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

multivariate rational approximants for multiclass closed queuing networks closed markovian networks of queues with multiclass customers and having a product form equilibrium state probability distribution are useful in the performance evaluation and design of computer and telecommunication systems therefore the efficient computation of the normalizing function the key element of the solution in product form has attracted considerable effort we consider a network that consists of one infinite server is station and one processor sharing ps or fcfs single server station we use multivariate newton pade approximants computed from data for small numbers of customers in each class to estimate the normalizing function for a larger population in the network the effectiveness and tremendous gain in computing time of this procedure are illustrated through various numerical experiments

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

short time fourier transform two fundamental properties and an optimal implementation shift and rotation invariance properties of linear time frequency representations are investigated it is shown that among all linear time frequency representations only the short time fourier transform stft family with the hermite gaussian kernels satisfies both the shift invariance and rotation invariance properties that are satisfied by the wigner distribution wd by extending the time bandwidth product tbp concept to fractional fourier domains a generalized time bandwidth product gtbp is defined for mono component signals it is shown that gtbp provides a rotation independent measure of compactness similar to the tbp optimal stft the gtbp optimal stft that causes the least amount of increase in the gtbp of the signal is obtained finally a linear canonical decomposition of the obtained gtbp optimal stft analysis is presented to identify its relation to the rotationally invariant stft

on the self fractional fourier functions it is shown that a self fractional fourier function for an angle where n and m are indivisible integers n m is also a self fractional fourier function for angles j 1 2 this allows us to define a self fractional fourier function of order m an eigenvalue of a fractional fourier operator for angle is equal to where l is an integer

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

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

reversible joint hilbert and linear canonical transform without distortion generalized analytic signal associated with the linear canonical transform lct was proposed recently by fu and li generalized analytic signal associated with linear canonical transform opt commun vol 281 pp 1468 1472 2008 however most real signals especially for baseband real signals cannot be perfectly recovered from their generalized analytic signals therefore in this paper the conventional hilbert transform ht and analytic signal associated with the lct are concerned to transform a real signal into the lct of its ht two integral transforms i e the ht and lct are required the goal of this paper is to simplify cascades of multiple integral transforms which may be the ht analytic signal lct or inverse lct the proposed transforms can reduce the complexity when realizing the relationships among the following six kinds of signals a real signal its ht and analytic signal and the lct of these three signals most importantly all the proposed transforms are reversible and undistorted using the proposed transforms several signal processing applications are discussed and show the advantages and flexibility over simply using the analytic signal or the lct

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

computing the region of convergence for power series in many real variables a ratio like test abstract we give an elementary proof that the region of convergence for a power series in many real variables is a star convex domain but not in general a convex domain in doing so we deduce a natural higher dimensional analog of the so called ratio test from univariate power series from the constructive proof of this result we arrive at a method to approximate the region of convergence up to a desired accuracy while most results in the literature are for rather specialized classes of multivariate power series the method devised here is general as far as applications are concerned note that while theorems such as the cauchy kowalevski theorem and its generalizations to many variables grant the existence of a region of convergence for a multivariate taylor series to certain pdes under appropriate restrictions they do not give the actual region of convergence the determination of the maximal region of convergence for such a series solution to a pde is one application of our result

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

discrete fractional fourier transforms based on closed form hermite gaussian like dft eigenvectors in this paper we construct discrete fractional fourier transforms dfrft using recently introduced closed form hermite gaussian like hgl eigenvectors with respect to such eigenvectors we discuss the convergence of their components to samples of the corresponding continuous hermite gaussian functions and propose solutions to deal with some restrictions related to their construction this allows us to give new procedures for obtaining orthonormal bases of hgl eigenvectors which are used to fractionalize the discrete fourier transform we illustrate the application of the resulting dfrft in the scenario of filtering in the fractional domain and compare the results with existing dfrft approaches

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

power filtering of nth order in the fractional fourier domain the main properties of the power filtering operation in the fractional fourier domain and its relationship to the differentiation operation are considered the application of linear power filtering for solving the phase retrieval problem from intensity distributions only is proposed the optical configuration for the experimental realization of the method is discussed

tomography time frequency transform the paper shows that the fractional fourier transform frft of a signal is the radon transform of the time frequency distribution of the same signal therefore a time frequency distribution known as the tomography time frequency transform ttft is defined as the inverse radon transform of the frft of the signal because the computation of the ttft does not explicitly require any window or kernel function high resolutions in both the frequency and time domains can be achieved when the signal contains multiple components the cross terms can be effectively removed by an adaptive filtering process that is applied on the frft rather than the final result therefore distortions made by the filtering process on the desired signal components can be minimized

general order newton pade approximants for multivariate functions pade approximants are a frequently used tool for the solution of mathematical problems one of the main drawbacks of their use for multivariate functions is the calculation of the derivatives off x 1 x p therefore multivariate newton pade approximants are introduced their computation will only use the value off at some points in sect 1 we shall repeat the univariate newton pade approximation problem which is a rational hermite interpolation problem in sect 2 we sketch some problems that can arise when dealing with multivariate interpolation in sect 3 we define multivariate divided differences and prove some lemmas that will be useful tools for the introduction of multivariate newton pade approximants in sect 4 a numerical example is given in sect 5 together with the proof that forp 1 the classical newton pade approximants for a univariate function are obtained

interpolating between periodicity and discreteness through the fractional fourier transform periodicity and discreteness are fourier duals in the same sense as operators such as coordinate multiplication and differentiation and translation and phase shift the fractional fourier transform allows interpolation between such operators which gradually evolve from one member of the dual pair to the other as the fractional order goes from zero to one here we similarly discuss the interpolation between the dual properties of periodicity and discreteness showing how one evolves into the other as the order goes from zero to one we also discuss the concepts of partial discreteness and partial periodicity and relate them to fractional discreteness and periodicity

self fourier functions the gaussian and dirac comb are often quoted as the only functions which are their own fourier transform ft f u f u the authors shows that for an arbitrary transformable g x the function f x g x g x g x g x is its own ft he gives physically reasonable examples and discusses some optical consequences

an uncertainty principle for real signals in the fractional fourier transform domain the fractional fourier transform frft can be thought of as a generalization of the fourier transform to rotate a signal representation by an arbitrary angle spl alpha in the time frequency plane a lower bound on the uncertainty product of signal representations in two frft domains for real signals is obtained and it is shown that a gaussian signal achieves the lower bound the effect of shifting and scaling the signal on the uncertainty relation is discussed an example is given in which the uncertainty relation for a real signal is obtained and it is shown that this relation matches with that given by the uncertainty relation derived

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

parameterized centroid frequency chirp rate distribution for lfm signal analysis and mechanisms of constant delay introduction this paper presents an extension of the integrated time chirp rate analysis technique known as the parameterized centroid frequency chirp rate distribution pcfcrd for noisy multicomponent linear frequency modulated signals analysis the pcfcrd is based on a newly defined correlation function and its auto term accumulation is two dimensionally coherent the principle cross term characteristic implementation properties parameter selection criterion antinoise performance and estimation accuracy are analyzed for the pcfcrd in this paper comparisons with the coherently integrated cubic phase function maximum likelihood method and lv s distribution are performed with mathematic analyses and numerical simulations we demonstrate that the pcfcrd outperforms these three representative counterparts further by combining with analyses of the pcfcrd several unclear mechanisms of the constant delay introduction are mathematically interpreted especially quantitative influences on the chirp rate resolution and theoretical antinoise performance

discrete fractional wavelet transform and its application to multiple encryption the fractional wavelet transform is a useful mathematical transformation that generalizes the most prominent tool in signal and image processing namely wavelet transform by rotation of signals in the time frequency plane the definition of discrete fractional wavelet transform is not reported yet in the literature therefore a definition of the discrete fractional wavelet transform is consolidated by discretizing continuous fractional wavelet transform in the proposed work possible applications of the transform are in transient signal processing image analysis image transmission biometrics image compression etc in this paper image transmission is chosen as the primary application and hence a novel encryption scheme is proposed for securing multiple images during communication and transmission over insecure channel the proposed multiple image encryption scheme is consolidated by fractional wavelet transform and chaotic maps first all the images are encrypted followed by their sharing the sharing process is done considering numerical techniques by making the sharing process a system of linear equations experimental results and security analysis demonstrate the efficiency and robustness of the proposed primary application

uncertainty principles for linear canonical transform this correspondence investigates the uncertainty principles under the linear canonical transform lct first a lower bound on the uncertainty product of signal representations in two lct domains for complex signals is derived which can be achieved by a complex chirp signal with gaussian envelope then the tighter lower bound for real signals in two lct domains proposed by sharma and joshi is also proven to hold for arbitrary lct parameters based on the properties of moments for the lct the uncertainty principle for the fractional fourier transform is a special case of the achieved results

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

fractional differential mask a fractional differential based approach for multiscale texture enhancement in this paper we intend to implement a class of fractional differential masks with high precision thanks to two commonly used definitions of fractional differential for what are known as grumwald letnikov and riemann liouville we propose six fractional differential masks and present the structures and parameters of each mask respectively on the direction of negative x coordinate positive x coordinate negative y coordinate positive y coordinate left downward diagonal left upward diagonal right downward diagonal and right upward diagonal moreover by theoretical and experimental analyzing we demonstrate the second is the best performance fractional differential mask of the proposed six ones finally we discuss further the capability of multiscale fractional differential masks for texture enhancement experiments show that for rich grained digital image the capability of nonlinearly enhancing complex texture details in smooth area by fractional differential based approach appears obvious better than by traditional integral based algorithms

comparison geometry for the bakry emery ricci tensor for riemannian manifolds with a measure m g edvolg we prove mean curvature and volume comparison results when the bakry emery ricci tensor is bounded from below and f is bounded or rf is bounded from below generalizing the classical ones i e when f is constant this leads to extensions of many theorems for ricci curvature bounded below to the bakry emery ricci tensor in particular we give extensions of all of the major comparison theorems when f is bounded simple examples show the bound on f is necessary for these results

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

a four parameter atomic decomposition of chirplets a new four parameter atomic decomposition of chirplets is developed for compact and precise representation of signals with chirp components the four parameter chirplet atom is obtained from the unit gaussian function by successive applications of scaling fractional fourier transform frft and time shift and frequency shift operators the application of the frft operator results in a rotation of the wigner distribution of the gaussian in the time frequency plane by a specified angle the decomposition is realized by using the matching pursuit algorithm for this purpose the four parameter space is discretized to obtain a small but complete subset in the hilbert space a time frequency distribution tfd is developed for clear and readable visualization of the signal components it is observed that the chirplet decomposition and the related tfd provide more compact and precise representation of signal inner structures compared with the commonly used time frequency representations

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

the generalized continuous wavelet transform associated with the fractional fourier transform the main objective of this paper is to study the fractional fourier transform frft and the generalized continuous wavelet transform and some of their basic properties applications of the frft in solving generalized nth order linear nonhomogeneous ordinary differential equations and a generalized wave equation are given the generalized continuous wavelet transform and its inversion formula and the parseval relation are also studied using the fractional fourier transform

fast discrete linear canonical transform based on cm cc cm decomposition and fft in this paper a discrete lct dlct irrelevant to the sampling periods and without oversampling operation is developed this dlct is based on the well known cm cc cm decomposition that is implemented by two discrete chirp multiplications cms and one discrete chirp convolution cc this decomposition doesn t use any scaling operation which will change the sampling period or cause the interpolation error compared with previous works dlct calculated by direct summation and dlct based on center discrete dilated hermite functions cddhfs the proposed method implemented by ffts has much lower computational complexity the relation between the proposed dlct and the continuous lct is also derived to approximate the samples of the continuous lct simulation results show that the proposed method somewhat outperforms the cddhfs based method in the approximation accuracy besides the proposed method has approximate additivity property with error as small as the cddhfs based method most importantly the proposed method has perfect reversibility which doesn t hold in many existing dlcts with this property it is unnecessary to develop the inverse dlct additionally because it can be replaced by the forward dlct

novel reversible integer fourier transform with control bits this paper presents a novel concept of the reversible integer discrete fourier transform ridft of order 2r r 2 when the transform is split by the paired representation into a minimum set of short transforms i e transforms of orders 2k k r by means of the paired transform the signal is represented as a set of short signals which carry the spectral information of the signal at specific and disjoint sets of frequencies the paired transform based fast fourier transform fft involves a few operations of multiplication that can be approximated by integer transforms examples of 1 point transforms with one control bit are described control bits allow us to invert such approximations two control bits are required to perform the 8 point ridft and 12 or even 8 bits for the 16 point ridft of real inputs the proposed forward and inverse ridfts are fast and the computational complexity of these transforms is comparative with the complexity of the fft the 8 point direct and inverse ridfts are described in detail

how well can the concept of pade approximant be generalized to the multivariate case what we know about multivariate pade approximation has been developed in the last 25 years in the next sections we compare and discuss many of these results it will become clear that simple properties or requirements such as the uniqueness of the pade approximant and consequently its consistency property can play a crucial role in the development of the multivariate theory a separate section is devoted to a discussion of the convergence properties at the end we include an extensive reference list on the topic

short time fractional fourier methods for the time frequency representation of chirp signals the fractional fourier transform frft provides a valuable tool for the analysis of linear chirp signals this paper develops two short time frft variants which are suited to the analysis of multicomponent and nonlinear chirp signals outputs have similar properties to the short time fourier transform stft but show improved time frequency resolution the frft is a parameterized transform with parameter a related to chirp rate the two short time implementations differ in how the value of a is chosen in the first a global optimization procedure selects one value of a with reference to the entire signal in the second a values are selected independently for each windowed section comparative variance measures based on the gaussian function are given and are shown to be consistent with the uncertainty principle in fractional domains for appropriately chosen frft orders the derived fractional domain uncertainty relationship is minimized for gaussian windowed linear chirp signals the two short time frft algorithms have complementary strengths demonstrated by time frequency representations for a multicomponent bat chirp a highly nonlinear quadratic chirp and an output pulse from a finite difference sonar model with dispersive change these representations illustrate the improvements obtained in using frft based algorithms compared to the stft

discrete chirp fourier transform and its application to chirp rate estimation the discrete fourier transform dft has found tremendous applications in almost all fields mainly because it can be used to match the multiple frequencies of a stationary signal with multiple harmonics in many applications wideband and nonstationary signals however often occur one of the typical examples of such signals is chirp type signals that are usually encountered in radar signal processing such as synthetic aperture radar sar and inverse sar imaging due to the motion of a target the radar return signals are usually chirps and their chirp rates include the information about the target such as the location and the velocity in this paper we study discrete chirp fourier transform dcft which is analogous to the dft besides the multiple frequency matching similar to the dft the dcft can be used to match the multiple chirp rates in a chirp type signal with multiple chirp components we show that when the signal length n is prime the magnitudes of all the sidelobes of the dcft of a quadratic chirp signal are 1 whereas the magnitude of the mainlobe of the dcft is spl radic n with this result an upper bound for the number of the detectable chirp components using the dcft is provided in terms of signal length and signal and noise powers we also show that the n point dcft performs optimally when n is a prime

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

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

ricci curvature and volume convergence the purpose of this paper is to give a new integral estimate of distances and angles on manifolds with a given lower ricci curvature bound this will provide us with an integral version of the toponogov comparison triangle theorem for ricci curvature and almost extreme triangles see the earlier works cl and c2 for an analog of this when the manifold has positive ricci curvature using this we prove the anderson cheeger conjecture saying that the volume is a continuous function on the space of all closed n manifolds with ricci curvature greater or equal to n 1 equipped with the gromovhausdorff metric we also prove gromov s conjecture for n 57 3 saying that an almost nonnegatively ricci curved n manifold with first betti number equal to n is a torus further we prove a conjecture of anderson cheeger saying that an open n manifold with nonnegative ricci curvature whose tangent cone at infinity is in is itself in finally we prove a conjecture of fukaya yamaguchi we will now describe these results in more detail let dgh denote the gromov hausdorff distance glp first we have the following result which was conjectured by anderson cheeger

interference suppression in the wigner distribution using fractional fourier transformation and signal synthesis a novel method for the suppression of cross terms in the time frequency domain is introduced first interference is identified using a fractional fourier transform based technique then auto terms are detected synthesized and subtracted from the original signal the process is repeated until all signal components are extracted finally the wigner distributions of pure auto terms are superimposed to yield a high readability representation

exploiting early time response using the fractional fourier transform for analyzing transient radar returns this paper presents a new technique for estimating parameters of damped sinusoids and impulse like responses utilizing both early and late time transient scattering responses transient scattering responses are composed of damped sinusoids at late times and impulse like components at early times due to the impulse like components it is difficult to extract meaningful damped sinusoids when analyzing the complete data set in this paper the entire time domain response is used to extract the signal parameters of interest utilizing both the early and late times the fractional fourier transform frft especially the half fourier transform hft is used to analyze the data for parameter identification impulse or gaussian like pulses can be easily separated from the late time damped exponentials in the hft domain as they have similar functional representations in addition the damped exponentials have a turn on time which needs to be solved for results from two examples show that the new technique is applicable for signals that are composed of damped exponentials with a turn on time and short pulse like components

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

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

a class of fractional integral transforms a generalization of the fractional fourier transform the paper presents a systematic and unified approach to fractional integral transforms we introduce a new class of fractional integral transforms that includes the fractional fourier and hankel transforms and the fractional integration and differentiation operators as special cases these fractional transforms may also be viewed as angular transforms indexed by an angular parameter spl alpha since their kernels are obtained by taking the limits of analytic functions in the unit disc along a radius making an angle spl alpha with the x axis

weighted least squares implementation of cohen posch time frequency distributions with specified conditional and joint moment constraints a positivity constrained iterative weighted least squares wls method for constructing non negative joint time frequency distributions i e cohen posch 1985 tfds satisfying marginal joint moment conditional moment and generalized marginal constraints is developed the new algorithm solves the leakage problem of the least squares approach and is computationally faster it is also more computationally efficient than the mce implementation of these constraints developed by loughlin pitton and atlas 1994

eigenfunctions of linear canonical transform the linear canonical transform the lct is a useful tool for optical system analysis and signal processing it is parameterized by a 2 spl times 2 matrix a b c d many operations such as the fourier transform ft fractional fourier transform frft fresnel transform and scaling operations are all the special cases of the lct we discuss the eigenfunctions of the lct the eigenfunctions of the ft frft fresnel transform and scaling operations have been known and we derive the eigenfunctions of the lct based on the eigenfunctions of these operations we find for different cases that the eigenfunctions of the lct also have different forms when a d 2 the eigenfunctions become the chirp multiplication and chirp convolution of self similar functions fractals besides since many optical systems can be represented by the lct we can thus use the eigenfunctions of the lct derived in this paper to discuss the self imaging phenomena in optics we show that there are usually varieties of input functions that can cause the self imaging phenomena for an optical system

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

generalized self fourier functions by generalizing the definition of self fourier functions recently introduced by caola 1991 the authors show that any fourier transformable function is the linear combination of four self fourier functions whose explicit form can be found

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

sparse interpolation of multivariate rational functions consider the black box interpolation of a sparse n variate rational function f where is the maximum number of terms in either numerator or denominator when numerator and denominator are at most of degree d then the number of possible terms in f is o dn and explodes exponentially as the number of variables increases the complexity of our sparse rational interpolation algorithm does not depend exponentially on n anymore it still depends on d because we densely interpolate univariate auxiliary rational functions of the same degree we remove the exponent n and introduce the sparsity in the complexity by reconstructing the auxiliary function s coefficients via sparse multivariate interpolation the approach is new and builds on the normalization of the rational function s representation our method can be combined with probabilistic and deterministic components from sparse polynomial black box interpolation to suit either an exact or a finite precision computational environment the latter is illustrated with several examples running from exact finite field arithmetic to noisy floating point evaluations in general the performance of our sparse rational black box interpolation depends on the choice of the employed sparse polynomial black box interpolation if the early termination ben or tiwari algorithm is used our method achieves rational interpolation in o d black box evaluations and thus is sensitive to the sparsity of the multivariate f

on the structure of almost einstein manifolds in this paper we study the structure of the limit space of a sequence of almost einstein manifolds which are generalizations of einstein manifolds roughly speaking such manifolds are the initial manifolds of some normalized ricci flows whose scalar curvatures are almost constants over space time in the l1 sense ricci curvatures are bounded from below at the initial time under the non collapsed condition we show that the limit space of a sequence of almost einstein manifolds has most properties which is known for the limit space of einstein manifolds as applications we can apply our structure results to study the properties of kahler manifolds

a recursive computation scheme for multivariate rational interpolants we derive here a recursive computation scheme for the rational interpolation method introduced in 7 explicit formulas for these multivariate rational interpolants are repeated in 2 while the recursive algorithm is described in 3 a number of interesting special cases such as the univariate rational interpolation problem and the multivariate pade approximants introduced in 6 and 10 are dealt with in 4 for some of these rational approximants other recursive schemes were described previously finally 5 contains the numerical results the multivariate rational interpolants described here are compared with multivariate polynomial interpolants interpolating branched continued fractions introduced by cuyt and verdonk 8 interpolating branched continued fractions introduced by siemaszko 12 and several multivariate pade approximants 6 3 10 besides the fact that our multivariate rational interpolants allow a large degree of freedom in the choice for the numerator and denominator in ord

pade approximation for stochastic discrete event systems we show that pade approximation can be effectively used for approximation of performance functions in discrete event systems the method is 1 obtaining the maclaurin coefficients of the performance function and 2 finding a pade approximant from the maclaurin coefficients and use it to approximate the function we use the method with the expected number of renewals in a random interval gi g 1 systems and inventory systems the results are very good

self fractional fourier functions and selection of modes it is shown that any function from can be represented as a linear combination of m self fractional fourier functions of order m which are orthogonal to each other each of them contains a selection of hermite gauss modes of the generator function the physical meaning of the synthesis of self fractional fourier functions is discussed

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

uncertainty principle for real signals in the linear canonical transform domains the linear canonical transform lct is a generalization of the fractional fourier transform frft having applications in several areas of signal processing and optics in this paper we extend the uncertainty principle for real signals in the fractional fourier domains to the linear canonical transform domains giving us the tighter lower bound on the product of the spreads of the signal in two specific lct domains than the existing lower bounds in the lct domains it is seen that this lower bound can be achieved by a gaussian signal the effect of time shifting and scaling the signal on the uncertainty principle is also discussed it is shown here that a signal bandlimited in one lct domain can be bandlimited in some other lct domains also the exceptions to the uncertainty principle in the lct domains arising out of this are also discussed

closed form expressions for time frequency operations involving hermite functions the product convolution correlation wigner distribution function wdf and ambiguity function af of two hermite functions of arbitrary order n and m are derived and expressed as a bounded weighted sum of n m hermite functions it was already known that these mathematical operations performed on gaussians hermite functions of the zeroth order lead to a result which can be expressed as a gaussian function again we generalize this reciprocity to hermite functions of arbitrary order the product convolution correlation wdf and af operations performed on two hermite functions of arbitrary order lead to remarkably similar closed form expressions where the difference between the operations is primarily determined by distinct phase changes of the weights of the hermite functions in the result the closed form expressions are generalized to the class of square integrable functions a key insight from the closed form expressions is applied to the design of orthogonal time frequency localized communication signals which are characterized by an af with rotational symmetry in addition to this application the theoretical expressions may prove useful for signal analysis in fields ranging from communications radar and image processing to quantum mechanics

continuum quantum systems as limits of discrete quantum systems ii state functions in this second of four papers on the eponymous topic pointwise convergence of a discrete state function to a continuum state function is shown to imply the algebraic criterion for convergence that was introduced in the prequel as examples and as a prerequisite for the sequels the normal approximation theorem and the convergence of the kravchuk functions to the hermite gaussians are expressed in terms of the algebraic notion of convergence

linear frequency modulated signal detection using radon ambiguity transform a novel time frequency technique for linear frequency modulated lfm signal detection is proposed the design of the proposed detectors is based on the radon transform of the modulus square or the envelope amplitude of the ambiguity function af of the signal a practical assumption is made that the chirp rate is the only parameter of interest since the af of lfm signals will pass through the origin of the ambiguity plane the line integral of the radon transform is performed over all lines passing through the origin of the ambiguity plane the proposed detectors yield maxima over chirp rates of the lfm signals this reduces the two dimensional 2 d problem of the conventional wigner ville distribution wvd based detection or the radon wigner transform rwt based detector to a one dimensional 1 d problem and consequently reduces the computation load and keeps the feature of built in filtering related issues such as the finite length effect the resolution and the effect of noise are studied the result is a tool for lfm detection as well as the time varying filtering and adaptive kernel design for multicomponent lfm signals

sparse discrete fractional fourier transform and its applications the discrete fractional fourier transform is a powerful signal processing tool with broad applications for nonstationary signals in this paper we propose a sparse discrete fractional fourier transform sdfrft algorithm to reduce the computational complexity when dealing with large data sets that are sparsely represented in the fractional fourier domain the proposed technique achieves multicomponent resolution in addition to its low computational complexity and robustness against noise in addition we apply the sdfrft to the synchronization of high dynamic direct sequence spread spectrum signals furthermore a sparse fractional cross ambiguity function sfrcaf is developed and the application of sfrcaf to a passive coherent location system is presented the experiment results confirm that the proposed approach can substantially reduce the computation complexity without degrading the precision

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

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

large manifolds with positive ricci curvature the main purpose of this paper is to show that an n dimensional manifold with ricci curvature greater or equal to n 1 which is close in the gromov hausdor topology to the unit n sphere has volume close to that of the sphere this shows the converse of the theorem in c1 namely together with c1 it shows that an n manifold with ricci curvature greater or equal to n 1 is close to the sphere if and only if the volume is close to that of the sphere in particular by p such a manifold is homeomorphic to a sphere further as an application of this and the result of c1 we prove a radius theorem saying that if an n manifold with ricci curvature greater or equal to n 1 has radius almost equal to then the volume is close to that of the sphere in order to obtain these results we further develop and apply the estimates of c1 whereas the main concern in c1 were with the large scale geometry the main concern of this paper is with the small scale geometry let n be the volume of the round n sphere sn with sectional curvature one

generalized prolate spheroidal wave functions associated with linear canonical transform time limited and a b c d band limited signals are of great interest not only in theory but also in real applications in this paper we use the sampling theorem associated with linear canonical transform to investigate an operator whose effect on a signal is to produce its first time limited then b b c d band limited version first the eigenvalue problem for the operator is shown to be equivalent to a discrete eigenvalue problem for an infinite matrix then the eigenfunctions of the operator which are referred to as generalized prolate spheroidal wave functions gpswfs are shown to be first orthogonal over finite as well as infinite intervals and second complete over l2 l l and the class of a b c d band limited signals a simple method based on sampling theorem for computing gpswfs is presented and the definite parity of gpswfs is also given finally based on the dual orthogonality and completeness of gpswfs several applications of gpswfs to the representation of time limited and a b c d band limited signals are presented

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

on sampling of band limited signals associated with the linear canonical transform sampling is one of the fundamental topics in the signal processing community theorems proposed under this topic form the bridge between the continuous time signals and discrete time signals several sampling theorems which aid in the reconstruction of signals in the linear canonical transform lct domain have been proposed in the literature however two main practical issues associated with the sampling of the lct still remain unresolved the first one relates to the reconstruction of the original signal from nonuniform samples and the other issue relates to the fact that only a finite number of samples are available practically focusing on these issues this paper seeks to address the above from the lct point of view first we extend several previously developed theorems for signals band limited in the fourier domain to signals band limited in the lct domain followed by the derivation of the reconstruction formulas for finite uniform or recurrent nonuniform sampling points associated with the lct simulation results and the potential applications of the theorem are also proposed

linear canonical stockwell transform abstract the linear canonical transform lct has long been recognized as a competent tool for optics and signal processing however the lct cannot reveal the local lct frequency contents due to its global kernel and as such makes the lct incompetent in situations demanding a joint information of time and linear canonical domain frequency in this paper we propose the linear canonical stockwell transform lcst in order to rectify the limitations of the lct and the stockwell transform st firstly we examine the resolution of the novel transform in the time and lct domains and then derive some of its basic properties such as moyal s formula inversion formula and provide a characterization of the range furthermore we derive a direct relationship between the well known wigner distribution and the proposed lcst in sequel a discrete version of the linear canonical stockwell transform is also presented finally we extend the scope of the proposed work by studying the lcst in the realm of almost periodic functions

lower bounds on ricci curvature and the almost rigidity of warped products the basic rigidity theorems for manifolds of nonnegative or positive ricci curvature are the volume cone implies metric cone theorem the maximal diameter theorem cg and the splitting theorem cg each asserts that if a certain geometric quantity volume or diameter is as large as possible relative to the pertinent lower bound on ricci curvature then the metric on the manifold in question is a warped product metric of a particular type in this paper we provide quantitative generalizations of the above mentioned results among the applications are the splitting theorem for gromovhausdorff limit spaces x where mn x ricmn i see fy other applications include the assertion that for complete manifolds m with ricmf 0 and euclidean volume growth all tangent cones at infinity are metric cones compare bkn ct p1 via resealing arguments there are also strong consequences for the local structure of manifolds whose ricci curvature satisfies a fixed lower bound and for their gromov hausdorff limits some of these are announced in ccol for a more detailed discussion see cco2 cco3 cco4 our work further develops and significantly extends techniques which were introduced in col co2 and significantly extended in co3 in order to prove certain stability conjectures of anderson cheeger gromov and perelman the results of col co3 were announced in co4 we briefly review some of those results let dgh denote the gromov hausdorff distance between metric spaces see glp let s denote the unit sphere and recall that s is the unique complete

method of flow graph simplification for the 16 point discrete fourier transform an efficient method for the realization of the paired algorithm for calculation of the one dimensional 1 d discrete fourier transform dft by simplifying the signal flow graph of the transform is described the signal flow graph is modified by separating the calculation for real and imaginary parts of all inputs and outputs in the signal flow graph and using properties of the transform the examples for calculation of the eight and 16 point dfts are considered in detail the calculation of the 16 point dft of real data requires 12 real multiplications and 58 additions two multiplications and 20 additions are used for the eight point dft

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