beamforming using the fractional fourier transform we present a new method of beamforming using the fractional fourier transform frft this method encompasses the conventional minimum mean squared error mmse beamforming in the frequency domain or spatial domain as special cases it is especially useful for applications involving chirp signals such as signal enhancement problems with accelerating sinusoidal sources where the doppler effect generates chirp signals and a frequency shift and active radar problems where chirp signals are transmitted numerical examples demonstrate the potential advantage of the proposed method over the ordinary frequency or spatial domain beamforming for a moving source scenario

comparing models of the intensional typed l calculus abstract in this paper we compare powerset models of the typed lambda calculus although the interpretation of the type constructor is in general not uniquely determined for intensional lambda calculi we can make a canonical choice in this particular case there exists a minimal interpretation of with respect to a certain class of interpretations which yields models with a maximal theory in that class

generalized sampling expansions with multiple sampling rates for lowpass and bandpass signals in the fractional fourier transform domain the objective of generalized sampling expansion gse is the reconstruction of an unknown continuously defined function f left t right from samples of the responses from m linear time invariant lti systems that are each sampled using the 1 m th nyquist rate in this paper we investigate the gse for lowpass and bandpass signals with multiple sampling rates in the fractional fourier transform frft domain first we propose an improvement of papoulis gse which has multiple sampling rates in the frft domain based on the proposed gse we derive the periodic nonuniform sampling scheme and the derivative interpolation method by designing different fractional filters and selecting specific sampling rates in addition the papoulis gse and the previous gse associated with frft are shown to be special instances of our results second we address the problem of the gse of fractional bandpass signals a new gse for fractional bandpass signals with equal sampling rates is derived we show that the restriction of an even number of channels in the gse for fractional bandpass signals is unnecessary and perfect signal reconstruction is possible for any arbitrary number of channels further we develop the gse for a fractional bandpass signal with multiple sampling rates lastly we discuss the application of the proposed method in the context of single image super resolution reconstruction based on gse illustrations and simulations are presented to verify the validity and effectiveness of the proposed results

the hermite transform applications it is demonstrated how the hermite transform can be used for image coding and analysis hierarchical coding structures based on increasingly specified basic patterns i e general 2 d patterns general 1 d patterns and specific 1 d patterns such as edges and corners are presented in the image coding application the relation with existing pyramid coders is described a new coding scheme based on local one dimensional image approximations is introduced in the image analysis application the relation between the hermite transform and existing line edge detection schemes is described it is shown that by concentrating on more specific patterns the coding efficiency can be increased since fewer coefficients have to be coded meanwhile sufficient descriptive power can be maintained for approximating the most interesting features in natural images

short time fractional fourier transform and its applications the fractional fourier transform frft is a potent tool to analyze the chirp signal however it fails in locating the fractional fourier domain frfd frequency contents which is required in some applications the short time fractional fourier transform stfrft is proposed to solve this problem it displays the time and frfd frequency information jointly in the short time fractional fourier domain stfrfd two aspects of its performance are considered the 2 d resolution and the stfrfd support the time frfd bandwidth product tfbp is defined to measure the resolvable area and the stfrfd support the optimal stfrft is obtained with the criteria that maximize the 2 d resolution and minimize the stfrfd support its inverse transform properties and computational complexity are presented two applications are discussed the estimations of the time of arrival toa and pulsewidth pw of chirp signals and the stfrfd filtering simulations verify the validity of the proposed algorithms

flattening and analytic continuation of affinoid morphisms remarks on a paper of gardener and schoutens we give an example of an affinoid curve without analytic continuation we use this to produce an example of an affinoid morphism that cannot be flattened by a finite sequence of local blow ups thus the global rigid analogue of hironaka s complex analytic flattening theorem given by t gardener and h schoutens in theorem 2 3 of flattening and subanalytic sets in rigid analytic geometry proc london math soc 3 83 2001 681 707 is not true since this is a key step in the proof of the affinoid elimination theorem loc cit theorem 3 12 that proof contains a serious gap we also give an example of an affinoid subset of the plane that is not the image under a proper rigid analytic map of a set that is globally semianalytic in the domain of that map this clarifies the relationship among several natural categories of rigid subanalytic sets

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

the fractional fourier transform and time frequency representations the functional fourier transform frft which is a generalization of the classical fourier transform was introduced a number of years ago in the mathematics literature but appears to have remained largely unknown to the signal processing community to which it may however be potentially useful the frft depends on a parameter spl alpha and can be interpreted as a rotation by an angle spl alpha in the time frequency plane an frft with spl alpha spl pi 2 corresponds to the classical fourier transform and an frft with spl alpha 0 corresponds to the identity operator on the other hand the angles of successively performed frfts simply add up as do the angles of successive rotations the frft of a signal can also be interpreted as a decomposition of the signal in terms of chirps the authors briefly introduce the frft and a number of its properties and then present some new results the interpretation as a rotation in the time frequency plane and the frft s relationships with time frequency representations such as the wigner distribution the ambiguity function the short time fourier transform and the spectrogram these relationships have a very simple and natural form and support the frft s interpretation as a rotation operator examples of frfts of some simple signals are given an example of the application of the frft is also given

harmonic analysis with a real frequency function i aperiodic case an integral transform which converts a real spatial or temporal function into a real frequency function is introduced the properties of this transform are investigated it is concluded that this transform is parallel to the fourier transform and may be applied to all fields in which the fourier transform has been successfully applied

filtering in rotated time frequency domains with unknown noise statistics the concept of rotation in the joint time frequency plane can be exploited in order to generalize classical fourier based operations it is known that filtering in rotated time frequency domains can lead to significant performance advantages for certain types of signals as compared to conventional linear time invariant systems in this correspondence we revisit the design problem of such a scheme and derive a formulation that does not require knowledge of the statistics of the corrupting noise simulations have been used to confirm the validity of the proposed solution

discrete fractional fourier transform based on new nearly tridiagonal commuting matrices based on discrete hermite gaussian like functions a discrete fractional fourier transform dfrft which provides sample approximations of the continuous fractional fourier transform was defined and investigated recently in this paper we propose a new nearly tridiagonal matrix which commutes with the discrete fourier transform dft matrix the eigenvectors of the new nearly tridiagonal matrix are shown to be dft eigenvectors which are more similar to the continuous hermite gaussian functions than those developed before rigorous discussions on the relations between the eigendecomposition of the newly proposed nearly tridiagonal matrix and the dft matrix are described furthermore by appropriately combining two linearly independent matrices that both commute with the dft matrix we develop a method to obtain dft eigenvectors even more similar to the continuous hermite gaussian functions hgfs then new versions of dfrft produce their transform outputs closer to the samples of the continuous fractional fourier transform and their applications are described related computer experiments are performed to illustrate the validity of the works in this paper

wigner ville spectral analysis of nonstationary processes the wigner ville spectrum has been recently introduced as the unique generalized spectrum for time varying spectral analysis its properties are revised with emphasis on its central role in the analysis of second order properties of nonstationary random signals we propose here a general class of spectral estimators of the wigner ville spectrum this class is based on arbitrarily weighted covariance estimators and its formal description corresponds to the general class of conjoint time frequency representations of deterministic signals with finite energy classical estimators like short time periodograms and the recently introduced pseudo wigner estimators are shown to be special cases of the general class the generalized framework allows the calculation of the moments of general spectral estimators and comparing the results emphasizes the versatility of the new pseudo wigner estimators the effective numerical implementation by an n point fft of pseudo wigner estimators of 2n points is indicated and various examples are given

repeated filtering in consecutive fractional fourier domains and its application to signal restoration filtering in a single time domain or in a single frequency domain has been generalized to filtering in a single fractional fourier domain we generalize this to repeated filtering in consecutive fractional fourier domains and discuss its applications to signal restoration through an illustrative example

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

signal reconstruction from two close fractional fourier power spectra based on the definition of the instantaneous frequency signal phase derivative as a local moment of the wigner distribution we derive the relationship between the instantaneous frequency and the derivative of the squared modulus of the fractional fourier transform fractional fourier transform power spectrum with respect to the angle parameter we show that the angular derivative of the fractional power spectrum can be found from the knowledge of two close fractional power spectra it permits us to find the instantaneous frequency and to solve the phase retrieval problem up to a constant phase term if only two close fractional power spectra are known the proposed technique is noniterative and noninterferometric the efficiency of the method is demonstrated on several examples including monocomponent multicomponent and noisy signals it is shown that the proposed method works well for signal to noise ratios snrs higher than about 3 db the appropriate angular difference of the fractional power spectra used for phase retrieval depends on the complexity of the signal and can usually reach several degrees other applications of the angular derivative of the fractional power spectra for signal analysis are discussed the proposed technique can be applied for phase retrieval in optics where only the fractional power spectra associated with intensity distributions can be easily measured

harmonic analysis with a real function of frequency ii periodic and bounded cases a real representation which is different from but equivalent to the fourier complex representation for a periodic function by trigonometric series is described a generalized series expression of a function defined on a finite interval is formed and its properties are investigated

time frequency projection filters and time frequency signal expansions we consider the problems of designing a linear time varying filter with a specified time frequency tf pass region and of constructing an orthonormal basis for the parsimonious expansion of signals located in a given tf support region these problems of tf filtering and tf signal expansion are reduced to the problem of designing a tf subspace i e a linear signal space comprising all signals located in a given tf legion specifically the tf filter is taken to be the orthogonal projection operator on the tf subspace we present an optimum design of tf subspaces that is based on the wigner distribution of a linear signal space and is an extension of the well known signal synthesis problem the optimum tf subspace is shown to be an eigenspace of the tf region and some properties of eigenspaces are discussed the performance of tf projection filters and tf signal expansions is studied both analytically and via computer simulation

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

approximate signal reconstruction using nonuniform samples in fractional fourier and linear canonical transform domains approximate signal reconstruction formulas for the class of l 2 r signals in the fractional fourier and linear canonical transform lct domains are presented the results make use of the finite number of nonuniform samples of the signal in fractional fourier or lct domains taken at the positions determined by the zeros of the hermite polynomials the results provide exact representation for the class of signals that can be expressed in the form of polynomials of some finite order the truncation error bounds in the presented results are also discussed simulation results for some of the proposed theorems are also presented

a method for the discrete fractional fourier transform computation a new method for the discrete fractional fourier transform dfrft computation is given in this paper with the help of this method the dfrft of any angle can be computed by a weighted summation of the dfrfts with the special angles

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

representations and computations of 2 3 we investigate full rank representations of 2 3 and 2 4 inverses with prescribed rank as well as with prescribed range and null space in an indefinite inner product space in certain particular cases we obtain new results or generalizations related with 2 3 and 2 4 inverses in a hilbert unitary space direct and iterative methods for computing 2 3 and 2 4 inverses in an indefinite inner product space are developed numerical examples with different metric matrices are presented

second order time frequency synthesis of nonstationary random processes we present time frequency methods for the synthesis of finite energy nonstationary random processes the energetic characteristics of the process to be synthesized are specified in a joint time frequency domain via a time frequency model function the synthesis methods optimize the autocorrelation function of the process such that the process wigner ville spectrum is closest to the given model function an optional signal subspace constraint allows the incorporation of additional properties such as bandlimitation and also permits the reformulation of the synthesis methods in a discrete time setting the synthesized process is expressed either in terms of an orthonormal basis of the constraint subspace or via its karhunen loeve expansion an example involving the prolate spheroidal functions is given and computer simulation results are provided

fractional krawtchouk transform with an application to image watermarking this paper proposes a novel fractional transform denoted as the fractional krawtchouk transform frkt a generalization of the krawtchouk transform the derivation of the frkt uses the eigenvalue decomposition method we determine the eigenvalues and the corresponding multiplicity of the krawtchouk transform matrix moreover the orthonormal eigenvectors of the transform matrix are derived for validation purpose only and as a first illustration of the interest of frkt a watermarking example was chosen experimental results show that better watermark robustness and imperceptibility are achieved by adjusting the fractional orders in the frkt

matrices with multiple symmetry properties applications of centrohermitian and perhermitian matrices abstract twelve known symmetry patterns of matrices are combined with three modest patterns to form a steiner triple system we investigate matrices satisfying more than one symmetry pattern we show how a group of operators on gl n c gives rise to distinct types of matrices which satisfy sets of patterns and which give unique decompositions of matrices into components of each type these give a new characterization of normal and unitary matrices we extend symmetry patterns to vectors to study spectral properties of these matrices when a skew symmetric basis of eigenvectors exist we can infer symmetry properties of these matrices

fractional power spectrum in this paper by investigating the definitions of the fractional power spectrum and the fractional correlation for the deterministic process we consider the case associated with the random process in an explicit manner the fractional power spectral relations for the fractional fourier domain filter are derived and the expression for the fractional power spectrum in terms of the fractional correlation is obtained in addition the definitions and the properties of the fractional white noise and the chirp stationary process are presented simulation results verify the theoretical derivations and demonstrate the potential applications such as detection and parameter estimation of chirp signals fractional power spectral estimation and system identification in the fractional fourier domain

i categories as a framework for solving domain equations abstract an abstract notion of category of information systems or i category is introduced as a generalisation of scott s well known category of information systems as in the theory of partial orders i categories can be complete or algebraic and it is shown that algebraic i categories can be obtained from a certain completion of countable i categories the proposed axioms for a complete i category introduce a global partial order on the morphisms of the category making them a cpo an initial algebra theorem for a class of functors continuous on the cpo of morphisms is proved thus giving canonical solution of domain equations an effective version of these results for algebraic i categories is also provided some basic examples of i categories representing the categories of sets boolean algebras scott domains and continuous scott domains are constructed

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

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

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

characterization and properties of matrices with generalized symmetry or skew symmetry let r cn n be a nontrivial involution i e r r 1 i we say that a cn n is r symmetric r skew symmetric if rar a rar a there are positive integers r and s with r s n and matrices p cn r and q cn s such that p p ir q q is rp p and rq q we give an explicit representation of an arbitrary r symmetric matrix a in terms of p and q and show that solving az w and the eigenvalue problem for a reduce to the corresponding problems for matrices app cr r and aqq cs s we also express a 1 in terms of app 1 and aqq 1 under the additional assumption that r r we show that moore penrose inversion and singular value decomposition of a reduce to the corresponding problems for app and aqq we give similar results for r skew symmetric matrices these results are known for the case where r j i n j 1 i j 1n however our proofs are simpler even in this case we say that a cn n is r conjugate if rar a where r rn n and r r 1 i in this case r a is r symmetric and i a is r skew symmetric so our results provide explicit representations for r conjugate matrices in terms of p and q which are now in rn r and rn s respectively we show that solving az w inverting a and the eigenvalue problem for a reduce to the corresponding problems for a related matrix s rn n if rt r this is also true for moore penrose inversion and singular value decomposition of a

linear summation of fractional order matrices yeh and pei presented a computation method for the discrete fractional fourier transform dfrft that the dfrft of any order can be computed by a linear summation of dfrfts with special orders based on their work we investigate linear summation of fractional order matrices in a general and comprehensive manner in this paper we have found that for any diagonalizable periodic matrices linear summation of fractional order forms with special orders is related to the size and the period of the fractional order matrix moreover some properties and generalized results about linear summation of fractional order matrices are also presented

optimal filtering in fractional fourier domains for time invariant degradation models and stationary signals and noise the classical fourier domain wiener filter which can be implemented in o n log n time gives the minimum mean square error estimate of the original undistorted signal for time varying degradations and nonstationary processes however the optimal linear estimate requires o n sup 2 time for implementation we consider filtering in fractional fourier domains which enables significant reduction of the error compared with ordinary fourier domain filtering for certain types of degradation and noise especially of chirped nature while requiring only o n log n implementation time thus improved performance is achieved at no additional cost expressions for the optimal filter functions in fractional domains are derived and several illustrative examples are given in which significant reduction of the error by a factor of 50 is obtained

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

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

real eigenvalues of nonnegative matrices which commute with a symmetric matrix involution abstract it is well known that if p is a nonnegative matrix then its spectral radius is an eigenvalue of p perron frobenius theorem in this paper it is shown that if p is an n n nonnegative matrix and it commutes with a nonnegative symmetric involution when n 4 m 3 then 1 p has at least two real eigenvalues if n 4 m or 4 m 2 2 p has at least one real eigenvalue if n 4 m 1 and 3 p has at least three real eigenvalues if n 4 m 3 where m is a nonnegative integer and n 1 examples are given to show that these results are the best possible and nonnegative symmetric involutions are classified

information systems for continuous posets abstract the method of information systems is extended from algebraic posets to continuous posets by taking a set of tokens with an ordering that is transitive and interpolative but not necessarily reflexive this develops the results of raney on completely distributive lattices and of hoofman on continuous scott domains and also generalizes smyth s r structures various constructions on continuous posets have neat descriptions in terms of these continuous information systems here we describe hoffmann lawson duality which could not be done easily with r structures and vietoris power locales we also use the method to give a partial answer to a question of johnstone s in the context of continuous posets vietoris algebras are the same as localic semilattices

the fractional discrete cosine transform the extension of the fourier transform operator to a fractional power has received much attention in signal theory and is finding attractive applications the paper introduces and develops the fractional discrete cosine transform dct on the same lines discussing multiplicity and computational aspects similarities and differences with respect to the fractional fourier transform are pointed out

flattening and subanalytic sets in rigid analytic geometry let k be an algebraically closed field endowed with a complete non archimedean norm with valuation ring r let f colon y to x be a map of k affinoid varieties in this paper we study the analytic structure of the image f y subset x such an image is a typical example of a subanalytic set we show that the subanalytic sets are precisely the mathbf d semianalytic sets where mathbf d is the truncated division function first introduced by denef and van den dries this result is most conveniently stated as a quantifier elimination result for the valuation ring r in an analytic expansion of the language of valued rings to prove this we establish a flattening theorem for affinoid varieties in the style of hironaka which allows a reduction to the study of subanalytic sets arising from flat maps that is we show that a map of affinoid varieties can be rendered flat by using only finitely many local blowing ups the case of a flat map is then dealt with by a small extension of a result of raynaud and gruson showing that the image of a flat map of affinoid varieties is open in the grothendieck topology using embedded resolution of singularities we derive in the zero characteristic case a uniformization theorem for subanalytic sets a subanalytic set can be rendered semianalytic using only finitely many local blowing ups with smooth centres as a corollary we obtain the fact that any subanalytic set in the plane r2 is semianalytic 2000 mathematical subject classification 32p05 32b20 13c11 12j25 03c10

rigid subanalytic subsets of the line and the plane working over an algebraically closed complete non archimedean non trivially valued field we show that any subanalytic subset of the line is semialgebraic and that any subanalytic subset of the plane is semianalytic

fractional convolution and correlation via operator methods and an application to detection of linear fm signals using operator theory methods together with our previously introduced unitary fractional operator we derive explicit definitions of fractional convolution and correlation operations in a systematic and comprehensive manner via operator manipulations we also provide alternative formulations of those fractional operations that suggest efficient algorithms for discrete implementation through simulation examples we demonstrate how well the proposed efficient method approximates the continuous formulation of fractional autocorrelation it is also shown that the proposed fractional autocorrelation corresponds to radial slices of the ambiguity function af we also suggest an application of the fast fractional autocorrelation for detection and parameter estimation of linear fm signals

two dimensional affine generalized fractional fourier transform as the one dimensional 1 d fourier transform can be extended into the 1 d fractional fourier transform frft we can also generalize the two dimensional 2 d fourier transform sahin et al see appl opt vol 37 no 11 p 2130 41 1998 have generalized the 2 d fourier transform into the 2 d separable frft which replaces each variable 1 d fourier transform by the 1 d frft respectively and the 2 d separable canonical transform further replaces frft by the canonical transform sahin et al see appl opt vol 31 no 23 p 5444 53 1998 have also generalized it into the 2 d unseparable frft with four parameters in this paper we introduce the 1 d affine generalized fractional fourier transform agfft it has even further extended the 2 d transforms described above it is unseparable and has in total ten degrees of freedom we show that the 2 d agfft has many wonderful properties such as the relations with the wigner distribution shifting modulation operation and the differentiation multiplication operation although the 2 d agfft form seems very complex in fact the complexity of the implementation will not be more than the implementation of the 2 d separable frft besides we also show that the 2 d agfft extends many of the applications for the 1 d frft such as the filter design optical system analysis image processing and pattern recognition and will be a very useful tool for 2 d signal processing

solution of the energy concentration problem in reproducing kernel hilbert space the problem of maximizing the energy of a signal bandlimited to sigma sigma on an interval tau tau in the time domain is one of the important classical problems in signal processing this problem was solved by a group of mathematicians slepian landau and pollak at bell labs in the 1960s the solution involved the prolate spheroidal wave functions more recently pei and ding solved the energy problem for more general transforms than the fourier transform such as the fractional fourier transform and the linear canonical transform their solution involved what they called the generalized prolate spheroidal wave functions the goals of this article are 1 to show that the problem of finding a bandlimited signal with maximum energy concentration in a given interval is a special case of a more general problem in reproducing kernel hilbert spaces 2 to solve the problem in the setting of a hilbert space and then obtain the solutions for the fourier fractional fourier and linear canonica

time variant filtering of signals in the mixed time frequency domain a new approach to the problem of time variant filtering is presented this approach is based upon the generation of the mixed time frequency representation mtfr of a signal multiplication of that representation by a time frequency function h t and obtaining a filtered output by an inverse operation the resultant filter is linear if the time frequency representation used is the complex spectrogram in contrast the filter is nonlinear if the wigner distribution function is used not every function of two variables is an allowed mtfr of a signal some conditions must be satisfied if the function produced by the product of the signal mtfr and the filter function is not an allowed mtfr an approximation based on projection onto the space of allowed mtfr functions is investigated this approximation yields a filtered function whose mtfr is as close as possible in the least squared sense to the desired

method for defining a class of fractional operations the fractional fourier transform frt permits a variety of associated fractional operations this correspondence proposes a systematic method based on the structure of the frt which not only provides unambiguous extensions of ordinary operations but permits writing the applicable expressions simply by inspection the approach also exposes the possible paths for implementing such operations

fractional spectrum analysis for nonuniform sampling in the presence of clock jitter and timing offset this paper focuses on analyzing the fractional spectrum of nonuniform sampling which is affected by clock jitter and timing offset both cases of single channel and recurrent samplings with limited bandwidth in the fractional fourier domain are considered for which we study two types of discrete time fractional fourier transforms dtfrfts for the first type dtfrfts we perform their statistical analyses and meanwhile establish relationships between the fractional spectra of the input and output signals of non ideal single multi channel analog to digital converter such relationships indicate that the statistical mean of each fractional spectrum contains periodic replicas of the spectrum for the input signal and these replicas are particularly modulated by the characteristic function of the perturbations introduced by the clock jitter moreover we interpret the first type dtfrfts from the perspective of linear frequency modulation signal decomposition for nonuniform sampling the second type dtfrfts serve as approximate dtfrfts and they especially deal with the general situation when the nonuniform sampling instants are unknown to this end we study modified forms of the first type dtfrfts with known inputs in kernels based on which the statistical mean of the approximate fractional spectra for both cases are derived then we compensate the fractional spectrum bias with high accuracy by developing optimal filters that minimize the mean square error between the original and the compensated fractional spectra for both cases different from single channel sampling the compensation for recurrent sampling involves an additional step before optimal filtering which gives better accuracy simulation results show that the proposed methods outperform existing methods

fractional chirp scaling algorithm mathematical model in this paper the new fractional chirp scaling algorithm frcsa is mathematically derived and analyzed from high resolution sar imaging point of view the frcsa is based on the use of the fractional fourier transform within the chirp scaling algorithm csa the performance of the frcsa is compared to that of the classical csa that is based on the fast fourier transform fft simulation results for airborne sar systems show that the frcsa offers enhanced signal to noise ratio snr better focusing capabilities sharpened main lobe with extra power concentration and greater side lobe reduction ratio slrr and all of these without the need for any additional focusing or enhancement techniques that are required in the classical fft based csa

relations between gabor transforms and fractional fourier transforms and their applications for signal processing many useful relations between the gabor transform gt and the fractional fourier transform frft have been derived first we find that like the wigner distribution function wdf the frft is also equivalent to the rotation operation of the gt then we show that performing the scaled inverse fourier transform ift along an oblique line of the gt of f t can yield its frft since the gt is closely related to the frft we can use it for analyzing the characteristics of the frft compared with the wdf the gt does not have the cross term problem this advantage is important for the applications of filter design sampling and multiplexing in the frft domain moreover we find that if the gt is combined with the wdf the resultant operation called the gabor wigner transform gwt also has rotation relation with the frft we also derive the general form of the linear distribution that has rotation relation with the frft

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

persymmetric and bisymmetric nonnegative inverse eigenvalue problem abstract the nonnegative inverse eigenvalue problem niep is the problem of finding conditions for the existence of an n n entrywise nonnegative matrix a with prescribed spectrum this problem remains open for n 5 if the matrix a is required to be persymmetric bisymmetric the problem will be called persymmetric bisymmetric nonnegative inverse eigenvalue problem pniep bniep persymmetric and bisymmetric matrices are common in physical sciences and engineering they arise for instance in the control of mechanical and electric vibrations a persymmetric version of a perturbation result due to rado and presented by h perfect in 5 is developed and used to give sufficient conditions for the pniep to have a solution our results generate an algorithmic procedure to compute the solution matrix

a multicarrier system based on the fractional fourier transform for time frequency selective channels traditional multicarrier techniques perform a frequency domain decomposition of a channel characterized by frequency selective distortion in a plurality of subchannels that are affected by frequency flat distortion the distortion in each independent subchannel can then be easily compensated by simple gain and phase adjustments typically digital fourier transform schemes make the implementation of the multicarrier system feasible and attractive with respect to single carrier systems however when the channel is time frequency selective as it usually happens in the rapidly fading wireless channel this traditional methodology fails since the channel frequency response is rapidly time varying the optimal transmission reception methodology should be able to process nonstationary signals in other words the subchannel carrier frequencies should be time varying and ideally decompose the frequency distortion of the channel perfectly at any instant in time however this ideally optimal approach presents significant challenges both in terms of conceptual and computational complexity the idea disclosed in this work is that a nonstationary approach can be approximated using signal bases that are especially suited for the analysis synthesis of nonstationary signals we propose in fact the use of a multicarrier system that employs orthogonal signal bases of the chirp type that in practice correspond to the fractional fourier transform signal basis the significance of the methodology relies on the important practical consideration that analysis synthesis methods of the fractional fourier type can be implemented with a complexity that is equivalent to the traditional fast fourier transform

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

multiple 3d far field near field moving target localization using wideband echo chirp signals in this paper we propose a technique for active detection and estimation of the range radial velocity and direction of arrival doa of multiple moving targets in the three dimensional space for both far field and near field cases using wideband chirp signals the proposed technique is an extension of the method recently proposed by leong in multiple target localization using wideband echo chirp signals ieee trans signal process vol 61 no 16 pp 4077 4089 aug 2013 for the localization of multiple far field targets in the two dimensional space in our approach the received signals at a sensor array are first decomposed into the spherical harmonic components and a modal preprocessing step is then carried out on the received wideband signals so that they can be further processed using narrowband techniques for parameter estimation the application of fractional fourier transform on the modal preprocessed signals allows the proposed method to identify closely spaced targets this is demonstrated via computer simulations where the proposed technique achieved low root mean square errors in the parameters estimated in a scenario with closely spaced targets

time varying filtering and signal estimation using wigner distribution synthesis techniques the short time fourier transform stft the ambiguity function af and the wigner distribution wd are mixed time frequency signal representations that use fourier transform techniques to map a one dimensional function of time into a two dimensional function of time and frequency these mixed time frequency mappings have been used to analyze the local frequency characteristics of a variety of signals and systems although much work has also been done to develop stft and af synthesis algorithms that can be used to implement a variety of time varying signal processing operations no such synthesis techniques have thus far been developed for the wd in this paper a signal synthesis algorithm that works directly with the real valued high resolution wd will be derived examples of how this wd synthesis procedure can be used to perform time varying filtering operations or signal separation will be given

centrosymmetric nonnegative realization of spectra abstract a list 1 2 n of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix in this paper we intent to characterize those lists of complex numbers which are realizable by a centrosymmetric nonnegative matrix in particular we show that lists of nonnegative real numbers and lists of complex numbers of suleimanova type except in one particular case are always the spectrum of some centrosymmetric nonnegative matrix for the general lists we give sufficient conditions via a perturbation result we also show that for n 4 every realizable list of real numbers is also realizable by a nonnegative centrosymmetric matrix

multiple target localization using wideband echo chirp signals active target detection and localization is a classical signal processing problem that arises in various military and biomedical applications a novel method for the detection and estimation of the range velocity and direction of arrival doa of multiple far field targets using wideband chirp signals is proposed in this paper using the plane wave representation of the signals reflected from far field targets a modal preprocessing procedure is designed for the echo signals received at a concentric circular array following that the parameter estimation method for the multiple targets present is developed based on i the fractional fourier transform which projects a signal onto an orthonormal basis formed by chirps and ii the root music algorithm which is a high resolution doa estimation method originally proposed for narrowband signals received at a uniform linear array there is no limit on the number of targets detectable by the proposed method as long as the echo signals are separable in the chosen fractional fourier domain simulation results show that the proposed method demonstrates good performance with low root mean square errors in the parameters estimated under difficult conditions such as closely spaced targets and low signal to noise ratio

matrices that commute with a permutation matrix abstract let p be an n n permutation matrix and let p be the corresponding permutation let a be a matrix such that ap pa it is well known that when p is an n cycle a is permutation similar to a circulant matrix we present results for the band patterns in a and for the eigenstructure of a when p consists of several disjoint cycles these results depend on the greatest common divisors of pairs of cycle lengths

continuous information system abstract in this paper we generalise the notion of algebraic information system to continuous information system just as algebraic information systems are concrete representations of scott domains continuous information systems are concrete representations of continuous scott domains the class of continuous information systems can be generated from the subclass of qualitative information systems in the sense that the category of continuous information systems is equivalent to the karoubi envelope of the category of qualitative information systems because qualitative information systems correspond to the qualitative domains of girard theoret comput sci 45 159 192 1986 this implies that the category of continuous scott domains is equivalent to the karoubi envelope of the category of qualitative domains we show how constructions on qualitative information systems such as product and function space can be translated to constructions on continuous information systems among other things we prove that the category of qualitative information systems is weak cartesian closed finally we define two universal information systems

uniform properties of rigid subanalytic sets in the context of rigid analytic spaces over a non archimedean valued field a rigid subanalytic set is a boolean combination of images of rigid analytic maps we give an analytic quantifier elimination theorem for complete algebraically closed valued fields that is independent of the field in particular the analytic quantifier elimination is independent of the valued field s characteristic residue field and value group in close analogy to the algebraic case this provides uniformity results about rigid subanalytic sets we obtain uniform versions of smooth stratification for subanalytic sets and the lojasiewicz inequalities as well as a unfiorm description of the closure of a rigid semianalytic set

radon transformation of time frequency distributions for analysis of multicomponent signals the radon transform of a time frequency distribution produces local areas of signal concentration that facilitate interpretation of multicomponent signals the radon wigner transform can be efficiently implemented with dechirping in the time domain however only half of the possible projections through the time frequency plane can be realized because of aliasing we show here that the frequency dual to dechirping exists so that all of the time frequency plane projections can be calculated efficiently both time and frequency dechirping are shown to warp the time frequency plane rather rotating it producing an angle dependent dilation of the radon wigner projection axis we derive the discrete time equations for both time and frequency dechirping and highlight some practical implementation issues discrete dechirping is shown to correspond to line integration through the extended discrete rather than the discrete wigner ville distribution computationally dechirping is o 2n log 2n instead of o n sup 3 for direct projection and the computation is dominated by the fast fourier transform calculation the noise and cross term suppression of the radon wigner transform are demonstrated by several examples using dechirping and using direct radon wigner transformation

information systems revisited the general continuous case in this paper a new notion of continuous information system is introduced it is shown that the information systems of this kind generate exactly the continuous domains the new information systems are of the same logic oriented style as the information systems first introduced by scott in 1982 they consist of a set of tokens a consistency predicate and an entailment relation satisfying a set of natural axioms it is shown that continuous information systems are closely related to abstract bases indeed both categories are equivalent since it is known that the categories of abstract bases and or continuous domains are equivalent it follows that the category of continuous information systems is also equivalent to that of continuous domains in applications mostly subclasses of continuous domains are considered for example the domains have to be pointed algebraic bounded complete or fs conditions are presented that when fulfilled by a continuous information system force the generated domain to belong to the required subclass in most cases the requirements are not only sufficient but also necessary

re visiting axioms of information systems based on the investigation of the axioms used in the theory of information system the notion of simplified continuous information system is introduced which is considered as a generalization of continuous information systems just like the continuous information system it is also a concrete representation of continuous domains and categorically equivalent to the category of continuous domains and scott continuous functions what is more all types of information systems studied in the literature so far turn out to be special cases of the new type by defining linear information systems and their entailment functions we show that the states of a scott style information system are exactly the fixpoints of its entailment function some axioms about an entailment relation can be characterized by the properties of the corresponding entailment function and in particular the simplified continuous information systems are solely determined by the idempotence of entailment functions

relations between fractional operations and time frequency distributions and their applications the fractional fourier transform frft is a useful tool for signal processing it is the generalization of the fourier transform many fractional operations such as fractional convolution fractional correlation and the fractional hilbert transform are defined from it in fact the frft can be further generalized into the linear canonical transform lct and we can also use the lct to define several canonical operations in this paper we discuss the relations between the operations described above and some important time frequency distributions tfds such as the wigner distribution function wdf the ambiguity function af the signal correlation function and the spectrum correlation function first we systematically review the previous works in brief then some new relations are derived and listed in tables then we use these relations to analyze the applications of the frpt lct to fractional canonical filter design fractional canonical hilbert transform beam shaping and then we analyze the phase amplitude problems of the frft lct for phase amplitude problems we find as with the original fourier transform that in most cases the phase is more important than the amplitude for the frft lct we also use the wdf to explain why fractional canonical convolution can be used for space variant pattern recognition

the hermite transform theory the author introduces a scheme for the local processing of visual information called the hermite transform the problem is addressed from the point of view of image coding and therefore the scheme is presented as an analysis resynthesis system the objectives of the present work however are not restricted to coding the analysis part is designed so that it can also serve applications in the area of computer vision indeed derivatives of gaussians which have found widespread application in feature detection over the past few years play a central role in the hermite analysis it is also argued that the proposed processing scheme is in close agreement with current insight into the image processing that is carried out by the human visual system in particular it is demonstrated that the hermite transform is in better agreement with human visual modeling than gabor expansions

fractional fourier transform wigner distribution and filter design for stationary and nonstationary random processes in this paper we derive the relationship among the fractional fourier transform frft the linear canonical transform lct and the stationary and nonstationary random processes we find many interesting properties for example if we perform the frft for a stationary process although the result is no longer stationary the amplitude of the autocorrelation function is still independent of time we also find that the lct of a white noise is still a white one for the frft of a stationary process the ambiguity function af is a tilted line and the wigner distribution function wdf is invariant along a certain direction we also define the fractional stationary random process and find that a nonstationary random process can be expressed by a summation of fractional stationary random processes in addition after performing the filter designed in the frft domain for a white noise we can use the segment length of the axis on the wdf plane to estimate the power of the noise and use the area circled by cutoff lines to estimate its energy thus in communication to reduce the effect of the white noise the area of the wdf of the transmitted signal should be as small as possible

matrices a such that ra as 1r when rk i abstract this paper examines matrices a c n n such that ra a s 1 r where r k i the identity matrix and where s and k are nonnegative integers with k 2 spectral theory is used to characterize these matrices the cases s 0 and s 1 are considered separately since they are analyzed by different techniques

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

weak algebraic information systems and a new equivalent category of dom of domains abstract in this paper concepts of generalized algebraic information system in short ga inf and weak algebraic information system in short wa inf are introduced relationships among several algebraicities of information systems are explored main results are 1 a dcpo d is an algebraic domain iff d is represented by an algebraic information system in short a inf and iff d is represented by a ga inf 2 every a inf is a ga inf and a wa inf 3 every c inf s d b induced by a domain with a basis b is a wa inf 4 the category wainf of wa infs and approximable mappings is equivalent to the category dom of domains and scott continuous functions 5 every wa inf s satisfying condition mt is a retract of w s the information structure induced by s in category wainf of wa infs and approximable mappings particularly c inf s d b induced by a domain d with a basis b is a retract of a inf w s d b in category wainf

harmonic analysis with a real frequency function iii data sequence a continuous frequency spectrum w spectrum is formed by a data sequence the w transform of this spectrum is a continuous function which corresponds uniquely to the known data sequence

a representation of l domains by information systems in this paper we propose the notion of l information systems which provides a concrete representation of l domains in particular we prove that the category of l information systems with approximable mappings as morphisms is equivalent to that of l domains with scott continuous functions as morphisms

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

sampling and sampling rate conversion of band limited signals in the fractional fourier transform domain the fractional fourier transform frft has become a very active area in signal processing community in recent years with many applications in radar communication information security etc this study carefully investigates the sampling of a continuous time band limited signal to obtain its discrete time version as well as sampling rate conversion for the frft firstly based on product theorem for the frft the sampling theorems and reconstruction formulas are derived which explain how to sample a continuous time signal to obtain its discrete time version for band limited signals in the fractional fourier domain secondly the formulas and significance of decimation and interpolation are studied in the fractional fourier domain using the results the sampling rate conversion theory for the frft with a rational fraction as conversion factor is deduced which illustrates how to sample the discrete time version without aliasing the theorems proposed in this study are the generalizations of the conventional versions for the fourier transform finally the theory introduced in this paper is validated by simulations