algebraic signal processing theory sampling for infinite and finite 1 d space we derive a signal processing framework called space signal processing that parallels time signal processing as such it comes in four versions continuous discrete infinite finite each with its own notion of convolution and fourier transform as in time these versions are connected by sampling theorems that we derive in contrast to time however space signal processing is based on a different notion of shift called space shift which operates symmetrically our work rigorously connects known and novel concepts into a coherent framework most importantly it shows that the sixteen discrete cosine and sine transforms are the space equivalent of the discrete fourier transform and hence can be derived by sampling the platform for our work is the algebraic signal processing theory an axiomatic approach and generalization of linear signal processing that we recently introduced

dct algorithms for composite sequence lengths this paper presents fast algorithms for the type ii and iii discrete cosine transforms of composite sequence length in particular a radix q algorithm where q is an odd integer is derived for uniform or mixed radix decomposition of the discrete cosine transform by combining the radix q and radix 2 algorithms a general decomposition method for any composite length is developed reduction of computational complexity can be achieved for many sequence lengths compared with that needed by the well known radix 2 algorithm furthermore both the proposed and chan and siu s 1993 mixed radix algorithms achieve the same computational complexity for n 3 2 sup p and 9 2 sup p however our algorithm uses a simpler decomposition approach and provides a wider range of choices of sequence lengths

bottom up tree pushdown automata classification and connection with rewrite systems we define different types of bottom up tree pushdown automata and study their connections with rewrite systems along this line of research we complete and generalize the results of gallier book and salomaa we define the notion of a tail reduction free trf rewrite system using the decidability of ground reducibility we prove the decidability of the trf property monadic rewrite systems of book gallier and salomaa become a natural particular case of trf rewrite systems we associate a deterministic bottom up tree pushdown automaton with any left linear trf rewrite system finally we generalize monadic rewrite systems by introducing the notion of a semi monadic rewrite system and show that like a monadic rewrite system it preserves recognizability

shifted fourier transform based tensor algorithms for the 2 d dct in this paper tensor algorithms for calculating the two dimensional 2 d discrete cosine transform dct are presented the tensor approach is based on the concept of the covering revealing the transforms which yields in particular the splitting of the shifted 2 sup r spl times 2 sup r point fourier and cosine transforms into 2 sup r 1 3 one dimensional 1 d incomplete 2 sup r point transforms the multiplicative complexity of the 2 d 2 sup r spl times 2 sup r point discrete cosine transforms in terms of the tensor representation is 4 sup r 3 2 sup r 2 r sup 2 7r 14 which is reduced to 4 sup r 8 3 2 sup r r sup 2 7r 10 20 3 when using the improved tensor algorithm the multiplicative complexity in the general l sup r spl times l sup r case with a prime l 2 as well as in the l sub 1 l sub 2 spl times l sub 1 l sub 2 case with arbitrary co prime l sub 1 l sub 2 1 is provided the examples of the tensor algorithms for calculating the 8 spl times 8 point dct through 104 88 and 84 multiplications are given in detail based on the proposed concept the fast algorithm for calculating the 1 d dct i is also developed the multiplicative complexity of the 2 sup r point dct i is 2 sup r 1 r 2 r 5 2 8 the comparative estimates with the known algorithms are given

autoregressive modeling of temporal envelopes autoregressive ar models are commonly obtained from the linear autocorrelation of a discrete time signal to obtain an all pole estimate of the signal s power spectrum we are concerned with the dual frequency domain problem we derive the relationship between the discrete frequency linear autocorrelation of a spectrum and the temporal envelope of a signal in particular we focus on the real spectrum obtained by a type i odd length discrete cosine transform dct io which leads to the all pole envelope of the corresponding symmetric squared hilbert temporal envelope a compact linear algebra notation for the familiar concepts of ar modeling clearly reveals the dual symmetries between modeling in time and frequency domains by using ar models in both domains in cascade we can jointly estimate the temporal and spectral envelopes of a signal we model the temporal envelope of the residual of regular ar modeling to efficiently capture signal structure in the most appropriate domain

signal flow graph approach to efficient and forward stable dst algorithms abstract in this paper signal flow graphs are mainly addressed for the efficient and forward stable discrete sine transform dst algorithms having sparse scaled orthogonal rotational rotational reflection and butterfly matrices in electrical engineering theoretical computer science control theory system engineering etc we often use signal flow graphs as a modeling tool which interconnects the system components or represents the realization of a system as electronic devices the objective in this paper is to establish the connection between algebraic operations used in sparse and scaled orthogonal factorizations of dst i iv matrices with the signal flow graph building blocks this paper elaborates signal flow graphs for the foundational frameworks with n 8 and n 16 and use these together with the digital structures to describe the dst algorithms having a n 1 point signal flow graph for dst i and n point signal flow graphs for dst ii iv the presented dst algorithms are completely recursive and solely based on corresponding matrices dst i iv these dst algorithms have low arithmetic complexity especially the low number of multiplications and significant speed improvement factor as opposed to most existing algorithms finally this paper establishes that the presented algorithms are forward stable dst algorithms

the algebraic approach to the discrete cosine and sine transforms and their fast algorithms it is known that the discrete fourier transform dft used in digital signal processing can be characterized in the framework of the representation theory of algebras namely as the decomposition matrix for the regular module mathbb c z n mathbb c x x n 1 this characterization provides deep insight into the dft and can be used to derive and understand the structure of its fast algorithms in this paper we present an algebraic characterization of the important class of discrete cosine and sine transforms as decomposition matrices of certain regular modules associated with four series of chebyshev polynomials then we derive most of their known algorithms by pure algebraic means we identify the mathematical principle behind each algorithm and give insight into its structure our results show that the connection between algebra and digital signal processing is stronger than previously understood

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

decision procedures for elementary sublanguages of set theory ix unsolvability of the decision problem for a restricted subclass of the δ0 formulas in set theory etude des procedures de decision pour les sous langages elementaires de la theorie des ensembles insolubilite du probleme de decision pour une sous classe restreinte des 0 formules de la theorie des ensembles etablissement de bornes

algebraic signal processing theory 1 d space in our paper titled ldquoalgebraic signal processing theory foundation and 1 d timerdquo appearing in this issue of the ieee transactions on signal processing we presented the algebraic signal processing theory an axiomatic and general framework for linear signal processing the basic concept in this theory is the signal model defined as the triple a m phi where a is a chosen algebra of filters m an associated a module of signals and phi is a generalization of the z transform each signal model has its own associated set of basic sp concepts including filtering spectrum and fourier transform examples include infinite and finite discrete time where these notions take their well known forms in this paper we use the algebraic theory to develop infinite and finite space signal models these models are based on a symmetric space shift operator which is distinct from the standard time shift we present the space signal processing concepts of filtering or convolution ldquoz transform rdquo spectrum and fourier transform for finite length space signals we obtain 16 variants of space models which have the 16 discrete cosine and sine transforms dcts dsts as fourier transforms using this novel derivation we provide missing signal processing concepts associated with the dcts dsts establish them as precise analogs to the dft get deep insight into their origin and enable the easy derivation of many of their properties including their fast algorithms

fractional cosine sine and hartley transforms in previous papers the fourier transform ft has been generalized into the fractional fourier transform frft the linear canonical transform lct and the simplified fractional fourier transform sfrft because the cosine sine and hartley transforms are very similar to the ft it is reasonable to think they can also be generalized by the similar way we introduce several new transforms they are all the generalization of the cosine sine or hartley transform we first derive the fractional cosine sine and hartley transforms frct frst frht they are analogous to the frft then we derive the canonical cosine and sine transforms cct cst they are analogous to the lct we also derive the simplified fractional cosine sine and hartley transforms sfrct sfrst sfrht they are analogous to the sfrft and have the advantage of real input real output we also discuss the properties digital implementation and applications e g the applications for filter design and space variant pattern recognition of these transforms the transforms introduced in this paper are very efficient for digital implementation we can just use one half or one fourth of the real multiplications required for the frft and lct to implement them when we want to process even odd or pure real imaginary functions we can use these transforms instead of the frft and lct besides we also show that the frct frst cct cst and sfrct sfrst are also useful for the one sided t spl isin 0 spl infin signal processing

an iterative shrinkage approach to total variation image restoration the problem of restoration of digital images from their degraded measurements plays a central role in a multitude of practically important applications a particularly challenging instance of this problem occurs in the case when the degradation phenomenon is modeled by an ill conditioned operator in such a situation the presence of noise makes it impossible to recover a valuable approximation of the image of interest without using some a priori information about its properties such a priori information commonly referred to as simply priors is essential for image restoration rendering it stable and robust to noise moreover using the priors makes the recovered images exhibit some plausible features of their original counterpart particularly if the original image is known to be a piecewise smooth function one of the standard priors used in this case is defined by the rudin osher fatemi model which results in total variation tv based image restoration the current arsenal of algorithms for tv based image restoration is vast in this present paper a different approach to the solution of the problem is proposed based upon the method of iterative shrinkage aka iterated thresholding in the proposed method the tv based image restoration is performed through a recursive application of two simple procedures viz linear filtering and soft thresholding therefore the method can be identified as belonging to the group of first order algorithms which are efficient in dealing with images of relatively large sizes another valuable feature of the proposed method consists in its working directly with the tv functional rather then with its smoothed versions moreover the method provides a single solution for both isotropic and anisotropic definitions of the tv functional thereby establishing a useful connection between the two formulae finally a number of standard examples of image deblurring are demonstrated in which the proposed method can provide restoration results of superior quality as compared to the case of sparse wavelet deconvolution

multiplication free approximate algorithms for compressed domain linear operations on images we propose a method for devising approximate multiplication free algorithms for compressed domain linear operations on images e g downsampling translation filtering etc we demonstrate that the approximate algorithms give output images that are perceptually nearly equivalent to those of the exact processing while the computational complexity is significantly reduced

preservation of subband symmetry in multirate signal coding the article derives necessary and sufficient conditions for a multirate linear phase fir filter bank acting on a symmetrically extended finite length input signal to produce either symmetric or antisymmetric downsampled subbands such techniques have gained popularity for image coding applications and we provide the solution to a key technical problem in terms of the input and filter symmetries and the downsampling ratio

undecidability results for restricted universally quantified formulae of set theory the problem of establishing whether there are sets satisfying a formula in the first order set theoretic language based on which involves only restricted quantifiers and has an equivalent prenex form 0 formula is neither decidable nor semidecidable in fact given any model of zf fa where fa denotes the foundation axiom the set of existential closures of 0 formulae true in the model is a productive set undecidability arises even when dealing with restricted universal quantifiers only provided a predicate is a pair x meaning that x is a pair of distinct sets is added to if satisfiability refers to models of zf fa in which a form of boffa s antifoundation axiom holds then semidecidability fails as well in fact given any such model the set of existential closures of formulae involving only restricted quantifiers and the predicate is a pair which are true in it is a productive set these results are all proved by making use of appropriate codings of turing machine computations in the set theoretic language

the discretew transform four different versions of the discretew transform dwt are introduced the dwt may be decomposed into the discrete cosine transform dct and the discrete sine transform dst eight versions of both dct and dst are introduced for the decomposition of the dwt the relationship among different versions of dwt and their relation with the discrete fourier transform dft are given convolution theorems represented by different versions of the dwt are derived

a fast recursive algorithm for computing the discrete cosine transform the discrete cosine transform dct is widely applied in various fields including image data compression because it operates like the karhunen loeve transform for stationary random data this paper presents a recursive algorithm for dct with a structure that allows the generation of the next higher order dct from two identical lower order dct s as a result the method for implementing this recursive dct requires fewer multipliers and adders than other dct algorithms

linear generalized semi monadic rewrite systems effectively preserve recognizability abstract we introduce the notion of the generalized semi monadic rewrite system which is a generalization of well known rewrite systems the ground rewrite system the monadic rewrite system and the semi monadic rewrite system we show that linear generalized semi monadic rewrite systems effectively preserve recognizability we show that a tree language l is recognizable if and only if there exists a rewrite system r such that r r 1 is a linear generalized semi monadic rewrite system and that l is the union of finitely many r classes we show several decidability and undecidability results on rewrite systems effectively preserving recognizability and on generalized semi monadic rewrite systems for example we show that for a rewrite system r effectively preserving recognizability it is decidable if r is locally confluent moreover we show that preserving recognizability and effectively preserving recognizability are modular properties of linear collapse free rewrite systems finally as a consequence of our results on trees we get that restricted right left overlapping string rewrite systems effectively preserve recognizability

the eigenstructure of finite field trigonometric transforms in this paper we discuss the eigenstructure of finite field trigonometric transforms fftt this transform family includes different types of cosine and sine transforms that correspond in the finite field case to the well known discrete cosine and sine transforms defined over the real numbers more specifically we determine the eigenvalues of the fftt matrices and study their multiplicities we propose procedures for constructing the associated eigenvectors applications of the developed theory to multiuser communication systems and error correcting codes are also suggested

orthogonal modes of frequency modulation and the sturm liouville frequency modulation model sinusoidal signals and complex exponentials play a critical role in lti system theory in that they are eigenfunctions of the lti convolution operator while processing frequency modulated fm waveforms using lti systems restrictive assumptions must be placed on the system so that a quasi eigenfunction approximation holds upon deviation from these assumptions fm waveforms incur significant distortion in this paper a sturm liouville s l model for frequency modulation introduced by the author is extended to a study orthogonal modes of continuous and discrete frequency modulation and b to develop system theoretical underpinnings for fm waveforms these fm modes have the same special connection with respect to the fm s l system operator that complex exponentials have with lti systems and the convolution operator the finite s l fm spectrum or transform that measures the strength of the orthogonal fm modes present in a fm signal analogous to the discrete fourier spectrum for sinusoids is introduced finally similarities between the orthogonal s l fm modes and angular mathieu functions are exposed and a conjecture connecting the two is put forth

total nuclear variation and jacobian extensions of total variation for vector fields we explore a class of vectorial total variation vtv measures formed as the spatial sum of a pixel wise matrix norm of the jacobian of a vector field we give a theoretical treatment that indicates that while color smearing and affine coupling bias often reported as gray scale bias are typically cited as drawbacks for vtv these are actually fundamental to smoothing vector direction i e smoothing hue and saturation in color images additionally we show that encouraging different vector channels to share a common gradient direction is equivalent to minimizing jacobian rank we thus propose total nuclear variation tnv and since nuclear norm is the convex envelope of matrix rank we argue that tnv is the optimal convex regularizer for enforcing shared directions we also propose extended jacobians which use larger neighborhoods than the conventional finite difference operator and we discuss efficient vtv optimization algorithms in simple color image denoising experiments tnv outperformed other common vtv regularizers and was further improved by using extended jacobians tnv was also competitive with the method of non local means often outperforming it by 0 25 to 2 db when using extended jacobians

generalized discrete fourier transform with nonlinear phase constant modulus transforms like discrete fourier transform dft walsh transform and gold codes have been successfully used over several decades in several engineering applications including discrete multi tone dmt orthogonal frequency division multiplexing ofdm and code division multiple access cdma communications systems in this paper we present a generalized framework for dft called generalized dft gdft with nonlinear phase by exploiting the phase space we show that gdft offers sizable correlation improvements over dft walsh oppermann and gold codes leading to better performance in all multi carrier communications scenarios investigated we also highlight how known constant modulus orthogonal transforms are special solutions of the proposed gdft framework moreover we introduce practical design methods offering computationally efficient implementations of gdft as enhancements to dft we conclude the paper with examples of communications applications where gdft is shown to outperform dft and other known constant modulus bases

anti symmetric multivariate trigonometric functions and corresponding fourier transforms four families of special functions depending on n variables are studied we call them symmetric and antisymmetric multivariate sine and cosine functions they are given as determinants or antideterminants of matrices whose matrix elements are sine or cosine functions of one variable each these functions are eigenfunctions of the laplace operator satisfying specific conditions at the boundary of a certain domain f of the n dimensional euclidean space discrete and continuous orthogonality on f of the functions within each family allows one to introduce symmetrized and antisymmetrized multivariate fourier like transforms involving the symmetric and antisymmetric multivariate sine and cosine functions

restructured recursive dct and dst algorithms the discrete cosine transform dct and the discrete sine transform dst have found wide applications in speech and image processing as well as telecommunication signal processing for the purpose of data compression feature extraction image reconstruction and filtering in this paper we present new recursive algorithms for the dct and the dst the proposed method is based on certain recursive properties of the dct coefficient matrix and can be generalized to design recursive algorithms for the 2 d dct and the 2 d dst these new structured recursive algorithms are able to decompose the dct and the dst into two balanced lower order subproblems in comparison to previous research works therefore when converting our algorithms into hardware implementations we require fewer hardware components than other recursive algorithms finally we propose two parallel algorithms for accelerating the computation

on the interrelationships among a class of convolutions in this paper some interrelationships among a class of circular operations are investigated based on matrix formulation it is shown that a class of convolutions representing forward backward and convolution correlation of two periodic sequences may be related to each other in terms of discrete transforms having the circular convolution property the results obtained are useful in efficient realization of adaptive digital filters using fast transforms

eigenvalues and eigenvectors of generalized dft generalized dht dct iv and dst iv matrices in this paper the eigenvalues and eigenvectors of the generalized discrete fourier transform gdft the generalized discrete hartley transform gdht the type iv discrete cosine transform dct iv and the type iv discrete sine transform dst iv matrices are investigated in a unified framework first the eigenvalues and their multiplicities of the gdft matrix are determined and the theory of commuting matrices is applied to find the real symmetric orthogonal eigenvectors set that constitutes the discrete counterpart of hermite gaussian function then the results of the gdft matrix and the relationships among these four unitary transforms are used to find the eigenproperties of the gdht dct iv and dst iv matrices finally the fractional versions of these four transforms are defined and an image watermarking scheme is proposed to demonstrate the effectiveness of fractional transforms

fast algorithms for type iii dct of composite sequence lengths this correspondence presents an odd factor algorithm for the type iii discrete cosine transform dct for uniform or mixed radix decomposition by jointly using the old factor and the existing radix 2 algorithms a general decomposition method for arbitrarily composite sequence length is developed a reduction of computational complexity can be achieved compared with that needed by other reported algorithms for m 2 sup m the decomposition approach has a simple computational structure and supports a wider range of choices for different sequence lengths

fast imdct and mdct algorithms a matrix approach this paper presents a systematic investigation of the modified discrete cosine transform inverse modified discrete cosine transform mdct imdct algorithm using a matrix representation this approach results in new understanding of the mdct imdct enables the development of new algorithms and makes clear the connection between the algorithms we represent in a matrix form the imdct as the product of the type iv dct with simple scaling sign changing and permutation operations such that fast algorithms for the type iv dct can be simply modified for the imdct and vice versa then the simple symmetry and inversion properties of the type iv dct are used to develop new algorithms and establish the connection between existing fast imdct algorithms this approach also enables us to show that mdct and imdct share common core operation and present an efficient architecture for implementing both the mdct and the imdct in one hardware

fast construction of fejer and clenshaw curtis rules for general weight functions the main purpose of this paper is to compute the weights of clenshaw curtis and fejer type quadrature rules via dct and dst for general weight functions w the approach is different from that used by waldvogel 2006 in 10 where the author considered the computation of these sets only for the legendre weight w 1 using dft arguments

techniques of computable set theory with applications to proof verification the original goal was the design and implementation of a proof verifier based on the very expressive formalism of set theory with a powerful inferential engine containing a wide collection of decision procedures cf 38 most of the research concentrated though on theoretical aspects namely the decision problem for fragments of set theory as shown by the numerous publications culminated in the monograph 22 which gives a complete account of the state of the art of computable set theory up to 1989 in this paper we present some recent techniques and compare them with early approaches to the decision problem in giving applications of such techniques we sort of review the decision problem in set theory and related areas with respect to the classifications of the models one is looking for namely pure set models with no individuals or stratified models and with respect to the structure of the formulae considered namely whether explicit quantification is allowed or not it must be noted however that the present survey is far from being complete for instance decision procedures relative to weak axiomatizations of set theory cf 54 55 57 62 64 65 or in presence of the anti foundation axiom are not surveyed cf 56 61 also undecidability results are not reviewed here cf 19 59 63 the paper is organized as follows section 2 introduces basic concepts about set hierarchies models realizations and the decision problem

two dimensional symmetric and antisymmetric generalizations of sine functions the properties of two dimensional generalizations of sine functions that are symmetric or antisymmetric with respect to permutations of their two variables are described it is shown that the functions are orthogonal when integrated over a finite region f of the real euclidean space and that they are discretely orthogonal when summed up over a lattice of any density in f the decomposability of the products of functions into their sums is shown by explicitly decomposing products of all types the formalism is set up for fourier like expansions of the digital data over two dimensional lattices in f analogs of the common cosine transforms of types i iv are described continuous interpolation of the digital data is studied

fast dct domain filtering using the dct and the dst a method for efficient spatial domain filtering directly in the discrete cosine transform dct domain is developed and proposed it consists of using the discrete sine transform dst and the dct for transform domain processing on the in jpeg basis of the previously derived convolution multiplication properties of discrete trigonometric transforms the proposed scheme requires neither zero padding of the input data nor kernel symmetry it is demonstrated that in typical applications the proposed algorithm is significantly more efficient than the conventional filtered spatial domain and earlier proposed dct domain methods the proposed method is applicable to any dct based image compression standard such as jpeg mpeg and h 261

tridiagonal commuting matrices and fractionalizations of dct and dst matrices of types i iv v and viii in this paper we first establish new relationships in matrix forms among discrete fourier transform dft generalized dft gdft and various types of discrete cosine transform dct and discrete sine transform dst matrices two new independent tridiagonal commuting matrices for each of dct and dst matrices of types i iv v and viii are then derived from the existing commuting matrices of dft and gdft with these new commuting matrices the orthonormal sets of hermite like eigenvectors for dct and dst matrices can be determined and the discrete fractional cosine transform dfrct and the discrete fractional sine transform dfrst are defined the relationships among the discrete fractional fourier transform dfrft fractional gdft and various types of dfrct and dfrst are developed to reduce computations for dfrft and fractional gdft

a local spectral inversion of a linearized tv model for denoising and deblurring we propose a model for denoising and deblurring it consists of a system of linear partial differential equations with locally constant coefficients obtained as a local linearization of the total variation tv models see rudin l et al physica d vol 60 p 259 68 1992 the keypoint of our model is to get the local inversion of the laplacian operator which is done via the fast fourier transform fft two local schemes are developed a pointwise one and a piecewise one we analyze them both and their advantages and limitations

kronecker product approximations for image restoration with whole sample symmetric boundary conditions reflexive boundary conditions bcs assume that the array values outside the viewable region are given by a symmetry of the array values inside the reflection guarantees the continuity of the image in fact there are usually two choices for the symmetry symmetry around the meshpoint and symmetry around the midpoint the first is called whole sample symmetry in signal and image processing the second is half sample many researchers have developed some fast algorithms for the problems of image restoration with the half sample symmetric bcs over the years however little attention has been given to the whole sample symmetric bcs in this paper we consider the use of the whole sample symmetric boundary conditions in image restoration the blurring matrices constructed from the point spread functions psfs for the bcs have block toeplitz plus pseudohankel with toeplitz plus pseudohankel blocks structures recently regardless of symmetric properties of the psfs a technique of kronecker product approximations was successfully applied to restore images with the zero bcs half sample symmetric bcs and anti reflexive bcs respectively all these results extend quite naturally to the whole sample symmetric bcs since the resulting matrices have similar structures it is interesting to note that when the size of the true psf is small the computational complexity of the algorithm obtained for the kronecker product approximation of the resulting matrix in this paper is very small it is clear that in this case all calculations in the algorithm are implemented only at the upper left corner submatrices of the big matrices finally detailed experimental results reporting the performance of the proposed algorithm are presented

a fast 4 4 dct algorithm for the recursive 2 d dct the authors present an efficient algorithm for the computation of the 4 4 discrete cosine transform dct the algorithm is based on the decomposition of the 4 4 dct into four 4 point 1 d dcts thus only 1 d transformations and some additions are required it is shown that the proposed algorithm requires only 16 multiplications which is half the number needed for the conventional row column method since the 2 sup m 2 sup m dct can be computed using the 4 4 dct recursively for any m the proposed algorithm leads to a fast algorithm for the computation of the 2 d dct

low complexity multidimensional dct approximations for high order tensor data decorrelation in this paper we introduce low complexity multidimensional discrete cosine transform dct approximations 3d dct approximations are formalized in terms of high order tensor theory the formulation is extended to higher dimensions with arbitrary lengths several multiplierless 8 times 8 times 8 approximate methods are proposed and the computational complexity is discussed for the general multidimensional case the proposed methods complexity cost was assessed presenting considerably lower arithmetic operations when compared with the exact 3d dct the proposed approximations were embedded into 3d dct based video coding scheme and a modified quantization step was introduced the simulation results showed that the approximate 3d dct coding methods offer almost identical output visual quality when compared with exact 3d dct scheme the proposed 3d approximations were also employed as a tool for visual tracking the approximate 3d dct based proposed system performs similarly to the original exact 3d dct based method in general the suggested methods showed competitive performance at a considerably lower computational cost

generalized commuting matrices and their eigenvectors for dfts offset dfts and other periodic operations it is well known that some matrices such as dickinson steiglitz s matrix can commute with the discrete fourier transform dft and that one can use them to derive the complete and orthogonal dft eigenvector set recently candan found the general form of the dft commuting matrix in this paper we further extend the previous work and find the general form of the commuting matrix for any periodic quasi periodic and offset quasi periodic operations using the general commuting matrix we can derive the complete and orthogonal eigenvector sets for offset dfts dcts of types 1 4 5 and 8 dsts of types 1 4 5 and 8 discrete hartley transforms of types 1 and 4 the walsh transform and the projection operation the operation that maps a whole vector space into a subspace successfully moreover several novel ways of finding dft eigenfunctions are also proposed furthermore we also extend our theories to the continuous case i e if a continuous transform is periodic quasi periodic or offset quasi periodic such as the ft and some cyclic operations in optics we can find the general form of the commuting operation and then find the complete and orthogonal eigenfunctions set for the continuous transform

generalized new mersenne number transforms two new number theoretic transforms named as odd and odd squared new mersenne number transforms are introduced for incorporation into a generalized new mersenne number transforms gnmnts suite which are defined in finite fields modulo mersenne primes where arithmetic operations and residue reductions are simple to implement this suite is categorized by type with detailed instructions regarding their derivations an example is given which shows their suitability for the calculation of different types of convolutions along with an analysis of their arithmetic complexities for radix 2 and split radix algorithms this in turn shows that these new transforms are suitable for fast error free calculation of convolutions correlations for signal processing and other applications

fast algorithms for the dft and other sinusoidal transforms a new matrix factorization is proposed for dct iv which is the basis of fast algorithms for many sinusoidal transforms the new fast algorithm for dft based on the new factorization requires the same number of multiplications and far fewer additions than the preuss algorithm

diagonalizing properties of the discrete cosine transforms since its introduction in 1974 by ahmed et al the discrete cosine transform dct has become a significant tool in many areas of digital signal processing especially in signal compression there exist eight types of discrete cosine transforms dcts we obtain the eight types of dcts as the complete orthonormal set of eigenvectors generated by a general form of matrices in the same way as the discrete fourier transform dft can be obtained as the eigenvectors of an arbitrary circulant matrix these matrices can be decomposed as the sum of a symmetric toeplitz matrix plus a hankel or close to hankel matrix scaled by some constant factors we also show that all the previously proposed generating matrices for the dcts are simply particular cases of these general matrix forms using these matrices we obtain for each dct a class of stationary processes verifying certain conditions with respect to which the corresponding dct has a good asymptotic behavior in the sense that it approaches karhunen loeve transform performance as the block size n tends to infinity as a particular result we prove that the eight types of dcts are asymptotically optimal for all finite order markov processes we finally study the decorrelating power of the dcts obtaining expressions that show the decorrelating behavior of each dct with respect to any stationary processes

tensor tensor products with invertible linear transforms abstract research in tensor representation and analysis has been rising in popularity in direct response to a the increased ability of data collection systems to store huge volumes of multidimensional data and b the recognition of potential modeling accuracy that can be provided by leaving the data and or the operator in its natural multidimensional form in recent work 1 the authors introduced the notion of the t product a generalization of matrix multiplication for tensors of order three which can be extended to multiply tensors of arbitrary order 2 the multiplication is based on a convolution like operation which can be implemented efficiently using the fast fourier transform fft the corresponding linear algebraic framework from the original work was further developed in 3 and it allows one to elegantly generalize all classical algorithms from numerical linear algebra in this paper we extend this development so that tensor tensor products can be defined in a so called transform domain for any invertible linear transform in order to properly motivate this transform based approach we begin by defining a new tensor tensor product alternative to the t product we then show that it can be implemented efficiently using dcts and that subsequent definitions and factorizations can be formulated by appealing to the transform domain using this new product as our guide we then generalize the transform based approach to any invertible linear transform we introduce the algebraic structures induced by each new multiplication in the family which is that of c algebras and modules finally in the spirit of 4 we give a matrix algebra based interpretation of the new family of tensor tensor products and from an applied perspective we briefly discuss how to choose a transform we demonstrate the convenience of our new framework within the context of an image deblurring problem and we show the potential for using one of these new tensor tensor products and resulting tensor svd for hyperspectral image compression

fast algorithms for multidimensional dct to dct computation between a block and its associated subblocks in this paper we first propose an efficient algorithm for computing one dimensional 1 d discrete cosine transform dct for a signal block given its two adjacent subblocks in the dct domain and then introduce several algorithms for the fast computation of multidimensional m d dct with size n sub 1 spl times n sub 2 spl times spl times n sub m given 2 sup m subblocks of dct coefficients with size n sub 1 2 spl times n sub 2 2 spl times spl times n sub m 2 where n sub i i 1 2 m are powers of 2 obviously the row column method which employs the most efficient algorithms along each dimension reduces the computational complexity considerably compared with the traditional method which employs only the one dimensional 1 d fast dct and inverse dct idct algorithms however when m spl ges 2 the traditional method which employs the most efficient multidimensional dct idct algorithms has lower computational complexity than the row column method besides we propose a direct method by dividing the data into 2 sup m parts for independent fast computation in which only two steps of r dimensional r 1 2 m idct and additional multiplications and additions are required if all the dimensional sizes are the same the number of multiplications required for the direct method is only 2 sup m 1 m2 sup m 1 times of that required for the row column method and if n spl ges 2 sup 2m 1 the computational efficiency of the direct method is surely superior to that of the traditional method which employs the most efficient multidimensional dct idct algorithms

l m fold image resizing in block dct domain using symmetric convolution image resizing is to change an image size by upsampling or downsampling of a digital image most still images and video frames on digital media are given in a compressed domain image resizing of a compressed image can be performed in the spatial domain via decompression and recompression in general resizing of a compressed image in a compressed domain is much faster than that in the spatial domain we propose a novel approach to resize images with l m resizing ratio in the discrete cosine transform dct domain which exploits the multiplication convolution property of dct multiplication in the spatial domain corresponds to symmetric convolution in the dct domain when an image is given in terms of its 8 spl times 8 block dct coefficients its resized image is also obtained in 8 spl times 8 block dct coefficients the proposed approach is computationally fast and produces visually fine images with high psnr

symmetric convolution using unitary transform matrices we demonstrate relationships between the convolutional and unitary forms of certain trigonometric transform matrices we then use the new relationships to derive forms of the symmetric convolution multiplication property based on unitary rather than convolutional forms of the transform matrices

enhancement of color images by scaling the dct coefficients this paper presents a new technique for color enhancement in the compressed domain the proposed technique is simple but more effective than some of the existing techniques reported earlier the novelty lies in this case in its treatment of the chromatic components while previous techniques treated only the luminance component the results of all previous techniques along with that of the proposed one are compared with respect to those obtained by applying a spatial domain color enhancement technique that appears to provide very good enhancement the proposed technique computationally more efficient than the spatial domain based method is found to provide better enhancement compared to other compressed domain based approaches

symmetric convolution and the discrete sine and cosine transforms this paper discusses the use of symmetric convolution and the discrete sine and cosine transforms dsts and dcts for general digital signal processing the operation of symmetric convolution is a formalized approach to convolving symmetrically extended sequences the result is the same as that obtained by taking an inverse discrete trigonometric transform dtt of the product of the forward dtts of those two sequences there are 16 members in the family of dtts each provides a representation for a corresponding distinct type of symmetric periodic sequence the author defines symmetric convolution relates the dsts and dcts to symmetric periodic sequences and then use these principles to develop simple but powerful convolution multiplication properties for the entire family of dsts and dcts symmetric convolution can be used for discrete linear filtering when the filter is symmetric or antisymmetric the filtering will be efficient because fast algorithms exist for all versions of the dtts conventional linear convolution is possible if one first zero pad the input data symmetric convolution and its fast implementation using dtts are now an alternative to circular convolution and the dft

a comparison of dct based ofdm and dft based ofdm in frequency offset and fading channels a precise method for calculating the bit error probability bep of a discrete cosine transform dct based orthogonal frequency division multiplexing ofdm system on additive white gaussian noise awgn channels in the presence of frequency offset is derived these accurate results are used to examine and compare the bep performance of a dct ofdm system and the conventional discrete fourier transform dft based ofdm system in an awgn environment several signaling formats such as binary phase shift keying quaternary phase shift keying and 16 ary quadrature amplitude modulation are considered the performance of a dct ofdm with a zero padding guard interval scheme is then compared with a zero padded dft ofdm with the employment of minimum mean square error mmse detection and mmse decision feedback detection with ordering scheme over frequency selective fast rayleigh fading channels analysis and simulation results show that the dct ofdm system outperforms the dft ofdm system in the presence of frequency offset and in frequency selective fast fading environments

on algebras related to the discrete cosine transform abstract an algebraic theory for the discrete cosine transform dct is developed which is analogous to the well known theory of the discrete fourier transform dft whereas the latter diagonalizes a convolution algebra which is a polynomial algebra modulo a product of various cyclotomic polynomials the former diagonalizes a polynomial algebra modulo a product of various polynomials related to the chebyshev types when the dimension of the algebra is a power of 2 the dct diagonalizes a polynomial algebra modulo a product of chebyshev polynomials of the first type in both dft and dct cases the chinese remainder theorem plays a key role in the design of fast algorithms

fractional cosine and sine transforms over finite fields abstract in this paper we introduce fractional cosine and sine transforms over finite fields the basis for the definition of such transforms which are respectively referred by the acronyms gfrct and gfrst is the recently proposed fractional fourier transform over finite fields gfrft 1 more specifically we use the eigenvectors of the finite field fourier transform to construct eigenvectors of the finite field cosine and sine transforms such eigenvectors are then used in spectral expansions that allow to compute fractional powers of the finite field cosine and sine transform matrices

transitional probabilities for the four state random walk on a lattice in the presence of partially reflecting boundaries the four state random walk 4rw model wherein the particle is endowed with two states of spin and two states of directional motion in each space coordinate permits a stochastic solution of the schrodinger equation or the equivalent parabolic equation without resorting to the usual analytical continuation in complex space of the particle trajectories analytical expressions are derived here for the various transitional probabilities in a 4rw by employing generating functions and eigenfunction expansions when the particle moves on a 1 1 space time lattice with two point boundary conditions the most general case of dissimilar boundaries with partially reflecting boundary conditions is treated in this paper the transitional probabilities are all expressed in terms of a finite summation involving trigonometric functions and or chebyshev polynomials of the second kind that are characteristics of diffusion and schrodinger equations respectively in the 4rw model results for the special case of perfectly

multi wiener sure let deconvolution in this paper we propose a novel deconvolution algorithm based on the minimization of a regularized stein s unbiased risk estimate sure which is a good estimate of the mean squared error we linearly parametrize the deconvolution process by using multiple wiener filters as elementary functions followed by undecimated haar wavelet thresholding due to the quadratic nature of sure and the linear parametrization the deconvolution problem finally boils down to solving a linear system of equations which is very fast and exact the linear coefficients i e the solution of the linear system of equations constitute the best approximation of the optimal processing on the wiener haar threshold basis that we consider in addition the proposed multi wiener sure let approach is applicable for both periodic and symmetric boundary conditions and can thus be used in various practical scenarios the very competitive both in computation time and quality results show that the proposed algorithm which can be interpreted as a kind of nonlinear wiener processing can be used as a basic tool for building more sophisticated deconvolution algorithms

analysis of pruning in fast cosine transform the discrete cosine transform dct is now the established technique for image and video coding in most image coding set schemes a large number of high frequency transform coefficients can be neglected and need not be calculated therefore significant savings in computation can be obtained by applying pruning in the fast algorithms we analyze pruning in chan and ho s 1990 fast dct algorithm and evaluate the saving of operations

a unified approach to reduced redundancy transceivers superfast linear and block iterative generalized decision feedback equalizers this paper shows under general input data models how block memoryless equalizers should be formulated considering reduced redundancy transmissions for superfast detection we propose linear and dfe based both multicarrier mc and single carrier frequency domain sc fd transceivers along with efficient methods for the equalizer calculation in a unified manner we argue that under a one tap block decision feedback transmitted redundancy can be reduced below the minimum l 1 2 samples allowed in the linear case where l is the channel length even down to zero redundancy with improved ber performance this is quantified in light of the optimal reconstruction delay set for a minimum norm zero forcing feedforward matrix in terms of the channel zeros location the proposed mc and sc fd block dfes do not cancel inter block interference ibi via zeros jamming instead it removes ibi completely in part by decision feedback and in part by zero padding which allows for much lower redundancy transmissions the remaining isi is further eliminated through a one step block iterative generalized dfe bi gdfe obtained in the minimum mean square error mmse sense unlike computationally demanding block dfes that eliminate isi via successive cancelation the proposed dfe schemes are as efficient as a superfast block linear equalizer requiring at most 3 receive branches to realize the order m feedforward matrices in o m logm operations

two dimensional symmetric and antisymmetric generalizations of exponential and cosine functions the properties of the four families of the recently introduced special functions of two real variables denoted here by e and cos are studied the superscripts and refer to the symmetric and antisymmetric functions respectively the functions are considered in all details required for their exploitation in fourier expansions of digital data sampled on square grids of any density and for general position of the grid in the real plane relative to the lattice defined by the underlying group theory the quality of continuous interpolation resulting from the discrete expansions is studied exemplified and compared for some model functions

termination of term rewriting using dependency pairs we present techniques to prove termination and innermost termination of term rewriting systems automatically in contrast to previous approaches we do not compare left and right hand sides of rewrite rules but introduce the notion of dependency pairs to compare left hand sides with special subterms of the right hand sides this results in a technique which allows to apply existing methods for automated termination proofs to term rewriting systems where they failed up to now in particular there are numerous term rewriting systems where a direct termination proof with simplification orderings is not possible but in combination with our technique well known simplification orderings such as the recursive path ordering polynomial orderings or the knuth bendix ordering can now be used to prove termination automatically unlike previous methods our technique for proving innermost termination automatically can also be applied to prove innermost termination of term rewriting systems that are not terminating moreover as innermost termination implies termination for certain classes of term rewriting systems this technique can also be used for termination proofs of such systems

downsampling versus folding in the dct domain signals can be split into subbands using ideal bandpass filters in the discrete cosine transform dct domain we show that decimation by band folding is not equivalent to downsampling but results in two convolution operations that involve both the odd and even samples of the signal

a new algorithm to compute the discrete cosine transform a new algorithm is introduced for the 2m point discrete cosine transform this algorithm reduces the number of multiplications to about half of those required by the existing efficient algorithms and it makes the system simpler

insights from a pseudospectral approach to the elder problem 1 the aim of this paper is to clarify and circumvent the issue of multiple steady state solutions in the elder problem a pseudospectral method is used to avoid numerical error associated with spatial discretization the pseudospectral method is verified by comparison to an analytical solution at rayleigh number ra 0 and by reproducing the three stable steady state solutions that are known to exist at ra 400 a bifurcation diagram for 0 ra 400 which is free of discretization error confirms that multiple steady states are indeed an intrinsic characteristic of the elder problem the existence of multiple steady states makes the ra 400 elder problem less suitable for benchmarking numerical models to avoid the multiple steady states we propose a benchmark at ra 60 the results for this low rayleigh number elder problem are presented and compared to simulations with the commercial groundwater modeling package feflow correspondence between the pseudospectral model and feflow is excellent

nonparametric least squares estimation in derivative families cost function estimation often involves data on a function and a family of its derivatives such data can substantially improve convergence rates of nonparametric estimators we propose series type estimators which incorporate the various derivative data into a single nonparametric least squares procedure convergence rates are obtained and it is shown that for low dimensional cases much of the beneficial impact is realized even if only data on ordinary first order partials are available in instances where root n consistency is attained smoothing parameters can often be chosen very easily without resort to cross validation simulations and an illustration of cost function estimation are included

fast algorithm for the 3 d dct ii recently many applications for three dimensional 3 d image and video compression have been proposed using 3 d discrete cosine transforms 3 d dcts among different types of dcts the type ii dct dct ii is the most used in order to use the 3 d dcts in practical applications fast 3 d algorithms are essential therefore in this paper the 3 d vector radix decimation in frequency 3 d vr dif algorithm that calculates the 3 d dct ii directly is introduced the mathematical analysis and the implementation of the developed algorithm are presented showing that this algorithm possesses a regular structure can be implemented in place for efficient use of memory and is faster than the conventional row column frame rcf approach furthermore an application of 3 d video compression based 3 d dct ii is implemented using the 3 d new algorithm this has led to a substantial speed improvement for 3 d dct ii based compression systems and proved the validity of the developed algorithm

discrete gyrator transforms computational algorithms and applications as an extension of the 2d fractional fourier transform frft and a special case of the 2d linear canonical transform lct the gyrator transform was introduced to produce rotations in twisted space spatial frequency planes it is a useful tool in optics signal processing and image processing in this paper we develop discrete gyrator transforms dgts based on the 2d lct taking the advantage of the additivity property of the 2d lct we propose three kinds of dgts each of which is a cascade of low complexity operators these dgts have different constraints characteristics and properties and are realized by different computational algorithms besides we propose a kind of dgt based on the eigenfunctions of the gyrator transform this dgt is an orthonormal transform and thus its comprehensive properties especially the additivity property make it more useful in many applications we also develop an efficient computational algorithm to significantly reduce the complexity of this dgt at the end a brief review of some important applications of the dgts is presented including mode conversion sampling and reconstruction watermarking and image encryption

generating matrices for the discrete sine transforms in this correspondence we obtain a general form for the generating matrices of the eight types of discrete sine transforms these matrix forms can be used to study the performance of the different discrete sine transforms as substitutes of the karhunen loeve transform and will allow us to show in a very straightforward way that the discrete sine transforms as the discrete cosine transforms are asymptotically optimal for any finite order markov process

on computing the discrete fourier and cosine transforms an efficient algorithm based on real matrix decomposition is developed for computing a class of sinusoidal transforms that include the discrete fourier and cosine transform

symmetric convolution of asymmetric multidimensional sequences using discrete trigonometric transforms this paper uses the fact that the discrete fourier transform diagonalizes a circulant matrix to provide an alternate derivation of the symmetric convolution multiplication property for discrete trigonometric transforms derived in this manner the symmetric convolution multiplication property extends easily to multiple dimensions using the notion of block circulant matrices and generalizes to multidimensional asymmetric sequences the symmetric convolution of multidimensional asymmetric sequences can then be accomplished by taking the product of the trigonometric transforms of the sequences and then applying an inverse trigonometric transform to the result an example is given of how this theory can be used for applying a two dimensional 2 d finite impulse response fir filter with nonlinear phase which models atmospheric turbulence

autoregressive modeling of temporal spectral envelopes with finite length discrete trigonometric transforms the theory of autoregressive ar modeling also known as linear prediction has been established by the fourier analysis of infinite discrete time sequences or continuous time signals nevertheless for various finite length discrete trigonometric transforms dtts including the discrete cosine and sine transforms of different types the theory is not well established several dtts have been used in current audio coding and the ar modeling method can be applied to reduce coding artifacts or exploit data redundancies this paper systematically develops the ar modeling fundamentals of temporal and spectral envelopes for the sixteen members of the dtts this paper first considers the ar modeling in the generalized discrete fourier transforms gdfts then we derive the modeling to all the dtts by introducing the analytic transforms which convert the real valued vectors into complex valued ones through the process we build the compact matrix representations for the ar modeling of the dtts in both time domain and dtt domain these compact forms also illustrate that the ar modeling for the envelopes can be performed through the hilbert envelope and the power envelope these compact forms can be used to develop new coding technologies or examine the possible defects in the existing ar modeling methods for dtts we apply the forms to analyze the current temporal noise shaping tns tool in mpeg 2 4 advanced audio coding aac

fast unified computation of the multi dimensional discrete sinusoidal transforms based on a one to one index mapping approach conversions of two types of m dimensional discrete w transforms into the multiple sums involving a number of one dimensional discrete w transforms 1 d dwts are first presented a unified fast m 1 dimensional polynomial transform m 1 d pt and 1 d dwts algorithm for computing these two types of m d dwts are then proposed by revealing the relationships among the m dimensional discrete w transforms m d dwts the m dimensional generalized discrete fourier transforms m d gdfts the m dimensional discrete cosine transforms m d dcts and the m dimensional discrete sine transforms m d dsts the generalized fast algorithms for 14 types of the m d discrete sinusoidal transforms including four types of m d dwts and the same number of types of m d gdfts and three types of m d dcts and the same number of types of m d dsts are presented the number of multiplications for all 14 types of the m d discrete sinusoidal transforms needed by the proposed algorithm is only 1 m times that of the widely used corresponding row column methods the number of additions required by the proposed algorithm is also reduced considerably finally with the help of the software platform visual c 6 0 for windows 98 the computation time comparisons between the proposed and the reported algorithms are given the numerical experiments show that the proposed algorithms are not only highly efficient but are also very simple in computational structure

on the use of discrete cosine transforms for multicarrier communications in this correspondence the conditions to use any kind of discrete cosine transform dct for multicarrier data transmission are derived the symmetric convolution multiplication property of each dct implies that when symmetric convolution is performed in the time domain an element by element multiplication is performed in the corresponding discrete trigonometric domain therefore appending symmetric redundancy as prefix and suffix into each data symbol to be transmitted and also enforcing symmetry for the equivalent channel impulse response the linear convolution performed in the transmission channel becomes a symmetric convolution in those samples of interest furthermore the channel equalization can be carried out by means of a bank of scalars in the corresponding discrete cosine transform domain the expressions for obtaining the value of each scalar corresponding to these one tap per subcarrier equalizers are presented this study is completed with several computer simulations in mobile broadband wireless communication scenarios considering the presence of carrier frequency offset cfo the obtained results indicate that the proposed systems outperform the standardized ones based on the dft

accurate and efficient method for smoothly space variant gaussian blurring this paper presents a computationally efficient algorithm for smoothly space variant gaussian blurring of images the proposed algorithm uses a specialized filter bank with optimal filters computed through principal component analysis this filter bank approximates perfect space variant gaussian blurring to arbitrarily high accuracy and at greatly reduced computational cost compared to the brute force approach of employing a separate low pass filter at each image location this is particularly important for spatially variant image processing such as foveated coding experimental results show that the proposed algorithm provides typically 10 to 15 db better approximation of perfect gaussian blurring than the blended gaussian pyramid blurring approach when using a bank of just eight filters

the discrete cosine transform each discrete cosine transform dct uses n real basis vectors whose components are cosines in the dct 4 for example the j th component of boldv k is cos j frac 1 2 k frac 1 2 frac pi n these basis vectors are orthogonal and the transform is extremely useful in image processing if the vector boldx gives the intensities along a row of pixels its cosine series sum c k boldv k has the coefficients c k boldx boldv k n they are quickly computed from a fast fourier transform but a direct proof of orthogonality by calculating inner products does not reveal how natural these cosine vectors are we prove orthogonality in a different way each dct basis contains the eigenvectors of a symmetric second difference matrix by varying the boundary conditions we get the established transforms dct 1 through dct 4 other combinations lead to four additional cosine transforms the type of boundary condition dirichlet or neumann centered at a meshpoint or a midpoint determines the applications that are appropriate for each transform the centering also determines the period n 1 or n in the established transforms n frac 1 2 or n frac 1 2 in the other four the key point is that all these eigenvectors of cosines come from simple and familiar matrices

generalized eigenvectors and fractionalization of offset dfts and dcts the offset discrete fourier transform dft is a discrete transform with kernel exp j2 spl pi m a n b n it is more generalized and flexible than the original dft and has very close relations with the discrete cosine transform dct of type 4 dct iv dct viii discrete sine transform dst iv dst viii and discrete hartley transform dht iv in this paper we derive the eigenvectors eigenvalues of the offset dft especially for the case where a b is an integer by convolution theorem we can derive the close form eigenvector sets of the offset dft when a b is an integer we also show the general form of the eigenvectors in this case then we use the eigenvectors eigenvalues of the offset dft to derive the eigenvectors eigenvalues of the dct iv dct viii dst iv dst viii and dht iv after the eigenvectors eigenvalues are derived we can use the eigenvectors decomposition method to derive the fractional operations of the offset dft dct iv dct viii dst iv dst viii and dht iv these fractional operations are more flexible than the original ones and can be used for filter design data compression encryption and watermarking etc