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

random walk on fractals numerical studies in two dimensions monte carlo calculations are used to investigate some statistical properties of random walks on fractal structures two kinds of lattices are used the sierpinski gasket and the infinite percolation cluster in two dimensions among other problems the authors study i the range rn of the walker number of distinct visited sites during n steps average value sn variance sigma n and asymptotic distribution ii renewal theory return to the original site probability of return p0 n mean number of returns nu n the probability distribution of the walker position p n r after n steps is discussed the asymptotic behaviour n 1 of these quantities exhibits power laws with associated exponents the numerical values of these exponents are in good agreement with recent theoretical predictions alexander and orbach 1982 rammal and toulouse 1982

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

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

a multicarrier architecture based upon the affine fourier transform recently innovative multicarrier schemes have been proposed that exploit the transmission of chirp shaped waves e sup j2 spl pi ct2 by optimally choosing the chirp parameter c on the basis of the channel characteristics and are more robust to time varying channels than ordinary ofdm schemes this concept was applied to continuous time and to discrete time systems in the present paper we aim at developing those ideas using the affine fourier transform aft which is a very general formulation of chirp transforms we present a multicarrier modulation based upon the discrete form of the aft that is therefore inherently discrete and strictly invertible moreover it allows to define a circular prefix concept that is coherent with the chirp nature of the transmission the system can be efficiently implemented by adding a simple phase correction block to standard ofdm modulators demodulators and can effectively combat interchannel interference when the propagation channel is made of few multipath components affected by independent frequency offsets our discrete time multicarrier scheme is an improved version of martone s approach as we also show by simulation results and exhibits analogous characteristics to barbarossa s continuous time system

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

sliding discrete linear canonical transform the linear canonical transform lct has been shown to be one of the most powerful tools in signal processing and in this paper we propose an adaptive approach for the computation of the discrete lct dlct termed the sliding discrete linear canonical transform sdlct first we introduce a scheme for the single point dlct which can effectively calculate a single or a few linear canonical spectra second the sdlct is proposed based on an iterative algorithm to meet the requirements of online spectral analysis when only a subset of n frequencies are required from an tilde n hbox point discrete lct n leq tilde n the additivity and reversibility properties of the proposed algorithms are also discussed in detail third the dlct convolution operation is obtained to reduce the spectral leakage of the proposed algorithm and time domain windowing is implemented via frequency domain convolution finally we present two methods to assess performance with regard to computational complexity and precision and to show the correctness of the derived results

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

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

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

asymptotic theory of propeller noise ii supersonic single rotation propeller this paper gives analytical formulas for the far field harmonic components of the acoustic radiation from a b bladed supersonic single rotation propeller in the asymptotic limit b o the dominant source for radiation in a given direction is identified as located at the mach radius along the blade span the radius that approaches the observer at precisely sonic speed once in each rotation favorable agreement is found between the asymptotic prediction and the results of full numerical evaluation of the radiation integrals singular surfaces in the field are identified from the asymptotic formulas which are extended to include chordwise noncompactness effects these show among other things that supersonic propeller noise is an algebraically decreasing function of tip mach number once chordwise noncompactness effects are taken into account in contrast to the exponential increase with tip mach number of subsonic propeller noise

topological properties of percolation clusters percolation properties including the total cluster mass s the shell mass b and the linear geometrical size r are studied as a function of the topological chemical distance parameter l all critical exponents are shown to be related to an apparently new exponent nu defined by r approximately lv critical exponents are calculated exactly for percolation clusters on the cayley tree a model for 6d percolation for which s approximately l2 b approximately l and r2 approximately l for diffusion on such clusters one finds that l3 approximately t numerical estimates of the exponents are obtained for other dimensions a conjecture which relates nu to beta and nu is discussed

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

scaling theory for the dynamic density profile at criticality a scaling description for the dynamic density profile of kinetic clusters formed by the sites visited during random walks executed on critical percolation clusters is proposed the analysis is done for both euclidean and fractal spaces various percolative diffusion laws are correctly recovered from this dynamic density profile

the structure of rotating sound fields an account is given of the three dimensional structure of the sound field produced by a spinning sinusoidal distribution of thickness or loading sources particular attention is paid to the creeping evanescent waves in the near field and their physical interpretation as trapped edge waves an exponentially small amount of energy leaks or tunnels through them to emerge as far field acoustic radiation as it does through the inhomogeneous waves carried by a waveguide in a cylindrically layered medium the dependence of the structure on three parameters is investigated in detail the mach number m at the outermost radius of the source the harmonic number n defined so that the source strength is a function of n theta where theta denotes azimuthal angle and the type of source i e thickness or loading parameter values considered include those for subsonic sonic and supersonic motion and for high and low harmonics the field is calculated by reducing a special case of rayleigh s double integral to a single integral containing a function related to the chebyshev polynomials then integrating numerically to give contour plots of pressure as a function of position on various plane and cylindrical sections these show that the evanescent waves occupy a spherical or ellipsoidal region and consist of crescents of alternating high and low pressure shaped and arranged like the segments of an orange its peel marks the transition to the propagating spiral waves of the far field i e the radiation zone contour plots on meridional sections are similar to those for the oscillating hertzian electric dipole suggesting that the field is approximately that produced by a suitably phased arrangement of its acoustic counterpart when m 1 the source distribution straddles both the evanescent and the radiation zone at high supersonic m the meridional contour plots display an intense beaming pattern with side lobes between the main beam and source plane the results of the paper agree with previous work on propeller acoustics especially the asymptotic theory

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

series expansions of rotating two and three dimensional sound fields the cylindrical and spherical harmonic expansions of oscillating sound fields rotating at a constant rate are derived these expansions are a generalized form of the stationary sound field expansions the derivations are based on the representation of interior and exterior sound fields using the simple source approach and determination of the simple source solutions with uniform rotation numerical simulations of rotating sound fields are presented to verify the theory

transform order division multiplexing multiplexing in the fractional fourier domain frfd only transmits the signals in a single direction in the time frequency plane and the transform order of the fractional fourier transform frft is not used sufficiently in this paper we apply the perfect reconstruction transmultiplexer in the frfd to develop the transform order division multiplexing todm which transmits the original information in the multidirections in the time frequency plane simultaneously the proposed todm systems can be divided into two types one needs the multitransmission of the original information while the other needs the modifications of the original information by the interpolation and the inverse discrete frft the performance analysis of the todm systems is also presented including the system capacity the reconstruction accuracy the security and the limitation compared with the multiplexing in the frfd the todm is more flexible and the security of the system is highly strengthened finally simulation results confirm the obtained analytical results

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

the sound field of a rotor in a stationary duct a method is developed for the prediction of noise from rotors in an axisymmetric duct at rest the technique combines a previously developed model for noise from disc sources with a boundary integral equation method for scattering from the duct surface recent developments in the theory of gaussian quadrature and generalized elliptic integrals are used to simplify the implementation of the formulation the sound field of a simplified model ducted rotor is then calculated and its structure examined in the framework of known features of the field around rotating sources it is found that as for an open rotor the sonic radius plays an important role in the structure of the field

diffusion on percolation clusters at criticality the concept of fractal dimensionality is used to study the problem of diffusion on percolation clusters the authors find from monte carlo simulations that the fractal dimensionality of a random walk on a critical percolation cluster in three dimensional space is d 3 3 or 0 1 where the size of the cluster is restricted to be larger than the span of the walk and is d 3 9 or 0 1 for a walk on clusters not subject to this restriction for two dimensional space they find d approximately d approximately 2 7 or p0 1 the exponent d and d is related to the scaling of the average length r of n steps via rd varies as n the fracton dimensionality which is related to the density of states was found to be 1 26 or 0 1 these results are in good agreement with the predictions of alexander and orbach 1982

the asymptotic theory of rapidly rotating sound fields a rapidly rotating sound field such as that produced by a supersonic propeller may contain several types of diffraction pattern each in a different region of space this paper determines these patterns and their locations for the field around a propeller of simple type the method used is stationary phase analysis of a certain double integral and leads to asymptotic formulae valid when the number of blades is not too small or the high harmonics are being investigated physically the results describe propagation along rays each stationary phase point is a loud spot producing a ray which points directly at the observer most of the noise comes from these loud spots because extensive cancellation takes place everywhere else at most two interior and two boundary stationary points may be present the number and type depend on the position of the observer in relation to a cusped torus and two hyperboloids of one sheet as these surfaces are crossed the acoustic field changes in character for example when two stationary points coalesce and annihilate each other as they do at a caustic an airy function describes the transition from a loud zone of rapid oscillation to a quiet zone of exponential decay and when an interior stationary point crosses the boundary of the disc the transition region is described either by a fresnel integral or by a generalized airy function separate analyses are given for regions close to and well away from the transition surfaces and inner and outer limits are calculated for use in the method of matched asymptotic expansions in all cases an overlap region is found in which the leading terms agree the results of the paper determine completely the geometry of the acoustic field because the different regions have boundaries at known positions and cover the whole of space

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

diffusion on random systems above below and at their percolation threshold in two and three dimensions a detailed monte carlo study is presented for classical diffusion random walks on random l l triangular and l l l simple cubic lattices with l up to 4096 and 256 respectively the speed of a cyber 205 vector computer is found to be about one order of magnitude larger than that of a usual cdc cyber 76 computer to reach the asymptotic scaling regime walks with up to 10 million steps were simulated with about 10 jl steps in total for l 256 at the percolation threshold we review and extend the dynamical scaling description for the distance traveled as function of time the diffusivity above the threshold and the cluster radius below earlier discrepancies between scaling theory and computer experiment are shown to be due to insufficient monte carlo data the conductivity exponent is found to be 2 0 0 2 in three and 1 28 0 02 in two dimensions our data in three dimensions follow well the finite size scaling theory below the threshold the approach of the distance traveled to its asymptotic value is consistent with theoretical speculations and an exponent 2 5 independent of dimensionality the correction to scaling exponent at pc seems to be larger in two than in three dimensions

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

time delay estimation of chirp signals in the fractional fourier domain a new time delay estimator is presented in this paper it is evaluated based on the delay property of the fractional fourier transform with less computation and is suitable for chirp signals the statistical analysis in terms of signal to noise ratio snr and estimation accuracy for this estimation is also studied the proposed method for the time delay yields a variance which is theoretically equal to the cramer rao lower bound the validity of this estimation method is demonstrated via simulations

diffusion on random clusters and the parasite problem in the parasite problem a particle ant diffuses randomly on a random percolation cluster in the limit of concentration to 0 lattice animal monte carlo simulations and scaling arguments show that for large animals the distance r travelled by this parasite increases as t 1 za with time t the authors find za approximately 3 4 on the simple cubic lattice and za approximately 2 6 on the square lattice this anomalous diffusion is in rough agreement with a generalisation of a suggestion by alexander and orbach 1982 heuristic arguments in favour of this suggestion are given also they look at corrections to scaling for concentrations equal to the percolation threshold

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