types of cyclospectra are presented and their interrelationships are explored in the literature the problem of cyclospectrum transformation by linear systems was investigated only for some specific configurations and was usually developed with inordinate complexities due to lack of a systematic approach a general multirate system that encompasses most common systems linear time invariant systems and linear periodically time varying systems is proposed as the unifying framework more importantly it also includes many configurations that have not been investigated before e g fractional sample rate changers with cyclostationary inputs the blocking technique provides a systematic solution as it associates a multirate system with an equivalent linear time invariant system and cyclostationary signals with stationary signals thus the original problem is elegantly converted into a relatively simple one which is solved in the form of matrix multiplication

modeling analysis and optimum design of quantized m band filter banks this paper provides a rigorous modeling and analysis of quantization effects in m band subband codecs followed by optimal filter bank design and compensation the codec is represented by a polyphase decomposition of the analysis synthesis filter banks and an embedded nonlinear gain plus additive noise model for the pdf optimized scalar quantizers we construct an equivalent time invariant but nonlinear structure operating at the slow clock rate that allows us to compute the exact expression for the mean square quantization error in the reconstructed output this error is shown to consist of two components a distortion component and a dominant random noise component uncorrelated with the input signal we determine the optimal paraunitary and biorthogonal fir filter coefficients compensators and integer bit allocation to minimize this mse subject to the constraints of filter length average bit rate and perfect reconstruction pr in the absence of quantizers the biorthogonal filter bank results in a smaller mse but the filter coefficients are very sensitive to signal statistics and to average bit constraints by comparison the paraunitary structure is much more robust we also show that the null compensated design that eliminates the distortion component is more robust than the optimally compensated case that minimizes the total mse but only at nominal conditions both modeling and optimal design are validated by simulation in the two channel case

image coding using translation invariant wavelet transforms with symmetric extensions in this correspondence we address the problem of translation sensitivity of conventional wavelet transforms for two dimensional 2 d signals we propose wavelet transform algorithms that achieve the following desirable properties simultaneously i translation invariance ii reduced edge effects and iii size limitedness we apply this translation invariant biorthogonal wavelet transform with symmetric extensions to image coding applications with good results

structures factorizations and design criteria for oversampled paraunitary filterbanks yielding linear phase filters we present here a special class of oversampled filterbanks fbs namely paraunitary fbs with linear phase filters we propose some necessary conditions for the existence of such banks based on the repartition between type i ii and type ii iv linear phase filters in the bank for a subset of these fbs we develop a factorization that leads to a minimal implementation as well as a direct parameterization of the fbs in terms of elementary rotation angles this factorization is applied to some design examples with two different optimization criteria coding gain and reconstructibility of lost coefficients

robust convex optimization we study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set u yet the constraints must hold for all possible values of the data from u the ensuing optimization problem is called robust optimization in this paper we lay the foundation of robust convex optimization in the main part of the paper we show that if u is an ellipsoidal uncertainty set then for some of the most important generic convex optimization problems linear programming quadratically constrained programming semidefinite programming and others the corresponding robust convex program is either exactly or approximately a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods

polyphase networks block digital filtering lptv systems and alias free qmf banks a unified approach based on pseudocirculants the relationship between block digital filtering and quadrature mirror filter qmf banks is explored necessary and sufficient conditions for alias cancellation in qmf banks are expressed in terms of an associated matrix derived from the polyphase components of the analysis and synthesis filters these conditions called the pseudocirculant conditions make it possible to unite qmf banks with the framework of block digital filtering directly absence of amplitude distortion in an alias free qmf bank translates into the losslessness property of the pseudocirculant matrix involved

optimization of filter banks using cyclostationary spectral analysis this paper proposes a design method of optimal biorthogonal fir filter banks that minimize the time averaged mean squared error tamse when the high frequency subband signal is dropped to study filter banks from a statistical point of view cyclostationary spectral analysis is used since the output of the filter bank for a wide sense stationary input is cyclostationary first the cyclic spectral density of the output signal is derived and an expression for the tamse is presented then optimal filter banks are given by minimizing the tamse with respect to the coefficients of the filters under the biorthogonality condition by imposing the additional constraints on the coefficients the optimal biorthogonal linear phase filter bank can be obtained

the coding gain of multiplexed wavelet transforms evangelista 1994 1993 introduced the multiplexed wavelet transform mwt and pointed out its potential applications to the analysis synthesis processing and coding of pseudo periodic signals such as voiced speech and music coders based on the mwt have been shown to outperform the conventional subband coders when a reliable pitch parameter can be extracted from data in this paper we investigate the effects of uniform quantization of the mwt coefficients we compare the performance of the new coders with that of wt block dct and klt coders in terms of the coding gain achieved when optimal bit allocation schemes are adopted

minimum mean squared error reconstruction for generalized undersampling of cyclostationary processes the generalized sampling problem is considered using a filter bank model with a stochastic process framework a signal reconstruction solution minimizing the time averaged mean squared error is found using a filter bank optimization technique for cyclostationary signals the use of a stochastic model approach to find a minimized error solution rather than a perfect reconstruction solution allows consideration of sampling problems that deterministic approaches cannot solve for instance a particular sampling density is not required allowing reconstruction optimization in cases of undersampling an extension is also presented for optimal reconstruction in the presence of additive noise and interference the result is a generalized sampling model which can be used for an extensive variety of scenarios combining aspects of undersampling multiple input multiple output mimo sampling and periodic nonuniform sampling

a general parametric analysis approach and its implication to sensitivity analysis in interior point methods adler and monteiro 1992 developed a parametric analysis approach that is naturally related to the geometry of the linear program this approach is based on the availability of primal and dual optimal solutions satisfying strong complementarity in this paper we develop an alternative geometric approach for parametric analysis which does not require the strong complementarity condition this parametric analysis approach is used to develop range and marginal analysis techniques which are suitable for interior point methods two approaches are developed namely the lu factorization approach and the affine scaling approach

subband coding of images subband coding has become quite popular for the source encoding of speech this paper presents a simple yet efficient extension of this concept to the source coding of images we specify the constraints for a set of two dimensional quadrature mirror filters qmf s for a particular frequency domain partition and show that these constraints are satisfied by a separable combination of one dimensional qmf s bits are then optimally allocated among the subbands to minimize the mean squared error for dpcm coding of the subbands also an adaptive technique is developed to allocate the bits within each subband by means of a local variance mask optimum quantization is employed with quantizers matched to the laplacian distribution subband coded images are presented along with their signal to noise ratios snr s the snr performance of the subband coder is compared to that of the adaptive discrete cosine transform dct vector quantization and differential vector quantization for bit rates of 0 67 1 0 and 2 0 bits per pixel for 256 256 monochrome images the adaptive subband coder has the best snr performance

a design method of multidimensional linear phase paraunitary filter banks with a lattice structure a lattice structure of multidimensional md linear phase paraunitary filter banks lppufbs is proposed which makes it possible to design such systems in a systematic manner our proposed structure can produce md lppufbs whose filters all have the region of support spl nscr m spl xi where m and spl xi are the decimation and positive integer diagonal matrices respectively and spl nscr n denotes the set of integer vectors in the fundamental parallelepiped of a matrix n it is shown that if spl nscr m is reflection invariant with respect to some center then the reflection invariance of spl nscr m spl xi is guaranteed this fact is important in constructing md linear phase filter banks because the reflection invariance is necessary for any linear phase filter since our proposed system structurally restricts both the paraunitary and linear phase properties an unconstrained optimization process can be used to design md lppufbs our proposed structure is developed for both an even and an odd number of channels and includes the conventional 1 d system as a special case it is also shown to be minimal and the no dc leakage condition is presented some design examples show the significance of our proposed structure for both the rectangular and nonrectangular decimation cases

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

coding gain in paraunitary analysis synthesis systems a formal proof that bit allocation results hold for the entire class of paraunitary subband coders is presented the problem of finding an optimal paraunitary subband coder so as to maximize the coding gain of the system is discussed the bit allocation problem is analyzed for the case of the paraunitary tree structured filter banks such as those used for generating orthonormal wavelets the even more general case of nonuniform filter banks is also considered in all cases it is shown that under optimal bit allocation the variances of the errors introduced by each of the quantizers have to be equal expressions for coding gains for these systems are derived

cyclo period estimation for discrete time cyclo stationary signals this paper presents a new method named as the variability method to estimate the cyclo period of a discrete time cyclo stationary signal the method is essentially based on the time varying correlation and or the time varying mean whose estimators are associated with some statistics of blocked signals a plot of variability of these statistics as a function of the blocking operator index visually reveals a periodic pattern from which the cyclo period is obtained properties of the variability method are analyzed and compared with three existing cyclo period estimation methods via simulation and real life examples

equivalence of convex problem geometry and computational complexity in the separation oracle model consider the supposedly simple problem of computing a point in a convex set that is conveyed by a separation oracle with no further information e g no domain ball containing or intersecting the set etc the authors interest in this problem stems from fundamental issues involving the interplay of i the computational complexity of computing a point in the set ii the geometry of the set and iii the stability or conditioning of the set under perturbation under suitable definitions of these terms the authors show herein that problem instances with favorable geometry have favorable computational complexity validating conventional wisdom the authors also show a converse of this implication by showing that there exist problem instances that require more computational effort to solve in certain families characterized by unfavorable geometry this in turn leads under certain assumptions to a form of equivalence among computational complexity geometry and the conditioning of the set the authors measures of the geometry relative to a given reference point are based on the radius of a certain domain ball whose intersection with the set contains a certain inscribed ball the geometry of the set is then measured by the radius of the domain ball the radius of the inscribed ball and the ratio between these two radii the latter of which is called the aspect ratio the aspect ratio arises in the analysis of many algorithms for convex problems and its importance in convex algorithm analysis has been well known for several decades however the presence in our bound of terms involving only the radius of the domain ball and only the radius of the inscribed ball are a bit counterintuitive nevertheless we show that the computational complexity must involve these terms in addition to the aspect ratio even when the aspect ratio itself is small this lower bound complexity analysis relies on simple features of the separation oracle model if we instead assume that the set is conveyed by a self concordant barrier function then it is an open challenge to prove such a complexity lower bound

a delay dependent model with hiv drug resistance during therapy abstract the use of combination antiretroviral therapy has proven remarkably effective in controlling hiv disease progression and prolonging survival however the emergence of drug resistance can occur it is necessary that we gain a greater understanding of the evolution of drug resistance here we consider an hiv viral dynamical model with general form of target cell density drug resistance and intracellular delay incorporating antiretroviral therapy the model includes two strains wild type and drug resistant the basic reproductive ratio for each strain is obtained for the existence of steady states qualitative analysis of the model such as the well posedness of the solutions and the equilibrium stability is provided global asymptotic stability of the disease free and drug resistant steady states is shown by constructing lyapunov functions furthermore sufficient conditions related to the properties of the target cell density are obtained for the local asymptotic stability of the positive steady state numerical simulations are conducted to study the impact of target cell density and intracellular delay focusing on the stability of the positive steady state the occurrence of hopf bifurcation of periodic solutions is shown to depend on the target cell density

multirate filter banks and transform coding gain an analytical derivation of quantization error in filter banks using additive and signal dependent noise models is presented the coding gain over pcm is derived for both noise models interesting properties of coding gain in optimized filter banks with an ar 1 input signal model asymptotic performance of multirate systems and the analytical constraints required to achieve such performance are discussed connections to the recent developments of optimal subband systems are presented

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

improved technique for design of perfect reconstruction fir qmf banks with lossless polyphase matrices a technique is developed for the design of analysis filters in an m channel maximally decimated perfect reconstruction finite impulse response quadrature mirror filter fir qmf bank that has a lossless polyphase component matrix e z the aim is to optimize the parameters characterizing e z until the sum of the stopband energies of the analysis filters is minimized there are four novel elements in the procedure reported here the first is a technique for efficient initialization of one of the m analysis filters as a spectral factor of an mth band filter the factorization itself is done in an efficient manner using the eigenfilters approach without the need for root finding techniques the second element is the initialization of the internal parameters which characterize e z based on the above spectral factor the third element is a modified characterization mostly free from rotation angles of the fir e z the fourth is the incorporation of symmetry among the analysis filters so as to minimize the number of unknown parameters being optimized the resulting design procedure always gives better filter responses than earlier ones for a given filter length and converges much faster

fir principal component filter banks two dimensional 2 d principal component filter banks pcfbs of finite impulse response fir are proposed for 2 d signals among all uniform paraunitary fir analysis synthesis filter banks the fir pcfbs have the most energy compaction and maximize the arithmetic mean to geometric mean ratio am gm ratio of subband variances which is the theoretic coding gain tcc of the systems under proper assumptions the theoretic proof and design techniques are provided several special cases are discussed experimental results show the potential power of the fir pcfbs

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

performance of sigma delta quantizations in finite frames in this paper we consider sigma delta sd quantization of geometrically uniform gu finite frames in the first part we prove that under some conditions the variant i and ii permutation modulation pm codes first introduced by slepian 1968 1965 can belong to the class of gu frames then we focus essentially on a subclass of a gu frame namely cyclic geometrically uniform cgu frame family of frames containing finite harmonic frames both in bbc m and bbr m for first and second order sd quantizers we establish that the reconstruction minimum squares error mse behaves as 1 r 2 where r denotes the frame redundancy this result is shown to be true both under the deterministic quantization model used in benedetto 2004 benedetto 2006 and yilmaz 2001 as well as under the widely used additive white quantization noise assumption for the widely used l th order noise shaping filter g z 1 z 1 l we show that the mse behaves as 1 r 2 irrespectively of the filter order l more importantly we prove also that in the case of tight and normalized cgu frame when the frame length is too large compared to the filter order and under some conditions on the quantizer the reconstruction mse can decay as fast as o 1 r 2 l 1 finally it is shown that degrees of freedom in cgu frames when compared to harmonic frames result in a smaller mse albeit mse prop 1 r 2 in both cases

extended active constraints in linear optimization with applications in this paper we introduce different relaxations of the concept of active constraint at a given point with respect to a certain linear inequality system with an arbitrary number of constraints we show that these concepts provide useful local information in linear optimization for instance conditions for a given feasible solution to be a unique extreme point optimal solution or a strongly unique optimal solution

second order statistical characterization of the filter bank and its elements this correspondence proposes an alternative scheme for characterizing the wide sense cyclostationary wscs processes and applies it for the analysis of the filter banks whose output is wscs for wide sense stationary and wscs inputs the proposed representation clearly illustrates the kind of transformation taking place in multiscale stochastic systems these relationships are extended to the filter bank to make clear the effect of the various stages of the filter bank on wscs processes

algebraic theory of optimal filterbanks we introduce an optimality theory for finite impulse response fir filterbanks using a general algebraic point of view we consider an admissible set spl lscr of fir filterbanks and use scalability as the main notion based on which performance of the elements in spl lscr are compared we show that quantification of scalability leads naturally to a partial ordering on the set spl lscr an optimal solution is therefore represented by the greatest element in spl lscr it turns out that a greatest element does not necessarily exist in spl lscr hence one has to settle with one of the maximal elements that exist in spl lscr we provide a systematic way of finding a maximal element by embedding the partial ordering at hand in a total ordering this is done by using a special class of order preserving functions known as schur convex there is however a price to pay for achieving a total ordering there are infinitely many possible choices for schur convex functions and the optimal solution specified in spl lscr depends on this subjective choice an interesting aspect of the presented algebraic theory is that the connection between several concepts namely principal component filterbanks pcfbs filterbanks with maximum coding gain and filterbanks with good scalability is clearly revealed we show that these are simply associated with different extremal elements of the partial ordering induced on spl lscr by scalability

on bounds of shift variance in two channel multirate filter banks critically sampled multirate fir filter banks exhibit periodically shift variant behavior caused by nonideal antialiasing filtering in the decimation stage we assess their shift variance quantitatively by analysing changes in the output signal when the filter bank operator and shift operator are interchanged we express these changes by a so called commutator we then derive a sharp upper bound for shift variance via the operator norm of the commutator which is independent of the input signal its core is an eigensystem analysis carried out within a frequency domain formulation of the commutator leading to a matrix norm which depends on frequency this bound can be regarded as a worst case instance holding for all input signals for two channel fir filter banks with perfect reconstruction pr we show that the bound is predominantly determined by the structure of the filter bank rather than by the type of filters used moreover the framework allows to identify the signals for which the upper bound is almost reached as so called near maximizers of the frequency dependent matrix norm for unitary pr filter banks these near maximizers are shown to be narrow band signals to complement this worst case bound we derive an additional bound on shift variance for input signals with given amplitude spectra where we use wide band model spectra instead of narrow band signals like the operator norm this additional bound is based on the above frequency dependent matrix norm we provide results for various critically sampled two channel filter banks such as quadrature mirror filters pr conjugated quadrature filters wavelets and biorthogonal filters banks

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