new look on q2 r point fast fourier transforms in this paper the method of splitting orparallelization of calculation of the n point discrete fourier transform dft by the dfts of smaller orders is described for that the concept of partitions revealing the one dimensional 1 d dft of order q2 r where r 1and q 1 is a positive odd number is described two different partitions are considered and the corresponding effective algorithms of calculation of the q2 r point 1 d dft are described for the q 3 case the calculation of the transform is based on the paired representations when the signal is represented as a set of 1 d signals which define the 1 d dft in disjoint subsets of frequency points which cover the set of all frequencies the splittings of the q2 r point 1 d dft are performed by the 1 d discrete 2and qpaired transforms which allow for calculating with a minimum number of operations the examples of the paired transforms and computational complexity of the proposed algorithms for the n 6 and 12 cases are given

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

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

exact and asymptotic features of the edge density profile for the one component plasma in two dimensions there is a well known analogy between the laughlin trial wave function for the fractional quantum hall effect and the boltzmann factor for the two dimensional one component plasma the latter requires continuation beyond the finite geometry used in its derivation we consider both disk and cylinder geometry and focus attention on the exact and asymptotic features of the edge density at the special coupling gamma q 2 k bt 2 the system is exactly solvable in particular the k point correlation can be written as a k times k determinant allowing the edge density to be computed to first order in gamma 2 a double layer structure is found which in turn implies an overshoot of the density as the edge of the leading support is approached from the interior asymptotic analysis shows that the deviation from the leading order step function value is different for the interior and exterior directions for general gamma a gaussian fluctuation formula is used to study the large deviation form of the density for n large but finite this asymptotic form involves thermodynamic quantities which we independently study and moreover an appropriate scaling gives the asymptotic decay of the limiting edge density outside of the plasma

large deviation eigenvalue density for the soft edge laguerre and jacobi β ensembles we analyze the eigenvalue density for the laguerre and jacobi ensembles in the cases that the corresponding exponents are extensive in particular we obtain the asymptotic expansion up to terms o 1 in the large deviation regime outside the limiting interval of support as found in recent studies of the large deviation density for the gaussian ensemble and laguerre ensemble with fixed exponent there is a scaling from this asymptotic expansion to the right tail asymptotics for the distribution of the largest eigenvalue at the soft edge

efficient computation of the dft with only a subset of input or output points ways of efficiently computing the discrete fourier transform dft when the number of input and output data points differ are discussed the two problems of determining whether the length of the input sequence or the length of the output sequence is reduced can be found to be duals of each other and the same methods can to a large extent be used to solve both the algorithms utilize the redundancy in the input or output to reduce the number of operations below those of the fast fourier transform fft algorithms the usual pruning method is discussed and an efficient algorithm called transform decomposition is introduced it is based on a mixture of a standard fft algorithm and the horner polynomial evaluation scheme equivalent to the one in goertzel s algorithms it requires fewer operations and is more flexible than pruning the algorithm works for power of two and prime factor algorithms as well as for real input data

comments on split manageable efficient algorithm for fourier and hadamard transforms the paper split manageable efficient algorithm for fourier and hadamard transforms in ieee transactions on signal processing vol 48 no 1 pp 172 183 january 2000 presents a novel paired transform fast fourier transform fft algorithm that is claimed to require fewer total real additions and multiplications than previous algorithms we show this claim to be false although the algorithm correctly computes the discrete fourier transform an accurate count shows that it requires more real arithmetic operations than pre existing ffts

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

on the number of multiplications necessary to compute a length 2 n dft the number of multiplications necessary and sufficient to compute a length 2ndft is determined the method of derivation is shown to apply to the multiplicative complexity results of winograd for a length pndft for p an odd prime number the multiplicative complexity of the one dimensional dft is summarized for many possible lengths

duhamel hollmann like discrete fourier transform algorithm with the smallest multiplicative complexity over a finite field the new method for the discrete fourier transform computation over a finite field is introduced this method is a nontrivial generalization of the duhamel hollmann algorithm with replacement of the toeplitz convolution calculation by the normalized cyclic convolution calculation both algorithms have the smallest multiplicative complexity

split manageable efficient algorithm for fourier and hadamard transforms in this paper a general efficient manageable split algorithm to compute one dimensional 1 d unitary transforms by using the special partitioning in the frequency domain is introduced the partitions determine fast transformations that split the n point unitary transform into a set of n sub i point transforms i 1 n n sub 1 n sub n n here we introduce a class of splitting transformations the so called paired transforms based on these transforms the decompositions of the fourier transforms of arbitrary orders are given and the corresponding algorithms are considered comparative estimates revealing the efficiency of the proposed algorithms with respect to the known ones are given in particular a proposed method of calculating the 2 sup r point fourier transform requires 2 sup r 1 r 3 2 multiplications and 2 sup r 1 r 9 r sup 2 3r 6 additions in terms of the paired transforms the splitting of the 2 sup r point hadamard transform is described as a result the proposed algorithm for computing this transform uses on the average no more than six operations of additions per sample

variations of hodge structure legendre submanifolds and accessibility variations of hodge structure of weight two are integral manifolds for a distribution in the tangent bundle of a period domain this distribution has dimension h2 0h 1 and is nonintegrable for h2 0 1 in this case it is known that the dimension of an integral manifold does not exceed 1h2oh 2 here we give a new proof based on an analogy between griffiths horizontal differential system of algebraic geometry and the contact system of classical mechanics we show also that any two points in such a domain can be joined by a horizontal curve which is piecewise holomorphic

fluctuation formula for complex random matrices a gaussian fluctuation formula is proved for linear statistics of complex random matrices in the case where the statistic is rotationally invariant for a general linear statistic without this symmetry coulomb gas theory is used to predict that the distribution will again be a gaussian with a specific mean and variance the variance splits naturally into a bulk and surface contibution the latter resulting from the long range correlations at the boundary of the support of the eigenvalue density

scaled radix 2 8 algorithm for efficient computation of length n 2 m dfts this paper presents a scaled radix 2 8 fast fourier transform fft sr28fft algorithm for computing length n 2m discrete fourier transforms dfts scaled by complex number rotating factors the idea of the sr28fft algorithm is from the modified split radix fft msrfft algorithm and its purpose is to furnish other algorithms with high efficiency but without shortcomings of the msrfft algorithm a novel radix 2 4 fft nr24fft algorithm and a novel radix 2 8 fft nr28fft algorithm are proposed these two algorithms use sr28fft to calculate their sub dfts of odd indexed terms several aspects of the two algorithms such as computational complexity computation accuracy and coefficient evaluations or accesses to the lookup table all are improved the bit reverse method can be used for their order permutation and no extra memory is required to store their extra coefficients by the two novel algorithms which contribute significantly to the performance of the fft algorithms the sr28fft algorithm can also be applied to other algorithms whose decomposition contains sub dfts of powers of two the appendix presents an algorithm named sr28fft 2 for further reducing the number of arithmetic operations and nr24fft and nr28fft algorithms based on sr28fft 2 requires fewer real operations than that required by the msrfft algorithm

a fast algorithm with less operations for length n q times 2 m dfts discrete fourier transform dft is widely used in almost all fields of science and engineering fast fourier transform fft is an efficient tool for computing dft in this paper we present a fast fourier transform fft algorithm for computing length q times 2 m dfts the algorithm transforms all q points sub dfts into three parts in the second part the operations of subtransformation contain only multiplications by real constant factors by transformation length 2 m scaled dfts sdft are obtained an extension of scaled radix 2 8 fft sr28fft is presented for computing these sdfts in which the real constant factors of sdfts are attached to the coefficients of sub dfts to simplify multiplication operations the proposed algorithm achieves reduction of arithmetic complexity over the related algorithms it can achieve a further reduction of arithmetic complexity for computing a length n q times 2 m idft by 2n 4m real multiplications in addition the proposed algorithm is applied to real data fft and is extended to 6 m dfts

tensor transform based quaternion fourier transform algorithm we present a concept of the tensor representation of color images in quaternion algebra we present concepts of direction components composing the color image we propose a new method of the 2 d quaternion discrete fourier transform qdft we calculate the 2 d qdft with minimum number of operations of multiplication we examine the proposed method in rgb color image model the traditional fourier transform is a fundamental tool in signal and image processing which in color imaging transforms images either in monochrome or as separate color channels thereby ignoring interactions between the color channels the quaternion discrete fourier transform qdft a generalization of the fourier transform represents the three color channels of an image in the rgb color space as a vector field in the quaternion arithmetic allowing for simultaneous analysis of all color data it is expected that this new color imaging model which captures the interactions between the different color channels will improve upon existing image processing and recognition systems efficient qdft algorithms are derived by combining classical dft transforms enabling standard fast fourier transform fft software to provide a fast qdft numerical implementation we developed an open source quaternion fourier transform tool that calculates the 2 d qdft by the column row method which is performed by calculating multiple 1 d qdfts where each 1 d transform is calculated by two complex 1 d dfts the matrices involved in these calculations necessitate the inclusion of a discussion on symplectic decomposition our main contribution is the presentation a new algorithm for the tensor representation of the qdft that is structurally simpler and uses less multiplications than its predecessors for instance for the images of size n i n where n 2 r r 2 the tensor 2 d qdft saves about 2 n 2 r real multiplications

comments on method of flow graph simplification for the 16 point discrete fourier transform the paper method of flow graph simplification for the 16 point discrete fourier transform by grigoryan and bhamidipati presents a paired transform fast fourier transform fft algorithm that is claimed to perform the size 8 and size 16 complex data discrete fourier transform dft with 44 and 140 arithmetic operations respectively if true this count would be less than the 56 and 168 operations achieved in the best pre existing split radix methods grigoryan and bhamidipati s count of real additions is erroneous however and this comment shows that their algorithm actually has arithmetic complexity identical to that of standard split radix algorithms

efficient discrete spatial techniques for blur support identification in blind image deconvolution this paper proposes two discrete spatial techniques for identifying blur support in blind image deconvolution blur identification is a challenging problem in blind image deconvolution in particular if the blur support can be estimated reliably in the beginning of restoration the computational cost of many blind deconvolution schemes can be reduced significantly and their convergence performance improved this paper proposes two methods called maximum average square difference masd and maximum average absolute difference maad they are derived from the autoregressive ar model of the underlying images the efficiency and validity of the techniques are also analyzed in this paper a main advantage of the proposed techniques is their algorithmic and implementation simplicity experimental results show that they are effective in identifying the blur support reliably thus providing a sound foundation for further blind image deconvolution

finite size corrections to the free energy of coulomb systems with a periodic boundary condition classical coulomb systems ind dimensions d 2 with a periodic boundary condition periodw in the directionx d are considered with the other directions of the confining volume of lengthl it is shown that if the system is in a conducting phase then the strip free energyktf w f w liml l d 1 log z has the large w expansion f w sim wf infty frac d 2 1 gamma d 2 1 pi d 2 w d 1 zeta d o left frac 1 w d 1 right wherektf is the bulk free energy per unit volume x denotes the riemann zeta function and x denotes the gamma function with 1 w identified askt this result is precisely the low temperature behavior of the free energy of a d 1 dimensional debye solid this fact is explained in terms of an equivalence between the coulomb gas and quantum fields also the expansion is verified for some exactly solved models of coulomb systems in two dimensions

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

coupled contact systems and rigidity of maximal dimensional variations of hodge structure in this article we prove that locally griffiths horizontal distribution on the period domain is given by a generalized version of the familiar contact differential system as a consequence of this description we obtain strong local rigidity properties of maximal dimensional variations of hlodge structure for example we prove that if the weight is odd greater than one then there is a unique germ of maximal dimensional variation of hodge structure through every point of the period domain similar results hold if the weight is even with the exception of one case

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

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

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

translation symmetry breaking in the one component plasma on the cylinder the two dimensional one component plasma i e the system of point like charged particles embedded in a homogeneous neutralizing background is studied on the surface of a cylinder of finite circumference or equivalently in a semiperiodic strip of finite width the model has been solved exactly by choquardet al at the free fermion coupling 2 in the thermodynamic limit of an infinitely long strip the particle density turns out to be a nonconstant periodic function in space and the system exhibits long range order of the wigner crystal type the aim of this paper is to describe qualitatively as well as quantitatively the crystalline state for a larger set of couplings 2 1 2 a positive integer when the plasma is mappable onto a one dimensional fermionic theory the fermionic formalism supplemented by some periodicity assumptions reveals that the density profile results from a hierarchy of gaussians with a uniform variance but with different amplitudes the number and spatial positions of these gaussians within an elementary cell depend on the particular value of analytic results are supported by the exact solution at 1 2 and by exact finite size calculations at 2 3

bounded fluctuations and translation symmetry breaking a solvable model the variance of the particle number equivalently the total charge in a domain of length mathcal l of a one component plasma ocp on a cylinder of circumference w at the reciprocal temperature 2 is shown to remain bounded as mathcal l this exactly solvable system with average density has a density profile which is periodic with period w 1 along the axis of the infinitely long cylinder this illustrates the connection between bounded variance and periodicity in quasi one dimensional systems 1 when w the system approaches the two dimensional ocp and the variance in a domain grows like its perimeter in this limit the system is translation invariant with rapid decay of correlations

split radix algorithms for length p sup m dft s the split radix algorithm for the discrete fourier transform dft of length 2 sup m is considered first the reason why the split radix algorithm is better than any single radix algorithm on length 2 sup m dfts is given then the split radix approach is generalized to length p sup m dfts it is shown that whenever a radix p sup 2 outperforms a radix p algorithm then a radix p p sup 2 algorithm will outperform both of them as an example a radix 3 9 algorithm is developed for length 3 sup m dfts

large n expansion for the 2d dyson gas we discuss the 1 n expansion of the free energy of n logarithmically interacting charges in the plane in an external field for some particular values of the inverse temperature this system is equivalent to the eigenvalue version of certain random matrix models where it is referred to as the dyson gas of eigenvalues to find the free energy at large n and the structure of 1 n corrections we first use the effective action approach and then confirm the results by solving the loop equation the results obtained give some new representations of the mathematical objects related to the dirichlet boundary value problem complex analysis and spectral geometry of exterior domains they also suggest interesting links with bosonic field theory on riemann surfaces gravitational anomalies and topological field theories

hypersurface variations are maximal ii the variation of hodge structure defined by the natural family of hypersurfaces of degreed and dimensionn is maximal if the cohomology has hodge level 1 there is a small list of hypersurfaces of level one which give non maximal variations plane curves of degreed 5 cubics of dimension 3 and 5 and quartic threefolds

kronecker products unitary matrices and signal processing applications discrete unitary transforms are extensively used in many signal processing applications and in the development of fast algorithms kronecker products have proved quite useful in this semitutorial paper we briefly review properties of kronecker products and direct sums of matrices which provide a compact notation in treating patterned matrices a generalized matrix product which inherits some useful algebraic properties from the standard kronecker product and allows a large class of discrete unitary transforms to be generated from a single recursion formula is then introduced the notation is intimately related to sparse matrix factorizations and examples are included illustrating the utility of the new notation in signal processing applications finally some novel characteristics of hadamard transforms and polyadic permutations are derived in the framework of kronecker products

basis vector decomposition based two stage computational algorithms for dft and dht discrete fourier transform dft discrete hartley transform dht algorithms based on the basis vector decomposition of the corresponding transform matrices are derived the computations of dft are divided into two stages an add subtract preprocessing and a block diagonal postprocessing both stages can be computed effectively it can be proved that the computational complexity of the proposed dft algorithm is identical to that of the most popular split radix fft yet requires real number arithmetics only generation and storage of the real multiplicative coefficients are easier than that in conventional ffts connections of the proposed approach with several well known dft algorithms are included furthermore many variations of the proposed algorithm are also pointed out

vicious random walkers in the limit of a large number of walkers the vicious random walker problem on a line is studied in the limit of a large number of walkers the multidimensional integral representing the probability that thep walkers will survive a timet denotedp is shown to be analogous to the partition function of a particular one component coulomb gas by assuming the existence of the thermodynamic limit for the coulomb gas one can deduce asymptotic formulas forp in the large p large t limit a straightforward analysis gives rigorous asymptotic formulas for the probability that after a timet the walkers are in their initial configuration this event is termed a reunion consequently asymptotic formulas for the conditional probability of a reunion given that all walkers survive are derived also an asymptotic formula for the conditional probability density that any walker will arrive at a particular point in timet given that allp walkers survive is calculated in the limitt p

fast computation of the discrete walsh and hadamard transforms the discrete walsh and hadamard transforms are often used in image processing tasks such as image coding pattern recognition and sequency filtering a new discrete walsh transform dwt algorithm is derived in which a modified form of the dwt relation is decomposed into smaller sized transforms using vectorized quantities a new sequency ordered discrete hadamard transform dhat algorithm is also presented the proposed approach results in more regular algorithms requiring no independent data swapping and fewer array index updating and bit reversal operations an analysis of the computational complexity and the execution time performance are provided the results are compared with those of the existing algorithms

transform based image enhancement algorithms with performance measure this paper presents a new class of the frequency domain based signal image enhancement algorithms including magnitude reduction log magnitude reduction iterative magnitude and a log reduction zonal magnitude technique these algorithms are described and applied for detection and visualization of objects within an image the new technique is based on the so called sequency ordered orthogonal transforms which include the well known fourier hartley cosine and hadamard transforms as well as new enhancement parametric operators a wide range of image characteristics can be obtained from a single transform by varying the parameters of the operators we also introduce a quantifying method to measure signal image enhancement called eme this helps choose the best parameters and transform for each enhancement a number of experimental results are presented to illustrate the performance of the proposed algorithms

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

generic integral manifolds for weight two period domains we define the notion of a generic integral element for the griffiths distribution on a weight two period domain draw the analogy with the classical contact distribution and then show how to explicitly construct an infinite dimensional family of integral manifolds tangent to a given element

2 d and 1 d multipaired transforms frequency time type wavelets a concept of multipaired unitary transforms is introduced these kinds of transforms reveal the mathematical structure of fourier transforms and can be considered intermediate unitary transforms when transferring processed data from the original real space of signals to the complex or frequency space of their images considering paired transforms we analyze simultaneously the splitting of the multidimensional fourier transform as well as the presentation of the processed multidimensional signal in the form of the short one dimensional 1 d signals that determine such splitting the main properties of the orthogonal system of paired functions are described and the matrix decompositions of the fourier and hadamard transforms via the paired transforms are given the multiplicative complexity of the two dimensional 2 d 2 sup r spl times 2 sup r point discrete fourier transform by the paired transforms is 4 sup r 2 r 7 3 8 3 12 r 3 which shows the maximum splitting of the 5 d fourier transform into the number of the short 1 d fourier transforms the 2 d paired transforms are not separable and represent themselves as frequency time type wavelets for which two parameters are united frequency and time the decomposition of the signal is performed in a way that is different from the traditional haar system of functions

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

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

16 point reversible integer discrete fourier transform with 12 control bits this correspondence discusses the reversible integer 16 point discrete fourier transform ridft which uses integer operations with control bits the decomposition of the ridft is based on the paired representation when the fourier transform is split recursively into a set of short transforms of orders 8 4 2 and 1 control bits allow for inverting the integer approximations of multiplications by twiddle factors the proposed 16 point ridft uses 16 operations of real multiplication and 62 additions the integer approximation of the transform with eight control bits with additional two lifting schemes which requires two more multiplications is also considered

fluctuations of eigenvalues of random normal matrices in this article we consider a fairly general potential in the plane and the corresponding boltzmann gibbs distribution of eigenvalues of random normal matrices as the order of the matrices tends to infinity the eigenvalues condensate on a certain compact subset of the plane the droplet we prove that fluctuations of linear statistics of eigenvalues of random normal matrices converge on compact subsets of the interior of the droplet to a gaussian field and we discuss various ramifications of this result

the two dimensional one component plasma at γ 2 the semiperiodic strip the one component two dimensional plasma is studied in a strip of finite width replicated periodically parallel to the long axis of the strip exact results for the one and two particle distribution functions are found at coupling q2 kt 2 the system is inhomogeneous the one and two particle distribution functions show long range order

split vector radix fast fourier transform the split radix approach for computing the discrete fourier transform dft is extended for the vector radix fast fourier transform fft to two and higher dimensions it is obtained by further splitting the n 2 n 2 transforms with twiddle factors in the radix 2 2 fft algorithm the generalization of this split vector radix fft algorithm to higher radices and higher dimensions is also presented by introducing a general approach for constructing the fast hartley transform fht from the corresponding fft new vector and split vector radix fht algorithms with the same desirable properties as their fft counterparts are obtained

the variations of hodge structure of maximal dimension with associated hodge numbers ℎ 2 0 2 and ℎ 1 1 2 1 do not arise from geometry the specified variations are proved to be covered by a bounded contractible domain q2 after classifying the analytic boundary components of q with respect to a fixed realization the group of the biholomorphic automorphisms aut q2 and the aut q orbit structure of q2 are found explicitly then n is shown to admit no quasiprojective arithmetic quotients whereas the lack of geometrically arising variations covered by q2 the weight 1 period domain sp n r u arises from geometry in as much as it classifies the first cohomologies of the n dimensional abelian varieties with fixed polarization the geometric locus on the weight 2 period domain is shown to satisfy the griffiths horizontality cf 9

a method to calculate correlation functions for β 1 random matrices of odd size the calculation of correlation functions for 1 random matrix ensembles which can be carried out using pfaffians has the peculiar feature of requiring a separate calculation depending on the parity of the matrix size n this same complication is present in the calculation of the correlations for the ginibre orthogonal ensemble of real gaussian matrices in fact the methods used to compute the 1 n odd correlations break down in the case of n odd real ginibre matrices necessitating a new approach to both problems the new approach taken in this work is to deduce the 1 n odd correlations as limiting cases of their n even counterparts when one of the particles is removed towards infinity this method is shown to yield the correlations for n odd real gaussian matrices

fast fourier transform using matrix decomposition to reduce both the multiplicative complexity and total number of operations this paper introduces a modeling scheme of the fast fourier transform fft to decompose the discrete fourier transform dft matrix recursively into a set of sparse matrices integrating three orthogonal transforms the hadamard modified haar and hybrid transforms the proposed scheme is able to obtain different fft representations with less computation operations than state of the arts to investigate the applications of the proposed fft scheme a multi stage image encryption algorithm is also introduced experimental results and security analysis are provided to show its encryption performance

bounds on the dimension of variations of hodge structure we derive upper bounds on the dimension of a variation of hodge structure of weight two and show that these bounds are sharp using them we exhibit maximal geometric variations of hodge structure analogous results for higher weight are obtained in the presence of a nondegeneracy hypothesis and variations coming from hypersurfaces are shown to be nondegenerate maximal geometric variations of higher weight are also constructed

automatic generation of prime length fft programs describes a set of programs for circular convolution and prime length fast fourier transforms ffts that are relatively short possess great structure share many computational procedures and cover a large variety of lengths the programs make clear the structure of the algorithms and clearly enumerate independent computational branches that can be performed in parallel moreover each of these independent operations is made up of a sequence of suboperations that can be implemented as vector parallel operations this is in contrast with previously existing programs for prime length ffts they consist of straight line code no code is shared between them and they cannot be easily adapted for vector parallel implementations the authors have also developed a program that automatically generates these programs for prime length ftts this code generating program requires information only about a set of modules for computing cyclotomic convolutions

an algorithm for calculation of the discrete cosine transform by paired transform a new algorithm for splitting the one dimensional 1 d 2 sup r point discrete cosine transform dct into a set of short 2 sup k point type iv dcts k 1 r 1 is introduced the splitting is performed by means of paired transformation that is defined by the paired representation of signals with respect to the cosine transform a proposed method of calculating the 2 sup r point cosine transform requires 2 sup r 1 r multiplications and 2 sup r 1 3r 2 1 additions when r spl ges 2

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