algebraic signal processing theory cooley tukey type algorithms for dcts and dsts in this paper we systematically derive a large class of fast general radix algorithms for various types of real discrete fourier transforms real dfts including the discrete hartley transform dht based on the algebraic signal processing theory this means that instead of manipulating the transform definition we derive algorithms by manipulating the polynomial algebras underlying the transforms using one general method the same method yields the well known cooley tukey fast fourier transform fft as well as general radix discrete cosine and sine transform algorithms the algebraic approach makes the derivation concise unifies and classifies many existing algorithms yields new variants enables structural optimization and naturally produces a human readable structural algorithm representation based on the kronecker product formalism we show for the first time that the general radix cooley tukey and the lesser known bruun algorithms are instances of the same generic algorithm further we show that this generic algorithm can be instantiated for all four types of the real dft and the dht

transforming real valued sequences fast fourier versus fast hartley transform algorithms five programs for efficient computation of dft of real valued data are analyzed with respect to their operation counts vis a vis run times on weak and powerful floating point processors the results help dispose of the claims of superiority of fht over corresponding real valued fft the bruun 1978 algorithm turns out to be a best performer

systolic arrays for the discrete hartley transform recently r n bracewell 1983 introduced the discrete hartley transform dht as an alternative to the discrete fourier transform dft two linear systolic array models for the dht are derived one model requires o 2n 1 in the computational phase and o n in the preloading phase the other model requires o 2n 1 in the computational phase and o n in the output phase a square systolic array for two dimensional dht is also constructed by combining the individual advantages of each model the cordic algorithm is proposed as an alternative to conventional multipliers to speed up the systolic array two level pipelining with cordic is also possible

the fast hartley transform algorithm the fast hartley transform fht is similar to the cooley tukey fast fourier transform fft but performs much faster because it requires only real arithmetic computations compared to the complex arithmetic computations required by the fft through use of the fht discrete cosine transforms dct and discrete fourier transforms dft can be obtained the recursive nature of the fht algorithm derived in this paper enables us to generate the next higher order fht from two identical lower order fht s in practice this recursive relationship offers flexibility in programming different sizes of transforms while the orderly structure of its signal flow graphs indicates an ease of implementation in vlsi

an analysis of real fourier domain based adaptive algorithms implemented with the hartley transform using cosine sine symmetries the least mean squared lms algorithm and its variants have been the most often used algorithms in adaptive signal processing however the lms algorithm suffers from a high computational complexity especially with large filter lengths the fourier transform based block normalized lms fbnlms reduces the computation count by using the discrete fourier transform dft and exploiting the fast algorithms for implementing the dft even though the savings achieved with the fbnlms over the direct lms implementation are significant the computational requirements of fbnlms are still very high rendering many real time applications like audio and video estimation infeasible the hartley transform based bnlms hbnlms is found to have a computational complexity much less than and a memory requirement almost of the same order as that of the fbnlms this paper is based on the cosine and sine symmetric implementation of the discrete hartley transform dht which is the key in reducing the computational complexity of the fbnlms by 33 asymptotically with respect to multiplications the parallel implementation of the discrete cosine transform dct in turn can lead to more efficient implementations of the hbnlms

combined polynomial transform and radix q algorithm for md discrete w transform the type ii r dimensional discrete w transform rd dwt ii with size q sup l1 spl times q sup l2 spl times spl middot spl middot spl middot spl times sup lr q where q is an odd prime number is converted into a series of one dimensional 1 d reduced dwt iis by using the multidimensional polynomial transform and an index permutation then a radix q algorithm and a cyclic convolution algorithm are presented for the computation of the 1 d reduced dwt iis the new fast algorithm substantially reduces the overall computational complexity compared with the row column method especially the number of multiplications required by the proposed algorithm for computing an rd dwt ii is only 1 r times that needed by the commonly used row column method

on computing the split radix fft this paper presents an efficient fortran program that computes the duhamel hollmann split radix fft an indexing scheme is used that gives a three loop structure for the split radix fft that is very similar to the conventional cooley tukey fft both a decimation in frequency and a decimation in time program are presented an arithmetic analysis is made to compare the operation count of the cooley tukey fft fo several different radixes to that of the split radix fft the split radix fft seems to require the least total arithmetic of any power of two dft algorithm

novel fixed point roundoff analysis of the decimation in time fht a least upper bound for the increasing factor of the magnitude of the decimation in time fast hartley transform fht in fixed point arithmetic is developed and a new scaling model for the roundoff analysis in the fixed point arithmetic computation is proposed in this new scaling model the input data for each computing stage of the decimation in time fht only need to be divided by a constant of 2 and this can prevent overflow successfully hence the novel approach would result in a higher noise to signal ratio for the fixed point computation of fht

fast algorithms for the computation of sliding sequency ordered complex hadamard transform fast algorithms for computing the forward and inverse sequency ordered complex hadamard transforms scht in a sliding window are presented the first algorithm consists of decomposing a length n inverse scht ischt into two length n 2 ischts the second algorithm calculating the values of window i n 4 from those of window i and one length n 4 ischt and one length n 4 modified ischt mischt is implemented by two schemes to achieve a good compromise between the computation complexity and the implementation complexity the forward scht algorithm can be obtained by transposing the signal flow graph of the ischt the proposed algorithms require o n arithmetic operations and thus are more efficient than the block based algorithms as well as those based on the sliding fft or the sliding dft the application of the sliding ischt in transform domain adaptive filtering tdaf is also discussed with supporting simulation results

computing pseudo wigner distribution by the fast hartley transform the use of the fast hartley transform fht to evaluate the wigner distribution entirely in the real domain is proposed and the computational complexity is reduced from three complex ffts to three real fhts

an interior point cutting plane method for the convex feasibility problem with second order cone inequalities the convex feasibility problem in general is a problem of finding a point in a convex set that contains a full dimensional ball and is contained in a compact convex set we assume that the outer set is described by second order cone inequalities and propose an analytic center cutting plane technique to solve this problem we discuss primal and dual settings simultaneously two complexity results are reported the complexity of restoration procedure and complexity of the overall algorithm we prove that an approximate analytic center is updated after adding a second order cone cut socc ino 1 newton step and that the analytic center cutting plane method accpm with socc is a fully polynomial algorithm

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

fast computation of md dct iv md dst iv by md dwt or md dct ii this paper reveals relationships between the type iv multidimensional discrete cosine transform md dct iv and the type ii multidimensional discrete cosine transform md dct ii or the multidimensional discrete w transform md dwt based on the relationships md dct iv can be efficiently computed by using the fast algorithms for md dct ii or md dwt iii or md dwt iii efficient method for implementation of the post addition or pre addition process is also presented compared to the widely used row column method considerable savings on the number of arithmetic operations are achieved for example the number of multiplications for computing an r dimensional dct iv is only about frac 1 r times that needed by the row column method and the number of additions is also reduced numerical experiments are made to test the computational time and roundoff errors they show that the proposed fast algorithms save time

on a set of matrix algebras related to discrete hartley type transforms a set of fast real transforms including the well known hartley transform is fully investigated mixed radix splitting properties of hartley type transforms are examined in detail the matrix algebras diagonalized by the hartley type matrices are expressed in terms of circulant and 1 circulant matrices

a generalized approach to direction finding with reference to the recently developed estimation of signal parameters using a rotational invariance techniques esprit it is pointed out that other operators can be used to generate the matrix pencil by way of example the moving window operator is discussed

fast hartley transform pruning the discrete hartley transform dht is discussed as a tool for the processing of real signals fast hartley transform fht algorithms which compute the dht in a time proportional to n log sub 2 n exist in many applications such as interpolation and convolution of signals a significant number of zeros are padded to the nonzero valued samples before the transform is computed it is shown that for such situations significant savings in the number of additions and multiplications can be obtained by pruning the fht algorithm the modifications in the fht algorithm as a result of pruning are developed and implemented in an fht subroutine the amount of savings in the operation is determined

a multiple cut analytic center cutting plane method for semidefinite feasibility problems we consider the problem of finding a point in a nonempty bounded convex body gamma in the cone of symmetric positive semidefinite matrices cal s m assume that gamma is defined by a separating oracle which for any given m ti m symmetric matrix hat y either confirms that hat y in gamma or returns several selected cuts i e a number of symmetric matrices ai i 1 p p le p max such that gamma is in the polyhedron y in cal s m mid a i bullet y le a i bullet hat y i 1 ldots p we present a multiple cut analytic center cutting plane algorithm starting from a trivial initial point the algorithm generates a sequence of positive definite matrices which are approximate analytic centers of a shrinking polytope in cal s m the algorithm terminates with a point in gamma within o m 3p max epsilon 2 newton steps to leading order where epsilon is the maximum radius of a ball contained in gamma

extension of primal dual interior point algorithms to symmetric cones in this paper we show that the so called commutative class of primal dual interior point algorithms which were designed by monteiro and zhang for semidefinite programming extends word for word to optimization problems over all symmetric cones the machinery of euclidean jordan algebras is used to carry out this extension unlike some non commutative algorithms such as the xs sx method this class of extensions does not use concepts outside of the euclidean jordan algebras in particular no assumption is made about representability of the underlying jordan algebra as a special case we prove polynomial iteration complexities for variants of the short semi long and long step path following algorithms using the nesterov todd xs or sx directions

fast computation of the discrete cosine transform and the discrete hartley transform a new factorization of the discrete hartley transform dht is presented it is used to derive new algorithms for the dht and the discrete cosine transform dct with reduced number of multiplications

the vector split radix algorithm for 2d dht a decimation in frequency vector split radix algorithm is proposed to decompose an n n 2d discrete hartley transform dht into one n 2 n 2 dht and twelve n 4 dhts the proposed algorithm possesses the in place property and needs no matrix transpose its computational structure is very regular and is simpler than those of all existing nonseparable 2d dhts

a jordan algebraic approach to potential reduction algorithms we consider the linear monotone complementarity problem for domains obtained as the intersection of an affine subspace and the cartesian product of symmetric cones a primal dual potential reduction algorithm is described and its complexity estimates are established with the help of the jordan algebraic technique

fast discrete transforms by means of eigenpolynomials abstract let a a ij be an n n matrix consider the discrete transform u a u and associate with the j th column of a the eigenpolynomial a j x n 1 i 0 a ij x j the properties of eigenpolynomials play an important role in the case where a is a matrix of eigenvectors of a toeplitz matrix 1 2 here we consider the cases where a is the matrix defining the discrete fourier transform dft the discrete hartley transform dht the discrete sine transform dst and the discrete cosine transform dct in its two versions of 3 and 4 for each eigenpolynomial of each transform we explicitly determine all its zeros we use eigenpolynomials as a unifying tool for describing the decimation in frequency dif versions of the main known algorithms for dft moreover by using the information about the zeros of the eigenpolynomials we devise new algorithms for dft dht dst and dct which reach or improve in the case of dst and dct the record complexity bounds without requiring the use of the algorithm for complex multiplication with three multiplications and three additions 5 and of the implicit preconditioning

unified parallel lattice structures for time recursive discrete cosine sine hartley transforms unified efficient computations of the discrete cosine transform dct discrete sine transform dst discrete hartley transform dht and their inverse transforms employing the time recursive approach are considered unified parallel lattice structures that can dually generate the dct and dst simultaneously as well the dht are developed these structures can obtain the transformed data for sequential input time recursively with a throughput rate of one per clock cycle the total number of multipliers required is a linear function of the transform size n with no constraint on n the resulting architectures are regular modular and without global communication so that they are very suitable for vlsi implementation for high speed applications it is shown that the dct dst dht and their inverse transforms share an almost identical lattice structure the tradeoff between time and area for the block data processing is considered

radix 2 decimation in frequency algorithm for the computation of the real valued fft an efficient algorithm for computing the real valued fft of length n using radix 2 decimation in frequency dif approach has been introduced the fact that the odd coefficients are the dft values of an n 2 length linear phase sequence introduces a redundancy in the form of the symmetry x 2k 1 x sup n 2k 1 which can be exploited to reduce the arithmetic complexity and memory requirements the arithmetic complexity and memory requirements of the algorithm presented are exactly the same as the most efficient decimation in time dit algorithm for the real valued fft that exists to date a c program that implements this algorithm has been included

an analytic center cutting plane method for semidefinite feasibility problems semidefinite feasibility problems arise in many areas of operations research the abstract form of these problems can be described as finding a point in a nonempty bounded convex body g in the cone of symmetric positive semidefinite matrices assume that g is defined by an oracle which for any givenm xm symmetric positive semidefinite matrix y either confirms that y e g or returns a cut i e a symmetric matrixa such that g is in the half space y a y a y we study an analytic center cutting plane algorithm for this problem at each iteration the algorithm computes an approximate analytic center of a working set defined by the cutting plane system generated in the previous iterations if this approximate analytic center is a solution then the algorithm terminates otherwise the new cutting plane returned by the oracle is added into the system as the number of iterations increases the working set shrinks and the algorithm eventually finds a solution to the problem all iterates generated by the algorithm are positive definite matrices the algorithm has a worst case complexity of o m3 e 2 on the total number of cuts to be used where e is the maximum radius of a ball contained by g

a note on 2 d hartley transform in recent years the application of the hartley transform originally introduced by hartley 1942 has gained importance in seismic signal processing and interpretation saatcilar et al 1990 1992 the hartley transform is similar to the fourier transform but is computationally much faster than even the fast fourier transform bracewell 1983 bracewell et al 1986 sorensen et al 1985 pei and wu 1985 duhamel and vetterli 1987 zhou 1992 surprisingly we have not seen a clear definition of the 2 d hartley transform in the published literature

hartley transforms over finite fields a general framework is presented for constructing transforms in the field of the input which have a convolution like property the construction is carried out over finite fields but is shown to be valid over the real and complex fields as well it is shown that these basefield transforms can be viewed as projections of the discrete fourier transform dft and that they exist for all lengths n for which the dft is defined the convolution property of the basefield transforms is derived and a condition for such transforms to have the self inverse property is given also fast algorithms for these basefield transforms are developed showing gains when compared to computations using the fft application of the methodology to hartley transforms over r leads to a simple derivation of fast algorithms for computing real hartley transforms

improved fourier and hartley transform algorithms application to cyclic convolution of real data this paper highlights the possible tradeoffs between arithmetic and structural complexity when computing cyclic convolution of real data in the transform domain both fourier and hartley based schemes are first explained in their usual form and then improved either from the structural point of view or in the number of operations involved namely we first present an algorithm for the in place computation of the discrete fourier transform on real data a decimation in time split radix algorithm more compact than the previously published one second we present a new fast hartley transform algorithm with a reduced number of operations a more regular convolution scheme based on fft s is also proposed finally we show that hartley transforms belong to a larger class of algorithms characterized by their generalized convolution property

a weighted normalized frequency domain lms adaptive algorithm a general filtering scheme is presented for obtaining an input power estimate for setting the convergence parameter mu separately in each frequency bin of a frequency domain lms adaptive filter fdaf algorithm a linear filtering operation is performed on the magnitude square of the input data and incorporated directly into the algorithm as a data dependent time varying stochastic mu n the mean performance of the weighted normalized frequency domain lms algorithm wnfdaf is analyzed using independent and identically distributed gaussian data and the results are validated by the monte carlo simulations of the algorithm the simulations are also used to study the weight transient behavior the simulations suggest that sort smoothing times are sufficient for rapid weight convergence without large fluctuations in the power estimates significantly affecting transient weight behavior

assessing the hartley transform the fast algorithm for the real hartly transform is discussed in relation to the established fast algorithm for the complex fourier transform the two transforms are compared by timing comparably written programs on a given machine and the discipline of timing is discussed as an adjunct to complexity analysis with real data one hartley transform program can economically replace such packages as a complex valued unilateral fourier transform combined with a real valued unilateral inverse fourier transform the hartley transform is favorable for fast convolution of real data sets the utility of spectral analysis into fourier series throughout physics suggested that the hartley transform might have less physical significance but the construction of hartley diffraction planes in the microwave and optical laboratories where electromagnetic phase is encoded as real valued field amplitudes has revealed interesting complementarity

generalized commuting matrices and their eigenvectors for dfts offset dfts and other periodic operations it is well known that some matrices such as dickinson steiglitz s matrix can commute with the discrete fourier transform dft and that one can use them to derive the complete and orthogonal dft eigenvector set recently candan found the general form of the dft commuting matrix in this paper we further extend the previous work and find the general form of the commuting matrix for any periodic quasi periodic and offset quasi periodic operations using the general commuting matrix we can derive the complete and orthogonal eigenvector sets for offset dfts dcts of types 1 4 5 and 8 dsts of types 1 4 5 and 8 discrete hartley transforms of types 1 and 4 the walsh transform and the projection operation the operation that maps a whole vector space into a subspace successfully moreover several novel ways of finding dft eigenfunctions are also proposed furthermore we also extend our theories to the continuous case i e if a continuous transform is periodic quasi periodic or offset quasi periodic such as the ft and some cyclic operations in optics we can find the general form of the commuting operation and then find the complete and orthogonal eigenfunctions set for the continuous transform

comments on generalized discrete hartley transform the author comments on the paper by hu et al ieee trans signal processing vol 40 no 12 p 2951 60 1992 information is provided about prior published work that precedes the transforms and convolution procedures defined in the above paper

adaptive filtering using hartley transform and overlap save method a novel method of implementing fir finite impulse response adaptive filters using the discrete hartley transform dht and the overlap save method is proposed it not only has the advantage of fast convergence for highly correlated input signals but also provides a significant reduction in computation over the conventional time domain and frequency domain least mean square lms adaptive filters compared with the previously proposed unconstrained frequency domain flms algorithm the convergence performance is similar but an approximate saving of 30 in computation can be obtained for the usual filter length

unveiling and exemplifying the unitary equivalence of discrete time quantum walk models the two major discrete time formulations for quantum walks coined and scattering are unitarily equivalent for arbitrary position dependent transition amplitudes and any topology andrade et al 2009 phys rev a 80 052301 although the proof explicitly describes the mapping obtention its high technicality may hinder relevant physical aspects involved in the equivalence discussing concrete examples the most general constructions for the line square and honeycomb lattices here we unveil the similarities and differences of these two versions of quantum walks we moreover show how to derive the dynamics of one from the other by means of proper projections we perform calculations for different probability amplitudes such as hadamard grover discrete fourier transform and the uncommon in the area but interesting discrete hartley transform comparing the evolutions our study illustrates the models interplay an important issue for implementations and applications of such systems

polynomial convergence of primal dual algorithms for the second order cone program based on the mz family of directions in this paper we study primal dual path following algorithms for the second order cone programming socp based on a family of directions that is a natural extension of the monteiro zhang mz family for semidefinite programming we show that the polynomial iteration complexity bounds of two well known algorithms for linear programming namely the short step path following algorithm of kojima et al and monteiro and adler and the predictor corrector algorithm of mizuno et al carry over to the context of socp that is they have an o loge 1 iteration complexity to reduce the duality gap by a factor of e where n is the number of second order cones since the mz type family studied in this paper includes an analogue of the alizadeh haeberly and overton pure newton direction we establish for the first time the polynomial convergence of primal dual algorithms for socp based on this search direction

a comment on on prime factor mapping for the discrete hartley transform lun and sin see ieee trans signal processing vol 40 no 6 p 1399 1411 1982 describe a method of computing prime factor decomposed discrete hartley transform dht over n points with 2n less additions compared to the existing algorithms of bracewell and sorensen 1976 the author shows how the method of lun and siu ran be extended to eliminate the 2n extra additions in the algorithms of bracewell and sorenson

fast radix 3 9 discrete hartley transform an efficient algorithm for computing radix 3 9 discrete hartley transforms dhts is presented it is shown that the radix 3 9 fast hartley transform fht algorithm reduces the number of multiplications required by a radix 3 fht algorithm for nearly 50 for the computation of real valued discrete fourier transforms dfts with sequence lengths that are powers of 3 it is shown that the radix 3 9 fht algorithm reduces the number of multiplications by 16 2 over the fastest real valued radix 3 9 fast fourier transform fft algorithm

the use of the hartley transform in geophysical applications the hartley transform ht is an integral transform similar to the fourier transform ft it has most of the characteristics of the ft several authors have shown that fast algorithms can be constructed for the fast hartley transform fht using the same structures as for the fast fourier transform however the ht is a real transform and for this reason since one complex multiplication requires four real multiplications the discrete ht dht is computationally faster than the discrete ft dft consequently any process requiring the dft such as amplitude and phase spectra can be performed faster by using the dht the general properties of the dht are reviewed first and then an attempt is made to use the fht in some seismic data processing techniques such as one dimensional filtering forward seismic modeling and migration the experiments show that the hartley transform is two times faster than the fourier transform

radix 2 spl times 2 spl times 2 algorithm for the 3 d discrete hartley transform the discrete hartley transform dht has proved to be a valuable tool in digital signal image processing and communications and has also attracted research interests in many multidimensional applications although many fast algorithms have been developed for the calculation of one and two dimensional 1 d and 2 d dht the development of multidimensional algorithms in three and more dimensions is still unexplored and has not been given similar attention hence the multidimensional hartley transform is usually calculated through the row column approach however proper multidimensional algorithms can be more efficient than the row column method and need to be developed therefore it is the aim of this paper to introduce the concept and derivation of the three dimensional 3 d radix 2 spl times 2 spl times 2 algorithm for fast calculation of the 3 d discrete hartley transform the proposed algorithm is based on the principles of the divide and conquer approach applied directly in 3 d it has a simple butterfly structure and has been found to offer significant savings in arithmetic operations compared with the row column approach based on similar algorithms

data compression properties of the hartley transform the data compression performance of the hartley transform on a markov 1 signal is theoretically compared to that of the fourier transform covariance distribution and residue correlation measurements have been computed for the hartley fourier and cosine transforms the hartley transform achieves better coding performance than the fourier transform but is inferior to the cosine transform

second order cone programming second order cone programming socp problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the cartesian product of second order lorentz cones linear programs convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as socp problems as can many other problems that do not fall into these three categories these latter problems model applications from a broad range of fields from engineering control and finance to robust optimization and combinatorial optimization on the other hand semidefinite programming sdp that is the optimization problem over the intersection of an affine set and the cone of positive semidefinite matrices includes socp as a special case therefore socp falls between linear lp and quadratic qp programming and sdp like lp qp and sdp problems socp problems can be solved in polynomial time by interior point methods the computational effort per iteration required by these methods to solve socp problems is greater than that required to solve lp and qp problems but less than that required to solve sdp s of similar size and structure because the set of feasible solutions for an socp problem is not polyhedral as it is for lp and qp problems it is not readily apparent how to develop a simplex or simplex like method for socp while socp problems can be solved as sdp problems doing so is not advisable both on numerical grounds and computational complexity concerns for instance many of the problems presented in the survey paper of vandenberghe and boyd vb96 as examples of sdps can in fact be formulated as socps and should be solved as such in 2 3 below we give socp formulations for four of these examples the convex quadratically constrained quadratic programming qcqp problem problems involving fractional quadratic functions rutcor rutgers university e mail alizadeh rutcor rutgers edu research supported in part by the u s national science foundation grant ccr 9901991 ieor columbia university e mail gold ieor columbia edu research supported in part by the department of energy grant de fg02 92er25126 national science foundation grants dms 94 14438 cda 97 26385 and dms 01 04282

on computing the discrete hartley transform the discrete hartley transform dht is a real valued transform closely related to the dft of a real valued sequence bracewell has recently demonstrated a radix 2 decimation in time fast hartley transform fht algorithm in this paper a complete set of fast algorithms for computing the dht is developed including decimation in frequency radix 4 split radix prime factor and winograd transform algorithms the philosophies of all common fft algorithms are shown to be equally applicable to the computation of the dht and the fht algorithms closely resemble their fft counterparts the operation counts for the fht algorithms are determined and compared to the counts for corresponding real valued fft algorithms the fht algorithms are shown to always require the same number of multiplications the same storage and a few more additions than the real valued fft algorithms even though computation of the fht takes more operations in some situations the inherently real valued nature of the discrete hartley transform may justify this extra cost

the hartley transform in seismic imaging phase shift migration techniques that attempt to account for lateral velocity variations make substantial use of the fast fourier transform fft generally the hermitian symmetry of the complex valued fourier transform causes computational redundancies in terms of the number of operations and memory requirements in practice a combination of the fft with the well known real to complex fourier transform is often used to avoid such complications as an alternative means to the fourier transform we introduce the inherently real valued non symmetric hartley transform into phase shift migration techniques by this we automatically avoid the hermitian symmetry resulting in an optimized algorithm that is comparable in efficiency to algorithms based on the real to complex fft we derive the phase shift operator in the hartley domain for migration in two and three dimensions and formulate phase shift plus interpolation split step migration and split step double square root prestack migration in terms of the

a prime factor fast w transform algorithm a method for converting any nesting dft algorithm to the type i discrete w transform dwt i is introduced a nesting algorithm that differs from either the windograd fourier transform algorithm wfta or the prime factor fft algorithm pfa is presented new small n dets which are suitable for this nesting algorithm are developed based on using sparse matrix decomposition the proposed algorithm is more efficient that either wfta or pfa for large n and it is more flexible for the choice of transform length because 32 points are used for 2d processing the proposed algorithm is more efficient than the polynomial transform

systolic architectures for the computation of the discrete hartley and the discrete cosine transforms based on prime factor decomposition two dimensional systolic array implementations for computing the discrete hartley transform dht and the discrete cosine transform dct when the transform size n is decomposable into mutually prime factors are proposed the existing two dimensional formulations for dht and dct are modified and the corresponding algorithms are mapped into two dimensional systolic arrays the resulting architecture is fully pipelined with no control units the hardware design is based on bit serial left to right msb most significant bit to lsb least significant bit binary arithmetic

calculation of multidimensional hartley transforms using one dimensional fourier transforms in the processing of real valued data a purely real transform such as the hartley transform is more desirable than the complex fourier transform because it avoids unnecessary complex computations this advantage is most significant in multidimensional transformations where a large amount of data has to be processed a multidimensional fast hartley transform algorithm is described that successively applies 1d fourier transforms redundant operations are reduced to a minimum special indexing schemes parity operators are introduced to avoid unscrambling procedures

semisystolic array implementation of circular skew circular and linear convolutions semisystolic array implementation of circular and linear convolutions in one and multidimensions are discussed the common feature of the various architectures studied is the broadcasting of the input sequence to the cells of the array in the case of circular convolutions there is also circular communication between the cells a circular convolution of period n can be calculated in n time steps whereas the response time for the computation of n outputs of linear convolution with finite weight and data vectors is also n time steps without initial delay

an analytic center cutting plane approach for conic programming we analyze the problem of finding a point strictly interior to a bounded convex and fully dimensional set from a finite dimensional hilbert space we generalize the results obtained for the linear programming lp semidefinite programming sdp and second order core programming socp cases the cuts added by our algorithm are central and conic in our analysis we find an upper bound for the number of newton steps required to compute an approximate analytic center also we provide an upper bound for the total number of cuts added to solve the problem this bound depends on the quality of the cuts the dimensionality of the problem and the thickness of the set we are considering

on prime factor mapping for the discrete hartley transform the authors propose a new prime factor mapping scheme which requires no extra arithmetic operations for the realization of prime factor mapping for the computation of the discrete hartley transform dht it is achieved by embedding all the extra arithmetic operations into the subsequent short length computations with the computational complexities of these embedded short lengths remaining unchanged consequently the present approach significantly eliminates the burden which is introduced by the extra arithmetic operations with this mapping scheme it is further demonstrated that a prime factor mapped dht would have superb performance compared with other fast dht algorithms

generalized discrete hartley transforms the discrete hartley transform is generalized into four classes in the same way as the generalized discrete fourier transform fast algorithms for the resulting transforms are derived the generalized transforms are expected to be useful in applications such as digital filter banks fast computation of the discrete hartley transform for any composite number of data points fast computations of convolution and signal representation the fast computation of skew circular convolution by the generalized transforms for any composite number of data points is discussed in detail

a strongly polynomial algorithm for linear systems having a binary solution this paper proposes a strongly polynomial algorithm which either finds a solution of a linear system ax b 0 x 1 or correctly decides that the system has no 0 1 solutions the algorithm can be used as the basis for the construction of a polynomial algorithm for linear programming

automatic generation of fast discrete signal transforms this paper presents an algorithm that derives fast versions for a broad class of discrete signal transforms symbolically the class includes but is not limited to the discrete fourier and the discrete trigonometric transforms this is achieved by finding fast sparse matrix factorizations for the matrix representations of these transforms unlike previous methods the algorithm is entirely automatic and uses the defining matrix as its sole input the sparse matrix factorization algorithm consists of two steps first the symmetry of the matrix is computed in the form of a pair of group representations second the representations are stepwise decomposed giving rise to a sparse factorization of the original transform matrix we have successfully demonstrated the method by computing automatically efficient transforms in several important cases for the dft we obtain the cooley tukey 1965 fft for a class of transforms including the dct type ii the number of arithmetic operations for our fast transforms is the same as for the best known algorithms our approach provides new insights and interpretations for the structure of these signal transforms and the question of why fast algorithms exist the sparse matrix factorization algorithm is implemented within the software package arep

split radix fast hartley transform in one and two dimensions a simple matrix decomposition is provided as a derivation of the one dimensional split radix fast hartley transform fht the method is then extended in a straightforward manner to the two dimensional fht 2 d fht by implementing the vector radix approach as a result the proposed algorithm requires many fewer computational operations than the nonsplit vector radix 2 d fht method

an extension of chubanov s algorithm to symmetric cones in this work we present an extension of chubanov s algorithm to the case of homogeneous feasibility problems over a symmetric cone mathcal k as in chubanov s method for linear feasibility problems the algorithm consists of a basic procedure and a step where the solutions are confined to the intersection of a half space and mathcal k following an earlier work by kitahara and tsuchiya on second order cone feasibility problems progress is measured through the volumes of those intersections when they become sufficiently small we know it is time to stop we never have to explicitly compute the volumes it is only necessary to keep track of the reductions between iterations we show this is enough to obtain concrete upper bounds to the minimum eigenvalues of a scaled version of the original feasibility problem another distinguishing feature of our approach is the usage of a spectral norm that takes into account the way that mathcal k is decomposed as simple cones in several key cases including semidefinite programming and second order cone programming these norms make it possible to obtain better complexity bounds for the basic procedure when compared to a recent approach by pena and soheili finally in the appendix we present a translation of the algorithm to the homogeneous feasibility problem in semidefinite programming

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

structured fast hartley transform algorithms the flowgraph originally proposed for the computation of the radix 2 fast hartley transform fht is restructured for clarity and ease of implementation with hardware and software by applying the transposition theorem to the newly formed flowgraph which corresponds to a decimation in time algorithm a decimation in frequency algorithm with a similar flowgraph is obtained the number of operations for the two algorithms is the same because of an existing symmetry which is also identical to that of another fht algorithm recently proposed

fast dht algorithms for length n q 2 sup m this article presents an improved split radix algorithm that can flexibly compute the discrete hartley transforms dht of length q 2 sup m where q is an odd integer comparisons with previously reported algorithms show that savings on the number of arithmetic operations can be made furthermore a wider range of choices on different dht lengths is naturally provided

signal extrapolation in the real zak space a new formulation of the gerchberg papoulis 1974 1975 algorithm for extrapolation of bandlimited signals was introduced the new formulation was obtained by translating the fundamental operations of the gp procedure the truncation and the fourier transform into the language of the finite zak 1967 transform however the zak transform formulation of the gp algorithm assumes complex valued signals whereas the gp procedure is usually applied to real signals we present a new and more efficient algorithm that acts directly on a real signal via the real zak transform rzt relation between a signal and its hartley transform leading in effect to approximately a four fold reduction in the computational complexity of the complex zak space approach

an efficient prime factor algorithm for the discrete cosine transform and its hardware implementations the prime factor decomposition is a fast computational technique for many important digital signal processing operations such as the convolution the discrete fourier transform the discrete hartley transform and the discrete cosine transform dct the authors present a new prime factor algorithm for the dct they also design a prime factor algorithm for the discrete sine transform based on the prime factor dct algorithm hardware implementations for the prime factor dct are also studied they are especially interested in the hardware designs which are suitable for the vlsi implementations they show three hardware designs for the prime factor dct including a vlsi circuit fabricated directly according to the signal flow graph a linear systolic array and a mesh connected systolic array these three designs show the trade off between cost and performance the methodology which deals with general n sub 1 spl times n sub 2 point dcts where n sub 1 and n sub 2 are mutually prime is illustrated by converting a 15 point dct problem into a 3 spl times 5 point 2d dct problem

similarity and other spectral relations for symmetric cones abstract the similarity relations that are derived in this paper reduce to well known results in the special case of symmetric matrices in particular for two positive definite matrices x and y the square of the spectral geometric mean is known to be similar to the matrix product xy it is shown in this paper that this property carries over to symmetric cones more elementary similarity relations such as xy 2 x yx 2 y are generalized as well we also extend the result that the eigenvalues of a matrix product xy are less dispersed than the eigenvalues of the jordan product xy yx 2 the paper further contains a number of inequalities on norms and spectral values this type of inequality is often used in the analysis of interior point methods in optimization we also derive an extension of stein s theorem to symmetric cones

unconstrained frequency domain adaptive filter an unconstrained frequency domain least mean squares uflms algorithm is presented the algorithm is based on the overlap save technique used in frequency domain filtering the proposed algorithm converges to the wiener solution without the time domain constraint on the impulse response as proposed by ferrara for a large number of taps geq32 the uflms algorithm offers a significant reduction in computation another advantage is its fast convergence for highly correlated input signals the performance of the algorithm is illustrated by a simulation

new split radix algorithm for the discrete hartley transform this paper presents a split radix algorithm that can flexibly compute the discrete hartley transforms of various sequence lengths comparisons with previously reported algorithms are made in terms of the required number of additions and multiplications it shows that the length 3 2 sup m dhts need a smaller number of multiplications than the length 2 sup m dhts however they both require about the same computational complexity in terms of the total number of additions and multiplications optimized computation of length 12 16 and 24 dfts are also provided

quantization errors in the computation of the discrete hartley transform the principal objective of this paper is the study of the arithmetic roundoff error characteristics of several discrete hartley transform dht algorithms we first summarize a variety of efficient dht algorithms including bracewell s original decimation in time radix 2 algorithm statistical models for fixed and floating point arithmetic roundoff errors are then used as the basis for a theoretical study of roundoff noise characteristics of a number of the dht algorithms the results of a detailed experimental study of roundoff noise are compared to the theoretical predictions in fixed point implementation of the decimation in time and frequency radix 2 algorithms it is found that the noise to signal ratio increases approximately 1 1 bits per stage for the floating point implementation the number of bits of rms noise to signal ratio for all the algorithms increase as sqrt log 2 n so that doubling the number of points produced a mild increase in the output noise

fast algorithm for the 3 d dct ii recently many applications for three dimensional 3 d image and video compression have been proposed using 3 d discrete cosine transforms 3 d dcts among different types of dcts the type ii dct dct ii is the most used in order to use the 3 d dcts in practical applications fast 3 d algorithms are essential therefore in this paper the 3 d vector radix decimation in frequency 3 d vr dif algorithm that calculates the 3 d dct ii directly is introduced the mathematical analysis and the implementation of the developed algorithm are presented showing that this algorithm possesses a regular structure can be implemented in place for efficient use of memory and is faster than the conventional row column frame rcf approach furthermore an application of 3 d video compression based 3 d dct ii is implemented using the 3 d new algorithm this has led to a substantial speed improvement for 3 d dct ii based compression systems and proved the validity of the developed algorithm

a new class of quasi newtonian methods for optimal learning in mlp networks in this paper we present a new class of quasi newton methods for an effective learning in large multilayer perceptron mlp networks the algorithms introduced in this work named lqn utilize an iterative scheme of a generalized bfgs type method involving a suitable family of matrix algebras l the main advantages of these innovative methods are based upon the fact that they have an o nlogn complexity per step and that they require o n memory allocations numerical experiences performed on a set of standard benchmarks of mlp networks show the competitivity of the lqn methods especially for large values of n

comments on fast computation of the discrete cosine transform and the discrete hartley transform by h s malvar the commenter shows that the algorithm presented in the above paper ibid vol assp 35 no 10 p 1484 5 oct 1987 requires exactly the same number of multiplications as but more additions than prevalent methods for the computation of the dht discrete hartley transform by existing methods contrary to the author s claim

fast algorithms for the computation of sliding discrete hartley transforms fast algorithms for computing generalized discrete hartley transforms in a sliding window are proposed the algorithms are based on second order recursive relations between subsequent local transform spectra new efficient inverse algorithms for signal processing in a sliding window are also presented the computational complexity of the algorithms is compared with that of known fast discrete hartley transforms and running recursive algorithms

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

a polynomial projection algorithm for linear feasibility problems we propose a polynomial algorithm for linear feasibility problems the algorithm represents a linear problem in the form of a system of linear equations and non negativity constraints then it uses a procedure which either finds a solution for the respective homogeneous system or provides the information based on which the algorithm rescales the homogeneous system so that its feasible solutions in the unit cube get closer to the vector of all ones in a polynomial number of calls to the procedure the algorithm either proves that the original system is infeasible or finds a solution in the relative interior of the feasible set

2 d hartley transforms two different versions of kernels associated with the 2 d hartley transforms are investigated in relation to their fourier counterparts this newly emerging tool for digital signal processing is an alternate means of analyzing a given function in terms of sinusoids and is an offshoot of fourier transform being a real valued function and fully equivalent to the fourier transform the hartley transform is more efficient and economical than its progenitor hartley and fourier pairs of complete orthogonal transforms comprise mathematical twins having definite physical significance the direct and inverse hartley transforms possess the same kernel unlike the fourier transform and hence have the dual distinction of being both self reciprocal and having the convenient property of occupying the real domain some of the properties of the hartley transform differ marginally from those of the fourier transform

fast unified computation of the multi dimensional discrete sinusoidal transforms based on a one to one index mapping approach conversions of two types of m dimensional discrete w transforms into the multiple sums involving a number of one dimensional discrete w transforms 1 d dwts are first presented a unified fast m 1 dimensional polynomial transform m 1 d pt and 1 d dwts algorithm for computing these two types of m d dwts are then proposed by revealing the relationships among the m dimensional discrete w transforms m d dwts the m dimensional generalized discrete fourier transforms m d gdfts the m dimensional discrete cosine transforms m d dcts and the m dimensional discrete sine transforms m d dsts the generalized fast algorithms for 14 types of the m d discrete sinusoidal transforms including four types of m d dwts and the same number of types of m d gdfts and three types of m d dcts and the same number of types of m d dsts are presented the number of multiplications for all 14 types of the m d discrete sinusoidal transforms needed by the proposed algorithm is only 1 m times that of the widely used corresponding row column methods the number of additions required by the proposed algorithm is also reduced considerably finally with the help of the software platform visual c 6 0 for windows 98 the computation time comparisons between the proposed and the reported algorithms are given the numerical experiments show that the proposed algorithms are not only highly efficient but are also very simple in computational structure

the inverse galois problem and rational points on moduli spaces we reduce the regular version of the inverse galois problem for any finite group g to finding one rational point on an infinite sequence of algebraic varieties as a consequence any finite group g is the galois group of an extension l p x with l regular over any pac field p of characteristic zero a special case of this implies that g is a galois group over fp x for almost all primes p

fast hartley transforms for image processing the fast hartley transform fht is used to transform two dimensional image data because the hartley transform is real valued it does not require complex operations both spectra and autocorrelations of two dimensional ultrasound images of normal and abnormal livers were computed