linear phase cosine modulated maximally decimated filter banks with perfect reconstruction we propose a novel way to design maximally decimated fir cosine modulated filter banks in which each analysis and synthesis filter has a linear phase the system can be designed to have either the approximate reconstruction property pseudo qmf system or perfect reconstruction property pr system in the pr case the system is a paraunitary filter bank as in earlier work on cosine modulated systems all the analysis filters come from an fir prototype filter however unlike in any of the previous designs all but two of the analysis filters have a total bandwidth of 2 spl pi m rather than spl pi m where 2m is the number of channels in our notation a simple interpretation is possible in terms of the complex hypothetical analytic signal corresponding to each bandpass subband the coding gain of the new system is comparable with that of a traditional m channel system rather than a 2m channel system this is primarily because there are typically two bandpass filters with the same passband support correspondingly the cost of the system in terms of complexity of implementation is also comparable with that of an m channel system we also demonstrate that very good attenuation characteristics can be obtained with the new system

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

design of linear phase m channel perfect reconstruction fir filter banks we propose two approaches to design m channel nonparaunitary filter banks that satisfy perfect reconstruction pr and linear phase lp properties in the first approach the pr condition is imposed on only a high pass filter although this method does not require nonlinear optimization it has a demerit in that the order of a high pass filter becomes high in the second approach two filters are optimized simultaneously using a lagrange newton method we can design pr filter banks that have the same length the pr constraint is also formulated as a linear and nonlinear equation of the analysis filter coefficients finally some design examples are included

efficient reconstruction of band limited sequences from nonuniformly decimated versions by use of polyphase filter banks an efficient polyphase structure for the reconstruction of a band limited sequence from a nonuniformly decimated version is developed theoretically the reconstruction involves the implementation of a bank of multilevel filters and it is shown that how all these reconstruction filters can be obtained at the cost of one mth band low pass filter and a constant matrix multiplier the resulting structure is therefore more general than previous schemes in addition the method offers a direct means of controlling the overall reconstruction distortion t z by appropriate design of a low pass prototype filter p z extension of these results to multiband band limited signals and to the case of nonconsecutive nonuniform subsampling are also summarized along with generalizations to the multidimensional case design examples are included to demonstrate the theory and the complexity of the new method is seen to be much lower than earlier ones

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

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

an overview of wavelet based multiresolution analyses in this paper an overview of wavelet based multiresolution analyses is presented first the continuous wavelet transform in its simplest form is discussed then the definition of a multiresolution analysis is given and how wavelets fit into it is shown the authors take a closer look at orthogonal biorthogonal and semiorthogonal wavelets the fast wavelet transform wavelets on an interval multidimensional wavelets and wavelet packets are discussed several examples of wavelet families are introduced and compared finally the essentials of two major applications are outlined data compression and compression of linear operators

the generalized 1 1 dimensional and 2 1 dimensional ito equations the multiple exp function algorithm as a generalization of hirota s perturbation scheme is used to construct multiple wave solutions to the generalized 1 1 dimensional and 2 1 dimensional ito equations some of the resulting solutions involve generic phase shifts and wave frequencies

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

dirac equation in external vector fields separation of variables the method of separation of variables in the dirac equation in the external vector fields is developed through the search for exact solutions the essence of the method consists of the separation of the first order matricial differential operators that define the dependence of the dirac bispinor on the related variables but commutation of such operators with the operator of the equations or between them is not assumed this approach which is perfectly justified in the presence of gravitational fields permits one to prove rigorous theorems about necessary and sufficient conditions on the field functions that allow one to separate variables in the dirac equation in analogous investigations by other authors bagrov et al exact solutions of relativistic wave equations nauka novosibirst 1982 for electromagnetic fields an essential demand related to the operators that define the dependence of the bispinor on the separated variables is the demand for the commutation of a complete set of operators betw

on m channel linear phase fir filter banks and application in image compression this paper investigates the theory and structure of a large subclass of m channel linear phase perfect reconstruction fir filter banks systems with analysis and synthesis filters of length l sub i k sub i m spl beta where spl beta is an arbitrary integer 0 spl les spl beta m and k sub i is any positive integer for this subclass of systems we first investigate the necessary conditions for the existence of linear phase perfect reconstruction filter banks lpprfbs next we develop a complete and minimal factorization for all even channel linear phase paraunitary systems the most general lapped orthogonal transforms to date finally several design examples as well as comparisons with previous generalized lapped orthogonal transforms genlots in image compression are presented to confirm the validity of the theory

a new approach to the design of critically sampled m channel uniform band perfect reconstruction linear phase fir filter banks a new approach is presented for the design of uniform band m channel perfect reconstruction pr fir filter banks employing linear phase analysis and synthesis filters the technique designs on the impulse responses of the analysis filters directly the design problem is formulated as an optimization program the filter bank s pr feature can either be implicitly enforced through a set of mathematical relationships among the analysis filters coefficients or through a set of constraints in the optimization program the former approach results in a filter bank whose pr feature s dependency on hardware and software is eliminated or at least minimized the synthesis filters are then obtained by a set of relationships that describe each synthesis filter as a function of the analysis filters the criterion for optimality is least squares where the square of the difference between the ideal and actual frequency responses is integrated over the appropriate frequency bands for all m analysis filters and minimized

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

families of biorthogonal wavelets several families of biorthogonal wavelet bases are constructed with various properties in particular for a given filter fn of finite length 2n 1 a parametric family of dual filters fvn of length 2n 1 is constructed the parametric nature of the dual filters makes it possible to design the optimum dual filter fv0n corresponding to a fixed filter fn

a fast design method for perfect reconstruction uniform cosine modulated filter banks in this correspondence we present a new and fast design algorithm for perfect reconstruction pr maximally decimated uniform cosine modulated filter banks perfect reconstruction is obtained within arithmetic machine precision the new design does not need numerical optimization routines and is significantly faster than a competing method based on second order cone programming socp the proposed design algorithm finds the optimum solution by iteratively solving a quadratic programming problem with linear equality constraints by a special modification of the basic algorithm we obtain pr filter banks with high stopband attenuations in addition fast convergence is verified by designing pr filter banks with up to 128 channels

linc a common theory of transform and subband coding a common theory of lapped orthogonal transforms lots and critically sampled filter banks called l into n coding linc is presented the theory includes a unified analysis of both coding methods and identity relations between the transform inverse transform analysis filter bank and synthesis filter bank a design procedure for linc analysis synthesis systems which satisfy the conditions for perfect reconstruction is developed the common linc theory is used to define an ideal linc system which is used together with the power spectral density of the input signal to calculate theoretical bounds for the coding gain a generalized overlapping block transform obt with time domain aliasing cancellation tdac is used to approximate the ideal linc a generalization of the obt includes multiple block overlap and additional windowing a recursive design procedure for windows of arbitrary lengths is presented the coding gain of the generalized obt is higher than that of the karhunen loeve transform klt and close to the theoretical bounds for linc in the case of image coding the generalized obt reduces the blocking effects when compared with the dct

optimized nonorthogonal transforms for image compression the transform coding of images is analyzed from a common standpoint in order to generate a framework for the design of optimal transforms it is argued that all transform coders are alike in the way they manipulate the data structure formed by transform coefficients a general energy compaction measure is proposed to generate optimized transforms with desirable characteristics particularly suited to the simple transform coding operation of scalar quantization and entropy coding it is shown that the optimal linear decoder inverse transform must be an optimal linear estimator independent of the structure of the transform generating the coefficients a formulation that sequentially optimizes the transforms is presented and design equations and algorithms for its computation provided the properties of the resulting transform systems are investigated in particular it is shown that the resulting basis are nonorthogonal and complete producing energy compaction optimized decorrelated transform coefficients quantization issues related to nonorthogonal expansion coefficients are addressed with a simple efficient algorithm two implementations are discussed and image coding examples are given it is shown that the proposed design framework results in systems with superior energy compaction properties and excellent coding results

subband coding of images using asymmetrical filter banks we address the problem of the choice of subband filters in the context of image coding the ringing effects that occur in subband based compression schemes are the major unpleasant distortions a new set of two band filter banks suitable for image coding applications is presented the basic properties of these filters are linear phase perfect reconstruction asymmetric length and maximum regularity the better overall performances compared to the classical qmf subband filters are explained the asymmetry of the filter lengths results in a better compaction of the energy especially in the highpass subbands moreover the quantization error is reduced due to the short lowpass synthesis filter the undesirable ringing effect is considerably reduced due to the good step response of the synthesis lowpass filter the proposed design takes into account the statistics of natural images and the effect of quantization errors in the reconstructed images which explains the better coding performance

frame theoretic analysis of oversampled filter banks we provide a frame theoretic analysis of oversampled finite impulse response fir and infinite impulse response fir uniform filter banks fbs our analysis is based on a new relationship between the fbs polyphase matrices and the frame operator corresponding to an fb for a given oversampled analysis fb we present a parameterization of all synthesis fbs providing perfect reconstruction we find necessary and sufficient conditions for an oversampled fb to provide a frame expansion a new frame theoretic procedure for the design of paraunitary fbs from given nonparaunitary fbs is formulated we show that the frame bounds of an fb can be obtained by an eigen analysis of the polyphase matrices the relevance of the frame bounds as a characterization of important numerical properties of an fb is assessed by means of a stochastic sensitivity analysis we consider special cases in which the calculation of the frame bounds and synthesis filters is simplified finally simulation results are presented

subband decomposition of signals with generalized sampling the authors generalize down and up sampling operations by proposing block sampling perfect reconstruction conditions for two band subband coding with block sampling are derived by generalizing the sampling operation new degrees of freedom are introduced and as a result filter banks which were not previously possible become possible generalized down sampler introduces different aliasing components than that of the traditional down sampler this can be used to ease some the requirements of the filter bank design problem a constructive sampling method is proposed so that coprimeness of the transfer functions of the analysis filter banks is not only a necessary but also sufficient condition for perfect reconstruction the results are extended to the case where the filter banks are linear periodically time varying the multichannel case is analyzed and the relation between unimodular matrices and perfect reconstruction filter banks is discussed

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

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

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

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

reconstruction of signals after filtering and sampling rate reduction a wave digital filter wdf two port has two input and two output terminals by using these to appropriately connect a wdf to its transpose the filtering effect is fully compensated except for an all pass phase and this independently of any selectivity requirement this remains true if at the points of interconnection every second sample is reduced to zero provided that a certain condition easily to be satisfied exactly is fulfilled very efficient solutions are then possible the method is of interest e g for subband coding

efficient wide band fir lti systems derived via multi rate techniques and sparse bandpass filters this correspondence introduces efficient realizations of wide band lti systems they are single rate realizations but derived via multi rate techniques and sparse bandpass filters the realizations target mid band systems with narrow don t care bands near the zero and nyquist frequencies design examples for fractional order differentiators demonstrate substantial complexity savings as compared to the conventional minimax optimal direct form realizations

wronskian and grammian solutions to a 3 1 dimensional generalized kp equation abstract wronskian and grammian formulations are established for a 3 1 dimensional generalized kp equation based on the plucker relation and the jacobi identity for determinants generating functions for matrix entries satisfy a linear system of partial differential equations involving a free parameter examples of wronskian and grammian solutions are computed and a few particular solutions are plotted

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

optimal design of oversampled synthesis fbs with lattice structure constraints oversampled filter banks fbs with lattice structure allow fast implementation and guarantee the linear phase and perfect reconstruction property for such fbs the synthesis fb is not unique this nonuniqueness provides extra design freedom for optimal reduction of subband noise required in many applications however the existing design methods of synthesis fb for optimal reduction of subband noise cannot guarantee the linear phase and lattice structure property this paper studies the design of oversampled synthesis fbs with guaranteed lattice structure for optimal reduction of subband noise based on a state space parameterization of all synthesis fbs with lattice structure the explicit formulae are derived to find the optimal solution for noise reduction numerical examples are presented to demonstrate the effectiveness of the new design methods obtained

oversampled filter banks perfect reconstruction oversampled filter banks are equivalent to a particular class of frames in l sup 2 z these frames are the subject of this paper first the necessary and sufficient conditions of a filter bank for implementing a frame or a tight frame expansion are established as well as a necessary and sufficient condition for perfect reconstruction using fir filters after an fir analysis complete parameterizations of oversampled filter banks satisfying these conditions are given further we study the condition under which the frame dual to the frame associated with an fir filter bank is also fir and give a parameterization of a class of filter banks satisfying this property then we focus on non subsampled filter banks non subsampled filter banks implement transforms similar to continuous time transforms and allow for very flexible design we investigate the relations of these filter banks to continuous time filtering and illustrate the design flexibility by giving a procedure for designing maximally flat two channel filter banks that yield highly regular wavelets with a given number of vanishing moments

lagrange multiplier approaches to the design of two channel perfect reconstruction linear phase fir filter banks the authors present two approaches to the design of two channel perfect reconstruction linear phase finite impulse response fir filter banks both approaches analyze and design the impulse responses of the analysis filter bank directly the synthesis filter bank is then obtained by simply changing the signs of odd order coefficients in the analysis filter bank the approach deals with unequal length filter banks by designing the lower length filters first one can take advantage of the fact that the number of variables for designing the higher length filters is more than the number of perfect reconstruction constraint equations the second approach generalizes the first and covers the design for all parts of linear phase perfect reconstruction constraint equations

a general formulation of modulated filter banks this paper presents a general framework for maximally decimated modulated filter banks the theory covers the known classes of cosine modulation and relates them to complex modulated filter banks the prototype filters have arbitrary lengths and the overall delay of the filter bank is arbitrary within fundamental limits necessary and sufficient conditions for perfect reconstruction pr are derived using the polyphase representation it is shown that these pr conditions are identical for all types of modulation modulation based on the discrete cosine transform dct both dct iii dct iv and dct i dct ii and modulation based on the modified discrete fourier transform mdft a quadratic constrained design method for prototype filters yielding pr with arbitrary length and system delay is derived and design examples are presented to illustrate the tradeoff between overall system delay and stopband attenuation subchannelization

antipodal paraunitary matrices and their application to ofdm systems paraunitary pu matrices and filterbanks have played an important role in many applications this paper studies a special class of pu matrices called antipodal paraunitary apu matrices apu matrices are pu matrices whose coefficient matrices consist of spl plusmn 1 only several new methods will be introduced for the construction of apu matrices in particular a new method based on the butterfly structure that has a low cost implementation is proposed moreover one application of apu matrices in precoded orthogonal frequency division multiplexing ofdm systems will be considered by using an apu precoding matrix the complexity will be very low and experiments show that the use of apu matrices can greatly enhance the performance

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

non maximally decimated analysis synthesis filter banks applications in wideband digital filtering we present a new class of highly effective and low complexity digital filters for processing very wideband signals digital filtering for wideband signals is often limited by the number of arithmetic operations that has to be performed per input sampling interval we will show the new architecture permits filtering to be performed on partitioned spectral segments of the input signal at significantly reduced sample rate the digital filtering will be shown to include various tasks such as linear non linear phase finite impulse response fir filtering fractional delay filter among others the proposed technique utilizes the framework of non maximally decimated filter banks nmdfbs with perfect reconstruction pr property which makes the filter bank design simpler and more flexible compact representation for the generalized dft based nmdfbs as well as its efficient polyphase implementation will be provided the digital filtering is made possible by incorporating the desired intermediate processing elements in between the analysis and synthesis filter banks we will show this embedded intermediate processing elements can achieve spectral shaping signal manipulation task which is a generalization of the concept of manipulating a digital filter in the frequency domain we also analyze both analytically and experimentally the spectral shaping accuracy based on different design strategies

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

oversampled paraunitary dft filter banks a general construction algorithm and some specific solutions oversampled filter banks are at the center of essential signal processing systems either for source coding or for transmission applications among this class of filter banks fbs the case of oversampled discrete fourier transform dft filter banks is particularly important since it leads to efficient algorithm implementations however up to now no general construction algorithm has been proposed to exhaustively cover the large set of the various solutions offered by this family of fbs the paper recalls in detail the precise features of the rectangular paraunitary matrices that are specific of this type of systems a parametrization method is proposed involving transformations being either rotations or shifts that depend upon the oversampling ratio and upon the prototype filter length this shows that several factorizations exist leading to solutions of different dimension orthogonal and paraunitary patterns are introduced and using a sequence of transformations applied to a pattern stap an algorithm is derived that allows us to get the exhaustive set of solutions for various values of the oversampling ratio and prototype filter length furthermore for some oversampling ratios explicit expressions of some solutions are also provided that are valid whatever the maximum length of the prototype filter finally design examples for linear and nonlinear phase prototype filters are presented

dirac equation in external fields separation of variables in nondiagonal metrics the algebraic method of separation of variables in the dirac equation proposed in earlier works by one of the present authors theor math phys 70 204 1987 j math phys 30 2132 1989 is developed for the space time with nondiagonal metrics the essence of the method consists of the separation of the first order matricial differential operators that define the dependence of the dirac s bispinor on the related variables in contrast to some other authors the pairs of operators are commuted on each step of separation including the variables mixed by nondiagonal elements of fundamental tensor of space time there are reasons to believe that it must be some local similarity transformation connected these commuted operators with noncommuted corresponding operators of other authors although such transformation in view of mathematical difficulties of problem in general were not successfully found

least squares approximation of perfect reconstruction filter banks designing good causal filters for perfect reconstruction pr filter banks is a challenging task due to the unusual nature of the design constraints we present a new least squares ls design methodology for approximating prfbs that avoids most of these difficult constraints the designer first selects a set of subband analysis filters from an almost unrestricted class of rational filters then given some desired reconstruction delay this design procedure produces the causal and rational synthesis filters that result in the best least squares approximation to a prfb this technique is built on a multi input multi output mimo system model for filter banks derived from the filter bank polyphase representation using this model we frame the ls approximation problem for prfbs as a causal ls equalization problem for mimo systems we derive the causal ls solution to this design problem and present an algorithm for computing this solution the resulting algorithm includes a mimo spectral factorization that accounts for most of the complexity and computational cost for this design technique finally we consider some design examples and evaluate their performance

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

eigenvalues and eigenfunctions of the dirac operator on spheres and pseudospheres the dirac equation for an electron on a curved space time may be viewed as an eigenvalue problem for the dirac operator on the spinor fields of the space time a general eigenvalue problem for the dirac operator on a metric manifold m in terms of spinor and tangent fields defined via the clifford algebra is derived herein then it is shown how to solve the dirac eigenvalue problem on an imbedded submanifold n of codimension one in m by solving an eigenvalue equation on m this is applied to the case of a sphere or pseudosphere imbedded in flat space eigenvalues and eigenfunctions for the dirac operator on any sphere or pseudosphere are determined in particular when the pseudosphere is a space time the dirac equation for a free lepton in this space time can be solved the resulting mass spectrum is discrete and depends on the curvature of the space time

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

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

on the synthesis of very sharp decimators and interpolators using the frequency response masking technique decimation and interpolation are very common multirate signal processing operations conventional decimation or interpolation technique using polyphase filters has the advantage that for a given transition band sharpness the filter s computational complexity decreases with increasing interpolation or decimation factor nevertheless if the transition band of the decimation or interpolation filter is very sharp the complexity of the filter may still be very high the complexity of a very sharp filter may be reduced using the frequency response masking frm technique however as shown in this paper for a given transition band sharpness the computational complexity of the classical frm method does not reduce as rapidly as the increase in decimation or interpolation factor in this paper we present a novel variant of the frm technique for interpolation or decimation application in this new variant the computational complexity reduces as rapidly as the interpolation or decimation factor increases the reduction in computational complexity increases with decreasing transition width over an order of magnitude reduction in computational complexity is achieved when compared with conventional polyphase approach in a particular example presented in this paper

design of filter banks using transformation of variables new results the design problem of multirate filter banks can be divided into two parts the first part involves the issue of reconstruction errors the second part involves the issue of designing good quality subband filters there is a class of techniques that we refer to as transformation of variables which satisfactorily addresses the two parts of the design problem this class of filter banks is specified by its prototype filters and transformation kernel it is the flexibility and relative ease in designing the kernel that enables the technique to satisfactorily address the second part of the design problem we present two new methods of designing the kernel that will enhance the design techniques flexibility and effectiveness the first is the combination of the mcclellan transformation and rotation operators the second introduces and uses the concept of directional singular value decomposition svd

moving window based double haar wavelet transform for image processing image denoising is a lively research field the classical nonlinear filters used for image denoising such as median filter are based on a local analysis of the pixels within a moving window recently the research of image denoising has been focused on the wavelet domain compared to the classical nonlinear filters it is based on a global multiscale analysis of images apparently the wavelet transform can be embedded in a moving window thus a moving window based local multiscale analysis is obtained in this paper based on the haar wavelet a class of nonorthogonal multichannel filter bank with its corresponding wavelet shrinkage called lee shrinkage is derived as a special case of this filter bank the double haar wavelet transform is introduced examples show that it is suitable for a moving window based local multiscale analysis used for image denoising edge detection and edge enhancement

m channel nonlinear phase filter banks in image compression structure design and signal extension this paper describes a transform coding technique based on m channel perfect reconstruction filter banks with a nonlinear phase in contrast with linear phase filter banks the nonlinearity provides an extra degree of freedom that can be used to design a more efficient transform we present new lattice structures of paraunitary and perfect reconstruction biorthogonal filter banks which can be implemented with a lower computational complexity and or represented by a few free parameters through the decomposition of the lattice blocks and the displacement across the delay block we further discuss a smooth extension method for nonlinear phase filter banks to obtain a nonexpansive transform the promise of our proposed approaches is demonstrated through several design examples extended signals and compression results

bilinear backlund transformations kink periodic solitary wave and lump wave solutions of the bogoyavlenskii kadomtsev petviashvili equation abstract this paper investigates the bogoyavlenskii kadomtsev petviashvili equation by using the binary bell polynomials method and differential constraint technique several kinds of more general bilinear representations and baklund transformations are presented at the same time multiple wave solutions including the kink periodic solitary wave and bright dark lump wave solutions are constructed the resonant interactions between two kink solitary waves are investigated and exhibited mathematically and graphically lump wave solutions a kind of rational function localized in all directions of the space are obtained by the taylor expansion of the kink periodic solitary wave also we discuss the dynamical behavior of the lump wave solutions and their structures

narrowband interference suppression using undecimated wavelet packets in direct sequence spread spectrum receivers a new algorithm for narrowband interference suppression in direct sequence spread spectrum ds ss communications is presented the algorithm combines the undecimated wavelet packet transform uwpt with frequency shifts to confine the interference energy in a subband that is subsequently eliminated computer simulation shows a robust performance that appears to be independent of the interference frequency

linear periodic systems and multirate filter design we present a systematic procedure for the design of filters intended for multirate systems this procedure is motivated by viewing the equiripple design of filters in linear time invariant systems as a process of obtaining optimum minimax filters for a class of bounded energy input signals the philosophy of designing optimum minimax filters for classes of input signals is extended to multirate systems which are not time invariant we develop a generalized fourier analysis appropriate for linear periodic systems and use it to derive new error criteria for multirate filter design using such criteria yields optimum minimax multirate filters for the input signal class the utility of our method is demonstrated by using it to analyze several multirate systems we give numerical results on the design of a multirate implementation of a narrowband filter and compare our work to previous work on multirate filter design our numerical analysis is based upon a new formulation of the design as a semi infinite linear programming problem

kronecker products unitary matrices and signal processing applications discrete unitary transforms are extensively used in many signal processing applications and in the development of fast algorithms kronecker products have proved quite useful in this semitutorial paper we briefly review properties of kronecker products and direct sums of matrices which provide a compact notation in treating patterned matrices a generalized matrix product which inherits some useful algebraic properties from the standard kronecker product and allows a large class of discrete unitary transforms to be generated from a single recursion formula is then introduced the notation is intimately related to sparse matrix factorizations and examples are included illustrating the utility of the new notation in signal processing applications finally some novel characteristics of hadamard transforms and polyadic permutations are derived in the framework of kronecker products

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

multirate filter banks with block sampling multirate filter banks with block sampling were recently studied by khansari and leon garcia 1993 in this paper we want to systematically study multirate filter banks with block sampling by studying general vector filter banks where the input signals and transfer functions in conventional multirate filter banks are replaced by vector signals and transfer matrices respectively we show that multirate filter banks with block sampling studied by khansari and leon garcia are special vector filter banks where the transfer matrices are pseudocirculant we present some fundamental properties for the basic building blocks such as noble identities interchangeability of down up sampling polyphase representations of m channel vector filter banks and multirate filter banks with block sampling we then present necessary and sufficient conditions for the alias free property finite impulse response fir systems with fir inverses paraunitariness and lattice structures for paraunitary vector filter banks we also present a necessary and sufficient condition for paraunitary multirate filter banks with block sampling as an application of this theory we present all possible perfect reconstruction delay chain systems with block sampling we also show some examples that are not paraunitary for conventional multirate filter banks but are paraunitary for multirate filter banks with proper block sampling in this paper we also present a connection between vector filter banks and vector transforms studied by li vector filter banks also play important roles in multiwavelet transforms and vector subband coding

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

nonuniform multirate filter banks analysis and design with an spl hscr sub spl infin performance measure a general multirate filter bank system with multiple channels and nonuniform bands is studied with the reconstruction performance measured by the worst case energy gain spl hscr sub spl infin norm of the error system between the multirate system and a pure time delay system using blocking and polyphase decomposition we associate with the multirate system an equivalent linear time invariant lti system whose transfer matrix can be computed by a simple procedure based on this lti system the optimal design problem can be reduced to one of spl hscr sub spl infin optimization which can be solved by ready made software in many cases for illustration optimal synthesis filters are designed for a two channel example and their properties are studied in detail

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

multiple soliton solutions for the kp equation by hirota s bilinear method and by the tanh coth method abstract in this work we use the hirota s direct method combined with the simplified version of this method to determine the n soliton solutions n 1 2 3 4 5 for the kadomtsev petviashvili kp equation the one soliton solutions will be handled by the tanh coth method the work emphasizes the power of the two methods that will be used

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

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

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

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

a direct design of oversampled perfect reconstruction fir filter banks we address a problem to find optimal synthesis filters of oversampled uniform finite impulse response fir filter banks fbs yielding perfect reconstruction pr when we are given an analysis fb in the case where all the filters have the same length that is twice a factor of downsampling we show that in this class of fbs a synthesis fb that achieves pr can be found in closed form with elementary matrix operations unlike conventional design methods with numerical optimization this framework allows filter coefficients to be complex as well as real due to the extra degrees of freedom in a synthesis fb provided by oversampling we can determine optimal coefficients of synthesis filters that meet certain criteria we introduce in this paper two criteria variance of additive noise and stopband attenuation we show theoretical results of optimal synthesis filters that minimize these criteria and design examples of oversampled linear phase fir fbs and dft modulated fbs moreover we discuss applications to signal reconstruction from incomplete channel data in transmission and inverse transform of windowed discrete fourier transform with 50 overlapping

diversity of interaction solutions to the 2 1 dimensional ito equation abstract we aim to show the diversity of interaction solutions to the 2 1 dimensional ito equation based on its hirota bilinear form the proof is given through maple symbolic computations an interesting characteristic in the resulting interaction solutions is the involvement of an arbitrary function special cases lead to lump solutions lump soliton solutions and lump kink solutions two illustrative examples of the resulting solutions are displayed by three dimensional plots and contour plots