the discrete rotational fourier transform we define a discrete version of the angular fourier transform and present the properties of the transform that show it to be a rotation in time frequency space this new transform is a generalization of the dft efficient algorithms for its computation can then be based on the fft and the eigenstructure of the dft

closed form discrete fractional and affine fourier transforms the discrete fractional fourier transform dfrft is the generalization of discrete fourier transform many types of dfrft have been derived and are useful for signal processing applications we introduce a new type of dfrft which are unitary reversible and flexible in addition the closed form analytic expression can be obtained it works in performance similar to the continuous fractional fourier transform frft and can be efficiently calculated by the fft since the continuous frft can be generalized into the continuous affine fourier transform aft the so called canonical transform we also extend the dfrft into the discrete affine fourier transform daft we derive two types of the dfrft and daft type 1 is similar to the continuous frft and aft and can be used for computing the continuous frft and aft type 2 is the improved form of type 1 and can be used for other applications of digital signal processing meanwhile many important properties continuous frft and aft are kept in the closed form dfrft and daft and some applications such as filter design and pattern recognition are also discussed the closed form dfrft we introduce has the lowest complexity among all current dfrfts that is still similar to the continuous frft

eigenvectors and functions of the discrete fourier transform a method is presented for computing an orthonormal set of eigenvectors for the discrete fourier transform dft the technique is based on a detailed analysis of the eigenstructure of a special matrix which commutes with the dft it is also shown how fractional powers of the dft can be efficiently computed and possible applications to multiplexing and transform coding are suggested

on q extended eigenvectors of the integral and finite fourier transforms mehta has shown that eigenvectors of the n n finite fourier transform can be written in terms of the standard hermite eigenfunctions of the quantum harmonic oscillator 1987 j math phys 28 781 here we construct a one parameter family of q extensions of these eigenvectors based on the continuous q hermite polynomials of rogers in the limit when q 1 these q extensions coincide with mehta s eigenvectors and in the continuum limit as n they give rise to q extensions of eigenfunctions of the fourier integral transform

general multifractional fourier transform method based on the generalized permutation matrix group the paper studies the possibility of giving a general multiplicity of the fractional fourier transform frft with the intention of combining existing finite versions of the frft we introduce a new class of frft that includes the conventional fractional fourier transforms cfrfts and the weighted type fractional fourier transforms wfrfts as special cases the class is structurally well organized because these new frfts which are called general multifractional fourier transform gmfrfts are related with one another by the generalized permutation matrix group gpmg and their kernels are related with that of cfrfts as the finite combination by the recursion of matrix in addition we have computer simulations of some gmfrfts on a rectangular function as a simple application of gmfrfts to signal processing

direct batch evaluation of optimal orthonormal eigenvectors of the dft matrix the generation of hermite gaussian like orthonormal eigenvectors of the discrete fourier transform dft matrix f is an essential step in the development of the discrete fractional fourier transform dfrft most existing techniques depend on the generation of orthonormal eigenvectors of a nearly tridiagonal matrix s which commutes with matrix f more sophisticated methods view the eigenvectors of s as only initial ones and use them for generating final ones which better approximate the hermite gaussian functions employing a technique like the orthogonal procrustes algorithm opa here a direct technique for the collective batch evaluation of optimal hermite gaussian like eigenvectors of matrix f is contributed it is a direct technique in the sense that it does not require the generation of initial eigenvectors to be used for computing the final superior ones it is a batch method in the sense that it solves for the entire target modal matrix of f instead of the sequential generation of the eigenvectors the simulation results show that the proposed method is faster than the opa

discrete fourier invariant signals design and application in this paper two methods for the design of discrete fourier invariant signals are proposed the direct design method provides splitting between independent and dependent signal parts and calculation of the dependent part for any given independent part the iterative design method generates a family of discrete fourier invariant signals by a successive approach further we show how the proposed direct design method can be combined with the gabor uncertainty principle to generate discrete fourier invariant signals with the minimum product of their bandwidth b and their time width t we show that these signals as well as signal families generated with the iterative design method achieve the theoretical lower bt bound also it is shown that the bt product of discrete hermite gauss signals converges to the theoretical lower bound finally possible applications are illustrated in the case of time frequency spectral analysis using the obtained discrete fourier invariant signals as the window that provides isoresolution

variational exemplar based image colorization in this paper we address the problem of recovering a color image from a grayscale one the input color data comes from a source image considered as a reference image reconstructing the missing color of a grayscale pixel is here viewed as the problem of automatically selecting the best color among a set of color candidates while simultaneously ensuring the local spatial coherency of the reconstructed color information to solve this problem we propose a variational approach where a specific energy is designed to model the color selection and the spatial constraint problems simultaneously the contributions of this paper are twofold first we introduce a variational formulation modeling the color selection problem under spatial constraints and propose a minimization scheme which computes a local minima of the defined nonconvex energy second we combine different patch based features and distances in order to construct a consistent set of possible color candidates this set is used as input data and our energy minimization automatically selects the best color to transfer for each pixel of the grayscale image finally the experiments illustrate the potentiality of our simple methodology and show that our results are very competitive with respect to the state of the art methods

an uncertainty principle for finite frames abstract we consider the notion of uncertainty for finite frames using a difference operator inspired by the gauss hermite differential equation we obtain a time frequency measure for finite frames we then find the minimizer of the measure over all equal norm parseval frames dependent on the dimension of the space and the number of elements in the frame next we show that given a frame one can find the dual frame that minimizes this time frequency measure generalizing some work of daubechies landau and landau to the finite case and extending some recent work on sobolev duals for finite frames

a unified framework for the fractional fourier transform the paper investigates the possibility for giving a general definition of the fractional fourier transform frt for all signal classes one dimensional 1 d and multidimensional continuous and discrete periodic and aperiodic since the definition is based on the eigenfunctions of the ordinary fourier transform ft the preliminary conditions is that the signal domain periodicity be the same as the ft domain periodicity within these classes a general frt definition is formulated and the frt properties are established in addition the multiplicity which is intrinsic in a fractional operator is clearly developed the general definition is checked in the case in which the frt is presently available and moreover to establish the frt in new classes of signals

eigenvectors of ordinary generalized centered and offset discrete fourier transforms based on lookup table methods efficiency and approximation uses in this paper we first propose a methodology to construct an eigenvector of the n n discrete fourier transform dft matrix from any eigenvector of the k 2 n k 2 n dft matrix where k is a positive integer specifically using the procedure to down sample by k any k 2 n point dft eigenvector with zero as the initial index and then fold modulus n and add the resultant down sampled kn point vector we obtain an n point dft eigenvector computing an n point dft eigenvector using the proposed method is very efficient which requires only n k 1 additions and no multiplications with a pre stored lookup table of k 2 n point dft eigenvectors similar efficient methods are also developed to compute n point eigenvectors of other dft related transforms including the generalized dft gdft centered dft cdft and the offset dft odft from k 2 n point eigenvectors of their corresponding transforms or the dft as application examples we apply the proposed methods to compute n point dft gdft cdft and odft eigenvectors which are closer to samples of the continuous hermite gaussian function hgf than existing n point eigenvectors from their corresponding k 2 n point eigenvectors these improved n point hermite gaussian like eigenvectors can be used to define fractional versions of the various n n dft transforms whose outputs are closer to samples of the continuous fractional fourier transform than outputs of the corresponding existing fractional transforms finally we perform computer experiments to verify the effectiveness and demonstrate applications of our proposed methods

jacobi ϑ functions and discrete fourier transforms properties of the jacobi 3 function and its derivatives under discrete fourier transforms are investigated and several interesting results are obtained the role of modulo n equivalence classes in the theory of functions is stressed an important conjecture is studied

the discrete fractional cosine and sine transforms this paper is concerned with the definitions of the discrete fractional cosine transform dfrct and the discrete fractional sine transform dfrst the definitions of dfrct and dfrst are based on the eigen decomposition of dct and dst kernels this is the same idea as that of the discrete fractional fourier transform dfrft the eigenvalue and eigenvector relationships between the dfrct dfrst and dfrft can be established the computations of dfrft for even or odd signals can be planted into the half size dfrct and dfrst calculations this will reduce the computational load of the dfrft by about one half

the discrete harmonic oscillator harper s equation and the discrete fractional fourier transform certain solutions to harper s equation are discrete analogues of and approximations to the hermite gaussian functions they are the energy eigenfunctions of a discrete algebraic analogue of the harmonic oscillator and they lead to a definition of a discrete fractional fourier transform ft the discrete fractional ft is essentially the time evolution operator of the discrete harmonic oscillator

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

hermite gaussian like eigenvectors of the discrete fourier transform matrix based on the direct utilization of the orthogonal projection matrices on its eigenspaces a new version is proposed for the gram schmidt algorithm gsa the orthogonal procrustes algorithm opa and the sequential orthogonal procrustes algorithm sopa for generating hermite gaussian like orthonormal eigenvectors for the discrete fourier transform matrix f this version is based on the direct utilization of the orthogonal projection matrices on the eigenspaces of matrix f rather than the singular value decomposition of those matrices for the purpose of generating initial orthonormal eigenvectors the proposed version of the algorithms has the merit of achieving a significant reduction in the computation time

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

reconstruction of random colourings reconstruction problems have been studied in a number of contexts including biology information theory and statistical physics we consider the reconstruction problem for random k colourings on the ary tree for large k bhatnagar et al 2 showed non reconstruction when delta leq frac12 k log k o k log k we tighten this result and show non reconstruction when delta leq k log k log log k 1 log 2 o 1 which is very close to the best known bound establishing reconstruction which is delta geq k log k log log k 1 o 1

discrete fourier transform of prime order eigenvectors with small support we show how to construct an eigenvector basis of the discrete fourier transform of odd prime order the special feature of the new basis is that the basis vectors have small support

signal scaling by centered discrete dilated hermite functions we present a novel method to generate centered discrete dilated hermite functions by utilizing hermite gaussian differential equation and dft commuting matrix proposed by pei compared with the conventional methods our method provides much higher accuracy and wider working range based on expansion of compact support signals into discrete dilated hermite basis several examples of signal approximation and scaling are demonstrated our results in ecg signal compression can achieve lesser approximation error in comparison with discrete daubechies wavelet transform also our proposed discrete dilated hermite functions have better ability to do signal scaling than the existing method

digital computation of the fractional fourier transform an algorithm for efficient and accurate computation of the fractional fourier transform is given for signals with time bandwidth product n the presented algorithm computes the fractional transform in o nlogn time a definition for the discrete fractional fourier transform that emerges from our analysis is also discussed

analytic representation of finite quantum systems a transform between functions in r and functions in zd is used to define the analogue of number and coherent states in the context of finite d dimensional quantum systems the coherent states are used to define an analytic representation in terms of theta functions all states are represented by entire functions with growth of order 2 which have exactly d zeros in each cell the analytic function of a state is constructed from its zeros results about the completeness of finite sets of coherent states within a cell are derived

eigenvalues and eigenvectors of the finite fourier transform the eigenvalues and eigenvectors of the n n unitary matrix of finite fourier transform whose j k element is 1 n 1 2 exp 2 i n jk i 1 1 2 is determined in doing so a multitude of identities some of which may be new are encountered a conjecture is advanced

on continuous q hermite polynomials and the classical fourier transform we prove that the classical fourier transform operator intertwines two q difference equations for the continuous q hermite polynomials hn x q of rogers which are associated with the two distinct sets of values for the parameter q 0 q 1 and 1 q

closed form orthogonal number theoretic transform eigenvectors and the fast fractional ntt in this paper we propose a new method to find the closed form solution of number theoretic transform ntt eigenvectors we construct the complete generalized legendre sequence over the finite field cglsf and use it to solve the ntt eigenvector problem we derive the cglsf like ntt eigenvectors successfully including the case where the operation field is defined over the fermat and mersenne numbers the derived ntt eigenvector set is orthogonal and has a closed form it is suitable for constructing sub ntt building blocks for ntt implementation in addition with different eigenvalue assignment rule we can construct the fractional number theoretic transform fntt including the fractional fermat number transform ffnt the fractional complex mersenne number transform fcmnt and the fractional new mersenne number transform fnmnt they are the generalizations of the original transforms and all have the complexities of o nlog2n

discrete fractional fourier transform based on orthogonal projections the continuous fractional fourier transform frft performs a spectrum rotation of signal in the time frequency plane and it becomes an important tool for time varying signal analysis a discrete fractional fourier transform has been developed by santhanam and mcclellan see ibid vol 42 p 994 98 1996 but its results do not match those of the corresponding continuous fractional fourier transforms we propose a new discrete fractional fourier transform dfrft the new dfrft has dft hermite eigenvectors and retains the eigenvalue eigenfunction relation as a continous frft to obtain dft hermite eigenvectors two orthogonal projection methods are introduced thus the new dfrft will provide similar transform and rotational properties as those of continuous fractional fourier transforms moreover the relationship between frft and the proposed dfrft has been established in the same way as the conventional dft to continuous fourier transform

the discrete fractional fourier transform we propose and consolidate a definition of the discrete fractional fourier transform that generalizes the discrete fourier transform dft in the same sense that the continuous fractional fourier transform generalizes the continuous ordinary fourier transform this definition is based on a particular set of eigenvectors of the dft matrix which constitutes the discrete counterpart of the set of hermite gaussian functions the definition is exactly unitary index additive and reduces to the dft for unit order the fact that this definition satisfies all the desirable properties expected of the discrete fractional fourier transform supports our confidence that it will be accepted as the definitive definition of this transform

multiplicity of fractional fourier transforms and their relationships the multiplicity of the fractional fourier transform frt which is intrinsic in any fractional operator has been claimed by several authors but never systematically developed the paper starts with a general frt definition based on eigenfunctions and eigenvalues of the ordinary fourier transform which allows us to generate all possible definitions the multiplicity is due to different choices of both the eigenfunction and the eigenvalue classes a main result obtained by a generalized form of the sampling theorem gives explicit relationships between the different frts

positive time frequency distributions based on joint marginal constraints the authors study the formulation of members of the cohen posch 1985 class of positive time frequency energy distributions minimization of cross entropy measures with respect to different priors and the case of no prior or maximum entropy were considered they conclude that in general the information provided by the classical marginal constraints is very limited and thus the final distribution heavily depends on the prior distribution to overcome this limitation joint time and frequency marginals are derived based on a direction invariance criterion on the time frequency plane that are directly related to the fractional fourier transform

dft commuting matrix with arbitrary or infinite order second derivative approximation recently candan introduced higher order dft commuting matrices whose eigenvectors are better approximations to the continuous hermite gaussian functions hgfs however the highest order 2k of the o h2k ntimesn dft commuting matrices proposed by candan is restricted by 2k 1 les n in this paper we remove this order upper bound restriction by developing two methods to construct arbitrary order dft commuting matrices computer experimental results show that the hermite gaussian like hgl eigenvectors of the new proposed dft commuting matrices outperform those of candan in addition the hgl eigenvectors of the infinite order dft commuting matrix are shown to be the same as those of the n2 dft commuting matrix recently discovered in the literature

an orthonormal class of exact and simple dft eigenvectors with a high degree of symmetry the paper presents a novel orthonormal class of eigenvectors of the discrete fourier transform dft whose order n is factored as n rm sup 2 the dft eigenvectors have the form e e spl alpha where spl alpha are eigenvectors of some spl lscr spl times spl lscr matrices given by or related to the dft matrix of order r with spl lscr r 2r or 4r and the matrix e expands spl alpha to the full dft size n rm sup 2 in particular when n is an arbitrarily large power of 2 r may be 1 or 2 the resulting eigenvectors are expressed exactly with simple exponential expressions have a considerable number of elements constrained to 0 and show a high degree of symmetry the derivation of such a class is based on a partition of the n dimensional linear space into subspaces of very small dimension r 2r or 4r

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