structures for anticausal inverses and application in multirate filter banks anticausal or time reversed inversion of digital filters has gained importance in the implementation of digital filter banks anticausal inversion has in the past been shown to be possible by using block processing with appropriate state initialization with a b c d denoting the state space description of a structure implementing a filter g z the anticausal inverse can be essentially regarded as a filter structure having an inverted state space description which we denote as a spl circ b spl circ c spl circ d spl circ it is usually not efficient to implement the state space equations given by a spl circ b spl circ c spl circ d spl circ directly because of excessive multiplier count rather one seeks to find an efficient structure having the inverse description a spl circ b spl circ c spl circ d spl circ while this can be done by inspection in simple cases such as the direct form structure systematic procedures for other important structures have yet to be developed we derive anticausal inverse structures corresponding to several standard iir filter structures such as the direct form cascade form coupled form and the entire family of iir lattice structures including the tapped cascaded lattice we introduce the notion of a causal dual which we find convenient in the derivations we show that the limit cycle free property of the original structure is inherited by the causal dual in some but not all cases

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

frame theory based analysis and design of oversampled filter banks direct computational method this paper studies the frames corresponding to oversampled filter banks fbs for this class of fb frames we present a state space parameterization of perfect reconstruction fb frames and explicit and numerically efficient formulas to compute the tightest frame bounds to obtain the dual fb frame and to construct a tight paraunitary fb frame from a given untight nonparaunitary fb frame the derivation uses well developed techniques from modern control theory which results in the unified formulas for generic infinite impulse response iir and finite impulse response fir fbs these formulas involve only algebraic manipulations of real matrices and can be computed efficiently reliably and exactly without the approximation required in the existing methods for generic fbs the results provide a unified framework for frame theory based analysis and systematic design of oversampled filter banks

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 generalized lapped pseudo biorthogonal transform oversampled linear phase perfect reconstruction filterbanks with lattice structures we investigate a lattice structure for a special class of n channel oversampled linear phase perfect reconstruction filterbanks with a decimation factor m smaller than n we deal with systems in which all analysis and synthesis filters have the same finite impulse response fir length and share the same center of symmetry we provide the minimal lattice factorization of a polyphase matrix of a particular class of these oversampled filterbanks fbs all filter coefficients are parameterized by rotation angles and positive values the resulting lattice structure is able to provide fast implementation and allows us to determine the filter coefficients by solving an unconstrained optimization problem we consider next the case where we are given the generalized lapped pseudo biorthogonal transform glpbt lattice structure with specific parameters and we a priori know the correlation matrix of noise that is added in the transform domain in this case we provide an alternative lattice structure that suppress the noise we show that the proposed systems with the lattice structure cover a wide range of linear phase perfect reconstruction fbs we also introduce a new cost function for oversampled fb design that can be obtained by generalizing the conventional coding gain finally we exhibit several design examples and their properties

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

rational sampling filter banks based on iir filters the problem of splitting the spectrum of a digital signal by using nonuniform infinite impulse response iir filter banks is addressed near perfect reconstruction npr is considered the method uses the modulation of different iir prototypes the cancellation of the main aliasing components constrains the prototypes to be dependent on each other by using this approach linear phase prototypes are needed and noncausal filtering is required numerical examples of filter bank design are given and the computational complexity is compared with the finite impulse response fir case

lapped unimodular transform and its factorization two types of lapped transforms have been studied in detail in the literature namely the lapped orthogonal transform lot and its extension the biorthogonal lapped transform bolt in this paper we will study the lapped unimodular transform lut all three transforms are first order matrices with finite impulse response fir inverses we will show that like lot and bolt all luts can be factorized into degree one unimodular matrices the factorization is both minimal and complete we will also show that all first order systems with fir inverses can be minimally factorized as a cascade of degree one lot bolt and lut building blocks two examples will be given to demonstrate that despite having a very small system delay the luts have a satisfactory performance in comparison with lot and bolt

linear phase perfect reconstruction filter bank lattice structure design and application in image coding a lattice structure for an m channel linear phase perfect reconstruction filter bank lpprfb based on the singular value decomposition svd is introduced the lattice can be proven to use a minimal number of delay elements and to completely span a large class of lpprfbs all analysis and synthesis filters have the same fir length sharing the same center of symmetry the lattice also structurally enforces both linear phase and perfect reconstruction properties is capable of providing fast and efficient implementation and avoids the costly matrix inversion problem in the optimization process from a block transform perspective the new lattice can be viewed as representing a family of generalized lapped biorthogonal transform glbt with an arbitrary number of channels m and arbitrarily large overlap the relaxation of the orthogonal constraint allows the glbt to have significantly different analysis and synthesis basis functions which can then be tailored appropriately to fit a particular application several design examples are presented along with a high performance glbt based progressive image coder to demonstrate the potential of the new transforms

recursive time varying filter banks for subband image coding filter banks subband wavelets and multiresolution decompositions that employ recursive filters have been considered previously and are recognized for their efficiency in partitioning the frequency spectrum this paper presents an analysis of a new infinite impulse response iir filter bank in which these computationally efficient filters may be changed adaptively in response to the input the new filter bank framework is presented and discussed in the context of subband image coding in the absence of quantization errors exact reconstruction can be achieved by the proper choice of an adaptation scheme it is shown that recursive linear time varying ltv filter banks can yield improvement over conventional ones

the tractability frontier of graph like first order query sets the focus of this work is first order model checking by which we refer to the problem of deciding whether or not a given first order sentence is satisfied by a given finite structure in particular we aim to understand on which sets of sentences this problem is tractable in the sense of parameterized complexity theory to this end we define the notion of a graph like sentence set the definition is inspired by previous work on first order model checking wherein the permitted connectives and quantifiers were restricted our main theorem is the complete tractability classification of such graph like sentence sets which is to our knowledge the first complexity classification theorem concerning a class of sentences that has no restriction on the connectives and quantifiers to present and prove our classification we introduce and develop a novel complexity theoretic framework that is built on parameterized complexity and includes new notions of reduction

nonsmooth hypersurfaces with smooth levi curvature abstract in r 3 lipschitz continuous viscosity solutions to the k prescribed levi curvature equation are smooth and strictly pseudoconvex if k is smooth and strictly positive see 5 we show here that in r 2 n 1 n 1 a similar result does not hold that is we prove the existence in c n 1 n 1 of nonsmooth pseudoconvex hypersurfaces with smooth levi curvature

role of anticausal inverses in multirate filter banks i system theoretic fundamentals in a maximally decimated filter bank with identical decimation ratios for all channels the perfect reconstructibility property and the nature of reconstruction filters causality stability fir property and so on depend on the properties of the polyphase matrix various properties and capabilities of the filter bank depend on the properties of the polyphase matrix as well as the nature of its inverse in this paper we undertake a study of the types of inverses and characterize them according to their system theoretic properties i e properties of state space descriptions mcmillan degree degree of determinant and so forth we find in particular that causal polyphase matrices with anticausal inverses have an important role in filter bank theory we study their properties both for the fir and iir cases techniques for implementing anticausal iir inverses based on state space descriptions are outlined it is found that causal fir matrices with anticausal fir inverses cafacafi have a key role in the characterization of fir filter banks

role of anticausal inverses in multirate filter banks ii the fir case factorizations and biorthogonal lapped transforms for pt i see ibid vol 43 no 5 p 1090 1990 in part i we studied the system theoretic properties of discrete time transfer matrices in the context of inversion and classified them according to the types of inverses they had in particular we outlined the role of causal fir matrices with anticausal fir inverses abbreviated cafacafi in the characterization of fir perfect reconstruction pr filter banks essentially all fir pr filter banks can be characterized by causal fir polyphase matrices having anticausal fir inverses in this paper we introduce the most general degree one cafacafi building block and consider the problem of factorizing cafacafi systems into these building blocks factorizability conditions are developed a special class of cafacafi systems called the biorthogonal lapped transform bolt is developed and shown to be factorizable this is a generalization of the well known lapped orthogonal transform lot examples of unfactorizable cafacafi systems are also demonstrated finally it is shown that any causal fir matrix with fir inverse can be written as a product of a factorizable cafacafi system and a unimodular matrix

time domain oversampled lapped transforms theory structure and application in image coding this paper generalizes time domain lapped transforms tdlts proposed by tran et al to oversampled systems thus leading to time domain oversampled lapped transforms tdolts these new transforms correspond to a subclass of oversampled linear phase perfect reconstruction filterbanks olpprfbs which can be implemented by adding a prefilter before the discrete cosine transform dct and a post filter after the inverse discrete cosine transform idct structures of the pre and post filters are developed and the frame theoretic properties of tdolts are analyzed a new parameterization of lattice matrices through the givens qr factorization is proposed for unconstrained optimization comparisons with other parameterization methods are also included several design examples along with some image coding results are presented to demonstrate the validity of the theory and the potential of tdolts in image coding especially in error resilient coding

instantaneous erasures in oversampled filter banks conditions for output perfect reconstruction in this paper we aim at finding the conditions that an oversampled filter bank ofb should satisfy in order to maintain its perfect reconstruction property when erasures happen in the subband domain this problem has been addressed before mainly from frame theoretic point of view and only for the case of what we call in this paper classic erasure in the frame theoretic approach the stable filter banks are associated with frames in l2 bbz and a subband erasure is defined as the deletion of the frame expansion coefficients corresponding to the frame vectors resulting from each filter and all its translated versions this is equivalent to assuming that all the samples of a subband have been completely lost classic erasure and this is why in this approach it is always assumed that each channel is either working perfectly or not at all in this paper we extend this notion of erasure to a situation where subband channels are allowed to be on or off arbitrarily in each time instance and we define a new type of erasure called instantaneous erasure using an approach based on the time domain analysis of perfect reconstruction property we introduce general conditions for perfect reconstruction of the output and also the sufficient conditions for two classes of filter banks causal ofbs with causal inverse and ofbs with maximum robustness against classic erasure

perfect reconstruction iir digital filter banks supporting nonexpansive linear signal extensions in this paper perfect reconstruction polyphase infinite impulse response iir filter banks involving causal and anticausal inverses are examined for finite length signals first a novel and efficient nonexpansive perfect reconstruction algorithm based on the state space implementation is presented then the proposed method is extended to support linear signal extensions at the boundaries in a nonexpansive manner the powerfulness of the proposed algorithm is demonstrated with the image compression results

lagrange wavelets for signal processing this paper deals with the design of interpolating wavelets based on a variety of lagrange functions combined with novel signal processing techniques for digital imaging halfband lagrange wavelets b spline lagrange wavelets and gaussian lagrange lagrange distributed approximating functional daf wavelets are presented as specific examples of the generalized lagrange wavelets our approach combines the perceptually dependent visual group normalization vgn technique and a softer logic masking slm method these are utilized to rescale the wavelet coefficients remove perceptual redundancy and obtain good visual performance for digital image processing

eigenstructure approach for characterization of fir pr filterbanks with order one polyphase a new approach for the characterization of m channel finite impulse response fir perfect reconstruction pr filterbanks is proposed by appropriately restricting the eigenstructure of the polyphase matrix of the bank a complete characterization of order one polyphase matrices is obtained in which the polynomial part is in a block diagonal form nilpotent matrices play a crucial role in the structure this structure allows imposing restrictions on the order of the inverse of the polyphase matrix and or analysis synthesis delay reconstruction delay next we derive an alternate complete characterization in terms of the degree of the determinant and the mcmillan degree of order one polyphase matrix which we call the dyadic based characterization the characterization of vaidyanathan and chen 1995 for matrices with anticausal inverse turns out to be a special case of the proposed characterization the dyadic based characterization is more suitable for design without any above mentioned restriction since it allows better initialization we finally present design examples with different cost functions

a general approach to coprime pairs of matrices based on minors in signal processing coprime pairs of integer matrices have an important role in the study of multidimensional multirate systems and in multidimensional array processing in this paper a general theorem for coprimality of pairs of matrices is first presented based on the minors of an associated composite matrix first a necessary and sufficient set of conditions for coprimality is presented based on minors this result applies to matrices of any size several useful consequences of the result are discussed and it is first applied to the 2 2 case including special cases such as toeplitz matrices commuting toeplitz matrices circulants and skew circulants the condition under which a 2 2 integer matrix and its ajdugate are coprime is also derived the minor based result is then applied extensively for the 3 3 case to derive new coprimality conditions for several matrix pairs in particular adjugates of matrices are considered in detail because adjugates always commute with the parent matrices the so called generalized 3 3 circulant matrix and its adjugate are then analyzed in detail generalized circulants also form a commuting family special cases such as 3 3 circulant pairs and skew circulant pairs are also elaborated

theory of paraunitary filter banks over fields of characteristic two motivated by our wavelet framework for error control coding we proceed to develop an important family of wavelet transforms over finite fields paraunitary pu filter banks that are realizations of orthogonal wavelets by multirate filters are an important subclass of perfect reconstruction pr filter banks a parameterization of pu filter banks that covers all possible pu systems is very desirable in error control coding because it provides a framework for optimizing the free parameters to maximize coding performance this paper undertakes the problem of classifying all pu matrices with entries from a polynomial ring where the coefficients of the polynomials are taken from finite fields it constructs householder transformations that are used as elementary operations for the realization of all unitary matrices then it introduces elementary pu building blocks and a factorization technique that is specialized to obtain a complete realization for all pu filter banks over fields of characteristic two this is proved for the 2 spl times 2 case and conjectured for the m spl times m case where m spl ges 3 using these elementary building blocks we can construct all pu filter banks over fields of characteristic two these filter banks can be used to implement transforms which in turn provide a powerful new perspective on the problems of constructing and decoding arbitrary rate error correcting codes

discrete wavelet transform based on cyclic convolutions a filterbank in which cyclic convolutions are used in place of linear convolutions will be referred to as a cyclic convolution filterbank ccfb a dyadic tree structured ccfb can be used to perform a discrete wavelet transform suitable for coding based on symmetric extension methods this paper derives two types of efficient implementation techniques for the tree structured ccfb one using complex arithmetic and one using only real arithmetic in addition the present paper analyzes in detail the perfect reconstruction pr condition for the two channel ccfb and shows that this condition is much less restrictive than that of usual two channel filterbanks when each of the two channel ccfbs constituting a tree structured ccfb is designed to be pr the whole system is pr a quadrature mirror filter qmf ccfb having the pr property is demonstrated to be easily designed using a standard filter design subroutine in contrast designing a linear phase pr fir qmf bank that has good frequency response is not possible when filters are realized by linear convolutions

joint channel and carrier offset estimation in cdma communications this paper addresses the problem of cdma multiuser detection in the presence of unknown multipath channels and residual carriers an analytic algorithm that provides closed form channel and carrier offset estimates is proposed the algorithm first converts the multiuser estimation problem into single user problems and then analytically solves the resulting nonlinear multivariate optimization problems using a polynomial matrix projection property the performance of the proposed algorithm is investigated through first order perturbation analysis we also calculate the cramer rao bound crb to illustrate the efficiency of the new algorithm the analyzes are supported by computer simulations

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