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

two dimensional period estimation by ramanujan s sum period estimation in one dimension 1 d has been studied for years however 2 d period estimation is still a hard problem since it has three parameters to determine length width and direction recently a special kind of 2 d function called 2d gcd delta function is proposed it has close relationship with ramanujan s sum and the 2 d periodicity matrix in this paper we will describe how to use this function to decompose an image into subband signals each subband signal will have its own periodicity matrix so we also provide a simple algorithm to calculate the least common multiple lcm of those subband signal periodicity matrices concrete experiments are given to prove the robustness of the proposed 2 d period estimation method

a modified split radix fft with fewer arithmetic operations recent results by van buskirk have broken the record set by yavne in 1968 for the lowest exact count of real additions and multiplications to compute a power of two discrete fourier transform dft here we present a simple recursive modification of the split radix algorithm that computes the dft with asymptotically about 6 fewer operations than yavne matching the count achieved by van buskirk s program generation framework we also discuss the application of our algorithm to real data and real symmetric discrete cosine transforms where we are again able to achieve lower arithmetic counts than previously published algorithms

recursive generation of the dif fft algorithm for 1 d dft it is shown by using the walsh hadamard matrix and a conversion matrix a dif fft algorithm can be developed for the radix 2 discrete fourier transform dft the recursive generation of the dft matrix leads to its sparse matrix factorization the flowgraph based on these sparse matrices is identical to that of the dft fft algorithm

multidimensional inverse lattice problem and a uniformly sampled arithmetic fourier transform the present work develops a unified and concise solution for inverse lattice problems also a uniformly sampled arithmetic fourier transform is presented in this work which uses ramanujan s sum rule

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

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

fast and numerically stable algorithms for discrete cosine transforms abstract in this paper we derive fast and numerically stable algorithms for discrete cosine transforms dct of radix 2 length which are based on real factorizations of the corresponding cosine matrices into products of sparse almost orthogonal matrices of simple structure these algorithms are completely recursive are simple to implement and use only permutations scaling with 2 butterfly operations and plane rotations rotation reflections our algorithms have low arithmetic costs which compare with known fast dct algorithms further a detailed analysis of the roundoff errors for the presented dct algorithms shows their excellent numerical stability which outperforms a real fast dct algorithm based on polynomial arithmetic

approximation metrics for discrete and continuous systems established system relationships for discrete systems such as language inclusion simulation and bisimulation require system observations to be identical when interacting with the physical world modeled by continuous or hybrid systems exact relationships are restrictive and not robust in this paper we develop the first framework of system approximation that applies to both discrete and continuous systems by developing notions of approximate language inclusion approximate simulation and approximate bisimulation relations we define a hierarchy of approximation pseudo metrics between two systems that quantify the quality of the approximation and capture the established exact relationships as zero sections our approximation framework is compositional for a synchronous composition operator algorithms are developed for computing the proposed pseudo metrics both exactly and approximately the exact algorithms require the generalization of the fixed point algorithms for computing simulation and bisimulation relations or dually the solution of a static game whose cost is the so called branching distance between the systems approximations for the pseudo metrics can be obtained by considering lyapunov like functions called simulation and bisimulation functions we illustrate our approximation framework in reducing the complexity of safety verification problems for both deterministic and nondeterministic continuous systems

separation of variables and the computation of fourier transforms on finite groups ii we present a general diagrammatic approach to the construction of efficient algorithms for computing the fourier transform of a function on a finite group by extending work which connects bratteli diagrams to the construction of fast fourier transform algorithms we make explicit use of the path algebra connection to the construction of gel fand tsetlin bases and work in the setting of quivers we relate this framework to the construction of a configuration space derived from a bratteli diagram in this setting the complexity of an algorithm for computing a fourier transform reduces to the calculation of the dimension of the associated configuration space our methods give improved upper bounds for computing the fourier transform for the general linear groups over finite fields the classical weyl groups and homogeneous spaces of finite groups while also recovering the best known algorithms for the symmetric group

analysis of the dissipation and dispersion properties of the multi domain chebyshev pseudospectral method the accuracy of the multi domain chebyshev pseudospectral method is investigated for wave propagation problems by examining the properties of the method in the wavenumber space theoretically in terms of dispersion and dissipation errors for a number of n 1 points in the subdomains used in the literature with n typically between 8 to 32 significant errors can be obtained for waves discretized by more than points per wavelength the dispersion and dissipation errors determined from the analysis in the wavenumber space are found to be in good agreement with those obtained in test cases accuracy limits based on arbitrary criteria are proposed yielding minimum resolutions of 7 7 5 2 and 4 0 points per wavelength for n 8 16 and 32 respectively the numerical efficiency of the method is estimated showing that it is preferable to choose n between 16 and 32 in practice the stability of the method is also assessed using the standard fourth order runge kutta algorithm finally 1 d and 2 d problems involving long range wave propagation are solved to illustrate the dissipation and dispersion errors for short waves the error anisotropy is studied in the 2 d case in particular for a hybrid fourier chebyshev configuration

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

the existence of regular orbits for nilpotent groups consider the following situation suppose r a cr where r is an r group and c is a cyclic group of order qe where r and q are distinct primes let k be a finite field of characteristic p and suppose v is a faithful irreducible k cr module this situation has been widely studied when p q it is similar to the situation occurring in the proof of theorem b of hall and higman 9 when p q it is similar to the situation occurring in the corresponding theorem of shult 131 in this paper a generalization of this situation is considered suppose g 4 ag where g is a solvable group a is nilpotent and ia g 1 again we assume v is a faithful irreducible k ag module two related questions arise in the study of this situation

a parallel fast multipole method for elliptic difference equations a new fast multipole formulation for solving elliptic difference equations on unbounded domains and its parallel implementation are presented these difference equations can arise directly in the description of physical systems e g crystal structures or indirectly through the discretization of pdes in the analog to solving continuous inhomogeneous differential equations using green s functions the proposed method uses the fundamental solution of the discrete operator on an infinite grid or lattice green s function fast solutions o n are achieved by using a kernel independent interpolation based fast multipole method unlike other fast multipole algorithms our approach exploits the regularity of the underlying cartesian grid and the efficiency of ffts to reduce the computation time our parallel implementation allows communications and computations to be overlapped and requires minimal global synchronization the accuracy efficiency and parallel performance of the method are demonstrated through numerical experiments on the discrete 3d poisson equation

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

observability and controllability of piecewise affine and hybrid systems we prove in a constructive way the equivalence between piecewise affine systems and a broad class of hybrid systems described by interacting linear dynamics automata and propositional logic by focusing our investigation on the former class we show through counterexamples that observability and controllability properties cannot be easily deduced from those of the component linear subsystems instead we propose practical numerical tests based on mixed integer linear programming

a flexible peak to average power ratio reduction scheme for ofdm systems by the adaptive projected subgradient method one of the main issues of the orthogonal frequency division multiplexing ofdm modulation is the high peak to average power ratio papr of the transmitted signal which adversely affects the complexity of power amplifiers in this paper we consider transmitters that reduce the papr by slightly disturbing the symbols in carriers used to transmit information and by sending dummy symbols i e symbols not conveying information in unused carriers the optimal choice of the data and dummy symbols is determined by the solution of a convex optimization problem to reduce the papr with low complexity we apply a modified version of the adaptive projected subgradient method to a sequence of convex cost functions closely related to the original optimization problem the resulting algorithm achieves near optimal papr in practical scenarios generalizes existing algorithms based on polyak s method and can easily handle multiple constraints

orthogonal exactly periodic subspace decomposition the detection and estimation of machine vibration multiperiodic signals of unknown periods in white gaussian noise is investigated new estimates for the subsignals signals making up the received signal and their periods are derived using an orthogonal subspace decomposition approach the concept of exactly periodic signals is introduced this in turn simplifies and enhances the understanding of periodic signals

recursive cyclotomic factorization a new algorithm for calculating the discrete fourier transform in this paper a new recursive algorithm for calculating the discrete fourier transformation is presented this new so called recursive cyclotomic factorization algorithm rcfa is more efficient than the fast fourier transformation fft algorithm moreover due to its recursive nature the rcfa can also be easily implemented using only a limited number of different computation cells

the techniques of the generalized fast fourier transform algorithm a general method of deriving dft discrete fourier transform algorithms generalised fast fourier transform algorithms is presented it is shown that a special case of the method is equivalent to nesting of ffts the application of the method to the case where n has mutually prime factors results in a new interpretation of the permutations characteristic of this class of algorithms it is shown that the equalization of ffts leads to results which are different from the widely used intuitive ones the high efficiency of split radix ffts is explained it is shown that the formulae of the method can be easily adapted for deriving algorithms for the cosine sine dft a set of ffts that has smaller arithmetical and or memory complexities than any algorithm known is presented in particular a method of deriving split radix 2 sup s ffts requiring n log sub 2 n 3n 4 real multiplications and 3n log sub 2 n 3n 4 additions for any s 1 is presented

efficient algorithms for discrete gabor transforms on a nonseparable lattice the discrete gabor transform dgt is the most commonly used transform for signal analysis and synthesis using a linear frequency scale it turns out that the involved operators are rich in structure if one samples the discrete phase space on a subgroup most of the literature focuses on separable subgroups in this paper we will survey existing methods for a generalization to arbitrary groups as well as present an improvement on existing methods comparisons are made with respect to the computational complexity and the running time of optimized implementations in the c programming language the new algorithms have the lowest known computational complexity for nonseparable lattices and the implementations are freely available for download by summarizing general background information on the state of the art this article can also be seen as a research survey sharing with the readers experience in the numerical work in gabor analysis

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

computation of an odd length dct from a real valued dft of the same length the discrete cosine transform dct is often computed from a discrete fourier transform dft of twice or four times the dct length dct algorithms based on identical length dft algorithms generally require additional arithmetic operations to shift the phase of the dct coefficients it is shown that a dct of odd length can be computed by an identical length dft algorithm by simply permuting the input and output sequences using this relation odd length dct modules for a prime factor dct are derived from corresponding dft modules the multiplicative complexity of the dct is then derived in terms of dft complexities

nested periodic matrices and dictionaries new signal representations for period estimation in this paper we propose a new class of techniques to identify periodicities in data we target the period estimation directly rather than inferring the period from the signal s spectrum by doing so we obtain several advantages over the traditional spectrum estimation techniques such as dft and music apart from estimating the unknown period of a signal we search for finer periodic structure within the given signal for instance it might be possible that the given periodic signal was actually a sum of signals with much smaller periods for example adding signals with periods 3 7 and 11 can give rise to a period 231 signal we propose methods to identify these hidden periods 3 7 and 11 we first propose a new family of square matrices called nested periodic matrices npms having several useful properties in the context of periodicity these include the dft walsh hadamard and ramanujan periodicity transform matrices as examples based on these matrices we develop high dimensional dictionary representations for periodic signals various optimization problems can be formulated to identify the periods of signals from such representations we propose an approach based on finding the least l 2 norm solution to an under determined linear system alternatively the period identification problem can also be formulated as a sparse vector recovery problem and we show that by a slight modification to the usual l 1 norm minimization techniques we can incorporate a number of new and computationally simple dictionaries

robust convex optimization we study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set u yet the constraints must hold for all possible values of the data from u the ensuing optimization problem is called robust optimization in this paper we lay the foundation of robust convex optimization in the main part of the paper we show that if u is an ellipsoidal uncertainty set then for some of the most important generic convex optimization problems linear programming quadratically constrained programming semidefinite programming and others the corresponding robust convex program is either exactly or approximately a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods

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

orthogonal and non orthogonal signal representations using new transformation matrices having npm structure in this article we introduce two types of real valued sums known as complex conjugate pair sums ccpss denoted as ccps 1 and ccps 2 and discuss a few of their properties using each type of ccpss and their circular shifts we construct two non orthogonal nested periodic matrices npms as npms are non singular this introduces two non orthogonal transforms known as complex conjugate periodic transforms ccpts denoted as ccpt 1 and ccpt 2 we propose another npm which uses both types of ccpss such that its columns are mutually orthogonal this transform is known as orthogonal ccpt occpt after a brief study of a few occpt properties like periodicity circular shift etc we present two different interpretations of it further we propose a decimation in time dit based fast computation algorithm for occpt termed as foccpt whenever the length of the signal is equal to 2 v v in mathbb n the proposed sums and transforms are inspired by ramanujan sums and ramanujan period transform rpt finally we show that the period both divisor and non divisor and frequency information of a signal can be estimated using the proposed transforms with a significant reduction in the computational complexity over discrete fourier transform dft

non linear reconstruction of delta sigma modulated signals randomized surrogate constraint decoding algorithm delta sigma a d conversion is a popular technique used to achieve high resolution data conversion for low to moderate bandwidth applications it is commonly accepted that the digital circuitry of the a d converters should perform a linear time invariant lti filtering on the output of the modulator in order to reconstruct the input signal however it has been shown that when higher order a d converters are operated at higher oversampling ratios osrs non linear reconstruction algorithms extract more information about the input signal than lti methods still use of non linear algorithms has been limited due to their complexity and stability issues two practical non linear methods presented in the literature are fast projection onto convex sets pocs and direct projection dp in this work we show that the non linear reconstruction process for modulated sequences can be treated as a linear feasibility problem lfp we then show that if the reconstruction process is defined as an lfp its implementation is less complex than in the case of the dp method we further describe how the lfp can be well conditioned and demonstrate the reconstruction process by using the surrogate constraint algorithm with random constraint selection the described algorithm considers a small number of constraints at a time which makes it suitable for serial implementation

implementation of split radix fft algorithms for complex real and real symmetric data a new algorithm is presented for the fast computation of the discrete fourier transform this algorithm belongs to that class of recently proposed 2n fft s which present the same arithmetic complexity the lowest among any previously published one moreover this algorithm has the advantage of being performed in place by repetitive use of a butterfly type structure without any data reordering inside the algorithm furthermore it can easily be applied to real and real symmetric data with reduced arithmetic complexity by removing all redundancy in the algorithm

algorithms meeting the lower bounds on the multiplicative complexity of length 2 sup n dfts and their connection with practical algorithms in a previous work see electron lett vol 20 no 17 p 690 1984 the author described an algorithm that computes a length 2 sup n discrete fourier transform using 2 sup 2 1 2n sup 2 4n 8 nontrivial i e not or j or square root 1 complex multiplications in the present work it is first shown that this algorithm actually provides the attainable lower bound on the number of complex multiplications a slight modification of the last step of this algorithm is also shown to provide the attainable lower bound on the number of real multiplications a connection with the split radix fft algorithm srfft is then explained showing that srfft is another variation of these optimal algorithms where the last step is computed recursively from shorter ffts in a suboptimal manner finally once the connection between the minimal complexity and srfft which is the best known practical algorithm is understood it provides useful information on the possibility of further improvements of the srfft

guaranteed stable sliding dft algorithm with minimal computational requirements the discrete fourier transform dft is the most widely used technique for determining the frequency spectra of digital signals however in the sliding transform scenario where the transform window is shifted one sample at a time and the transform process is repeated the use of dft becomes difficult due to its heavy computational burden this paper proposes an optimal sliding dft osdft algorithm that achieves both the lowest computational requirement and the highest computational accuracy among existing sliding dft algorithms the proposed osdft algorithm directly computes the dft bins of the shifted window by simply adding or subtracting the bins of a previous window and an updating vector we show that the updating vector can be efficiently computed with a low complexity in the sliding transform scenario our simulations demonstrate that the proposed algorithm outperforms the existing sliding dft algorithms in terms of computational accuracy and processing time

ramanujan fourier series the wiener khintchine formula and the distribution of prime pairs the wiener khintchine formula plays a central role in statistical mechanics it is shown here that the problem of prime pairs is related to autocorrelation and hence to a wiener khintchine formula experimental evidence is given for this c 1999 elsevier science b v all rights reserved

accuracy of the discrete fourier transform and the fast fourier transform fast fourier transform fft based computations can be far more accurate than the slow transforms suggest discrete fourier transforms computed through the fft are far more accurate than slow transforms and convolutions computed via fft are far more accurate than the direct results however these results depend critically on the accuracy of the fft software employed which should generally be considered suspect popular recursions for fast computation of the sine cosine table or twiddle factors are inaccurate due to inherent instability some analyses of these recursions that have appeared heretofore in print suggesting stability are incorrect even in higher dimensions the fft is remarkably stable

fast fir algorithms for the continuous wavelet transform from constrained least squares new algorithms for the continuous wavelet transform are developed that are easy to apply each consisting of a single pass finite impulse response fir filter and several times faster than the fastest existing algorithms the single pass filter named wt fir 1 is made possible by applying constraint equations to least squares estimation of filter coefficients which removes the need for separate low pass and high pass filters non dyadic two scale relations are developed and it is shown that filters based on them can work more efficiently than dyadic ones example applications to the mexican hat wavelet are presented

on the fixed point accuracy analysis of fft algorithms in this paper we investigate the effect of fixed point arithmetics with limited precision for different fast fourier transform fft algorithms a matrix representation of error propagation model is proposed to analyze the rounding effect an analytic expression of overall quantization loss due to the arithmetic quantization errors is derived to compare the performance with decimation in time dit and decimation in frequency dif configurations from the simulation results the radix 2 dit fft algorithm has better accuracy in term of signal to quantization noise ratio sqnr based on the results a simple criterion of wordlength optimization is proposed to yield comparable accuracy with fewer bit budget

efficient dealiased convolutions without padding algorithms are developed for calculating dealiased linear convolution sums without the expense of conventional zero padding or phase shift techniques for one dimensional in place convolutions the memory requirements are identical with the zero padding technique with the important distinction that the additional work memory need not be contiguous with the input data this decoupling of data and work arrays dramatically reduces the memory and computation time required to evaluate higher dimensional in place convolutions the technique also allows one to dealias the higher order convolutions that arise from fourier transforming cubic and higher powers implicitly dealiased convolutions can be built on top of state of the art fast fourier transform libraries vectorized multidimensional implementations for the complex and centered hermitian pseudospectral cases have been implemented in the open source software fftw

signal periodic decomposition with conjugate subspaces in this paper we focus on hidden period identification and the periodic decomposition of signals based on recent results on the ramanujan subspace we reveal the conjugate symmetry of the ramanujan subspace with a set of complex exponential basis functions and represent the subspace as the union of a series of conjugate subspaces with these conjugate subspaces the signal periodic model is introduced to characterize the periodic structure of a signal to achieve the decomposition of the proposed model conjugate subspace pursuit with periodicity and energy csppe algorithm is proposed based on two different greedy strategies the csppe is performed iteratively in two stages in the first stage the dominant hidden period is chosen with the periodicity strategy then the dominant conjugate subspace is chosen with the energy strategy in the second stage compared with the current state of the art methods for hidden period identification the main advantages provided by the csppe are the following 1 the capability of identifying all the hidden periods in the range from 1 to the maximum hidden period q of a signal of any length without truncating the signal 2 the ability to identify the time varying hidden period with its shifted version and 3 the low computational cost without generating and using a large over complete dictionary moreover we provide examples and applications to demonstrate the abilities of the proposed two stage csppe algorithm which include hidden period identification signal approximation time varying period detection and pitch detection of speech

h sub spl infin design with pole placement constraints an lmi approach this paper addresses the design of state or output feedback h sub spl infin controllers that satisfy additional constraints on the closed loop pole location sufficient conditions for feasibility are derived for a general class of convex regions of the complex plane these conditions are expressed in terms of linear matrix inequalities lmis and the authors formulation is therefore numerically tractable via lmi optimization in the state feedback case mixed h sub 2 h sub spl infin synthesis with regional pole placement is also discussed finally the validity and applicability of this approach are illustrated by a benchmark example

ellipsoidal techniques for reachability analysis internal approximation for the reach tube of a linear time varying system with ellipsoidal bounds on the control variable consider the following approximation problem find a tight ellipsoid valued tube inside the reach tube that touches it along any prespecified smooth curve on the boundary we construct this ellipsoidal tube and show that if the given curve is itself a system trajectory the tube can be calculated recursively with minimum computational burden

a two stage representation of dft and its applications a two stage representation in terms of preprocessing and postprocessing of dft is developed by vector transformation of sines and cosines into new basis functions using mobius inversion of number theory the preprocessing matrix with elements 1 1 and 0 is obtained by replacing cos 2 pin n and sin 2 pin n by mu n n 1 4 and mu n n respectively where mu cdot is the bipolar rectangular wave function the postprocessing matrix is block diagonal where each block is a circular correlation and consists of the new basis functions the two stage representation has been found very useful in applications such as parallel implementation of dft and signal image recognition

convolution idempotents with a given zero set we investigate the structure of n length discrete signals h satisfying h h h that vanish on a given set of indices we motivate this problem from examples in sampling fuglede s conjecture and orthogonal interpolation of bandlimited signals when n p m is a prime power we characterize all such h with a prescribed zero set in terms of base p expansions of nonzero indices in mathcal f 1 h

on polynomial multiplication in chebyshev basis in a recent paper lima panario and wang have provided a new method to multiply polynomials expressed in chebyshev basis which reduces the total number of multiplication for small degree polynomials although their method uses karatsuba s multiplication a quadratic number of operations are still needed in this paper we extend their result by providing a complete reduction to polynomial multiplication in monomial basis which therefore offers many subquadratic methods our reduction scheme does not rely on basis conversions and we demonstrate that it is efficient in practice finally we show a linear time equivalence between the polynomial multiplication problem under monomial basis and under chebyshev basis

optimization over state feedback policies for robust control with constraints this paper is concerned with the optimal control of linear discrete time systems subject to unknown but bounded state disturbances and mixed polytopic constraints on the state and input it is shown that the class of admissible affine state feedback control policies with knowledge of prior states is equivalent to the class of admissible feedback policies that are affine functions of the past disturbance sequence this implies that a broad class of constrained finite horizon robust and optimal control problems where the optimization is over affine state feedback policies can be solved in a computationally efficient fashion using convex optimization methods this equivalence result is used to design a robust receding horizon control rhc state feedback policy such that the closed loop system is input to state stable iss and the constraints are satisfied for all time and all allowable disturbance sequences the cost to be minimized in the associated finite horizon optimal control problem is quadratic in the disturbance free state and input sequences the value of the receding horizon control law can be calculated at each sample instant using a single tractable and convex quadratic program qp if the disturbance set is polytopic or a tractable second order cone program socp if the disturbance set is given by a 2 norm bound

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

adaptive comb filtering for harmonic signal enhancement a new algorithm is presented for adaptive comb filtering and parametric spectral estimation of harmonic signals with additive white noise the algorithm is composed of two cascaded parts the first estimates the fundamental frequency and enhances the harmonic component in the input and the second estimates the harmonic amplitudes and phases performance analysis provides new results for the asymptotic cramer rao bound crb on the parameters of harmonic signals with additive white noise results of simulations indicate that the variances of the estimates are of the same order of magnitude as the crb for sufficiently large data sets and illustrate the performance in enhancing noisy artificial periodic signals

orthogonal ramanujan sums its properties and applications in multiresolution analysis signal processing community has recently shown interest in ramanujan sums which were defined by s ramanujan in 1918 in this paper we propose a family of sequences termed here as orthogonal ramanujan sums c k q n and also derive their various properties we also prove two new properties of the ramanujan sums namely shifted multiplication property and interpolation property these properties are used to define higher order orthogonal ramanujan sums these sums are periodic in nature and are orthogonal for different values of k and q using orthogonal ramanujan sums a new representation of finite length signal is given termed here as orthogonal ramanujan periodic transform the coefficients obtained using this transform can be divided into two parts first coefficient gives smoothing component and the rest give the detail components of the signal further application of the orthogonal ramanujan sums in discrete wavelet transform is developed construction of ramanujan wavelet analysis matrix for applying discrete wavelet transform on a signal is described the application of this matrix on an image of size n times n where n is divisible by integer q will result in q 2 1 detail components and one average component in transform domain some of the applications of this transform are also demonstrated especially in the field of image processing

bounded gaps between primes it is proved that lim inf n 8 p n 1 p n 7 10 7 where p n is the n th prime our method is a refinement of the recent work of goldston pintz and yildirim on the small gaps between consecutive primes a major ingredient of the proof is a stronger version of the bombieri vinogradov theorem that is applicable when the moduli are free from large prime divisors only but it is adequate for our purpose

periodicity transforms this paper presents a method of detecting periodicities in data that exploits a series of projections onto periodic subspaces the algorithm finds its own set of nonorthogonal basis elements based on the data rather than assuming a fixed predetermined basis as in the fourier gabor and wavelet transforms a major strength of the approach is that it is linear in period rather than linear in frequency or linear in scale the algorithm is derived and analyzed and its output is compared to that of the fourier transform in a number of examples one application is the finding and grouping of rhythms in a musical score another is the separation of periodic waveforms with overlapping spectra and a third is the finding of patterns in astronomical data examples demonstrate both the strengths and weaknesses of the method

fast frequency template matching using higher order statistics this correspondence proposes a novel template matching technique using a fourth central moment the fourth central moment is an established estimator which uses higher order statistics theory important in the presence of an additive gaussian noise by use of some substitutions and complex arithmetic computation of the fourth central moment is derived from correlation functions of substituting functions the fourth central moment can be computed using the fast fourier transform fft approach simulation results show that the proposed algorithm performs better than the classical estimators in terms of robustness while the extra computational cost is negligible

solving the protein sequence metric problem biological sequences are composed of long strings of alphabetic letters rather than arrays of numerical values lack of a natural underlying metric for comparing such alphabetic data significantly inhibits sophisticated statistical analyses of sequences modeling structural and functional aspects of proteins and related problems herein we use multivariate statistical analyses on almost 500 amino acid attributes to produce a small set of highly interpretable numeric patterns of amino acid variability these high dimensional attribute data are summarized by five multidimensional patterns of attribute covariation that reflect polarity secondary structure molecular volume codon diversity and electrostatic charge numerical scores for each amino acid then transform amino acid sequences for statistical analyses relationships between transformed data and amino acid substitution matrices show significant associations for polarity and codon diversity scores transformed alphabetic data are used in analysis of variance and discriminant analysis to study dna binding in the basic helix loop helix proteins the transformed scores offer a general solution for analyzing a wide variety of sequence analysis problems

two dimensional efficient multiplier less structures of mobius function for ramanujan filter banks ramanujan s sum has been recently applied to detect different periodic components in 2d image by filtering in frequency domain and obtaining the least common multiplier lcm of periodicity matrices the basic 2d periodic pattern can be found this paper introduces a multiplier less structure to efficiently implement this process based on the well known number theoretic function called mobius function the relationship between impulse train and ramanujan s sum is also clearly described in both 1d and 2d cases

ramanujan sums in the context of signal processing part ii fir representations and applications the mathematician ramanujan introduced a summation in 1918 now known as the ramanujan sum c q n in a companion paper part i properties of ramanujan sums were reviewed and ramanujan subspaces s q introduced of which the ramanujan sum is a member in this paper the problem of representing finite duration fir signals based on ramanujan sums and spaces is considered first it is shown that the traditional way to solve for the expansion coefficients in the ramanujan sum expansion does not work in the fir case two solutions are then developed the first one is based on a linear combination of the first n ramanujan sums with n being the length of the signal the second solution is based on ramanujan subspaces with q 1 q 2 q k denoting the divisors of n it is shown that x n can be written as a sum of k signals x qi n s qi furthermore the i th signal x qi n has period q i and any pair of these periodic components is orthogonal the components x qi n can be calculated as orthogonal projections of x n onto ramanujan spaces s qi then the ramanujan periodic transform rpt is defined based on this and is useful to identify hidden periodicities it is shown that the projection matrices which compute x qi n from x n are integer matrices except for an overall scale factor the calculation of projections is therefore rendered easy to estimate internal periods n n of x n one only needs to know which projection energies are nonzero

music maximum likelihood and cramer rao bound the performance of the music and ml methods is studied and their statistical efficiency is analyzed the cramer rao bound crb for the estimation problems is derived and some useful properties of the crb covariance matrix are established the relationship between the music and ml estimators is investigated as well a numerical study is reported of the statistical efficiency of the music estimator for the problem of finding the directions of two plane waves using a uniform linear array an exact description of the results is included

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

a linear cost algorithm to compute the discrete gabor transform in this paper we propose an alternative efficient method to calculate the gabor coefficients of a signal given a synthesis window with a support of size much lesser than the length of the signal the algorithm uses the canonical dual of the window which does not need to be calculated beforehand and achieves a computational cost that is linear with the signal length in both analysis and synthesis this is done by exploiting the block structure of the matrices and using an ad hoc cholesky decomposition of the gabor frame matrix

fixed point analysis and parameter selections of msr cordic with applications to fft designs mixed scaling rotation msr coordinate rotation digital computer cordic is an attractive approach to synthesizing complex rotators this paper presents the fixed point error analysis and parameter selections of msr cordic with applications to the fast fourier transform fft first the fixed point mean squared error of the msr cordic is analyzed by considering both the angle approximation error and signal round off error incurred in the finite precision arithmetic the signal to quantization noise ratio sqnr of the output of the fft synthesized using msr cordic is thereafter estimated based on these analyses two different parameter selection algorithms of msr cordic are proposed for general and dedicated msr cordic structures the proposed algorithms minimize the number of adders and word length when the sqnr of the fft output is constrained design examples show that the fft designed by the proposed method exhibits a lower hardware complexity than existing methods

integer fast fourier transform a concept of integer fast fourier transform intfft for approximating the discrete fourier transform is introduced unlike the fixed point fast fourier transform fxpfft the new transform has the properties that it is an integer to integer mapping is power adaptable and is reversible the lifting scheme is used to approximate complex multiplications appearing in the fft lattice structures where the dynamic range of the lifting coefficients can be controlled by proper choices of lifting factorizations split radix fft is used to illustrate the approach for the case of 2 sup n point fft in which case an upper bound of the minimal dynamic range of the internal nodes which is required by the reversibility of the transform is presented and confirmed by a simulation the transform can be implemented by using only bit shifts and additions but no multiplication a method for minimizing the number of additions required is presented while preserving the reversibility the intfft is shown experimentally to yield the same accuracy as the fxpfft when their coefficients are quantized to a certain number of bits complexity of the intfft is shown to be much lower than that of the fxpfft in terms of the numbers of additions and shifts finally they are applied to noise reduction applications where the intfft provides significantly improvement over the fxpfft at low power and maintains similar results at high power

complexity reduction self completely recursive radix 2 dct i iv algorithms abstract this paper proposes fast self completely recursive radix 2 discrete cosine transform dct of types i iv algorithms implementations of these new algorithms are stated through signal flow graphs these algorithms are derived mainly using a matrix factorization technique to factor the dct i iv matrices into sparse and scaled orthogonal matrices this paper fills the gap of addressing the remaining dcts in connection to our former work in the lowest multiplication complexity self recursive radix 2 dct ii iii algorithms there has been no radix 2 dct i algorithm which is self recursive and has sparse and scaled orthogonal factors as proposed in this paper the paper also establishes a novel relationship between dct i and dct iii matrices having sparse factors for any n 2 t where t 1 this enables one to observe all traditional completely recursive dct i algorithms as the self recursive dct i algorithm proposed in this paper apart from the novel dct i algorithm we also establish a novel completely recursive dct iv algorithm which executes recursively with the lowest multiplication complexity self recursive radix 2 dct ii algorithm which was derived in our former paper the proposed dct iv algorithm attains the lowest possible multiplication counts

a general class of split radix fft algorithms for the computation of the dft of length 2 m in this paper a general class of split radix fast fourier transform fft algorithms for computing the length 2m dft is proposed by introducing a new recursive approach coupled with an efficient method for combining the twiddle factors this enables the development of higher split radix fft algorithms from lower split radix fft algorithms without any increase in the arithmetic complexity specifically an arbitrary radix 2 2s fft algorithm for any value of s 4les sles m is proposed and its arithmetic complexity analyzed it is shown that the number of arithmetic operations multiplications plus additions required by the proposed radix 2 2s fft algorithm is independent of s and is 2m 3 2m 1 8 regardless of whether a complex multiplication is carried out using four multiplications and two additions or three multiplications and three additions this paper thus provides a variety of choices and ways for computing the length 2m dft with the same arithmetic complexity

on tractable approximations of uncertain linear matrix inequalities affected by interval uncertainty we present efficiently verifiable sufficient conditions for the validity of specific np hard semi infinite systems of linear matrix inequalities lmis arising from lmis with uncertain data and demonstrate that these conditions are tight up to an absolute constant factor in particular we prove that given an n times n interval matrix cal u rho a mid a ij a ij leq rho c ij one can build a computable lower bound accurate within the factor pi over 2 on the supremum of those rho for which all instances of cal u rho share a common quadratic lyapunov function we then obtain a similar result for the problem of quadratic lyapunov stability synthesis finally we apply our techniques to the problem of maximizing a homogeneous polynomial of degree 3 over the unit cube

a unified theory of union of subspaces representations for period estimation several popular period estimation techniques use union of subspaces models to represent periodic signals the main idea behind these techniques is to compare the components of the signal along a set of subspaces representing different periods such techniques were shown to offer important advantages over traditional methods such as those based on dft so far most of these subspace techniques have been developed independent of each other and there has not been a unified theory analyzing them from a common perspective in this paper all such methods are first unified under one general framework further several fundamental aspects of such subspaces are investigated such as the conditions under which a generic set of subspaces offers unique periodic decompositions their minimum required dimensions etc a number of basic questions in the context of dictionaries spanning periodic signals are also answered for example what is the theoretically minimum number of atoms required in any type of dictionary in order to be able to represent periods 1 p p max for each period p what should be the minimum dimension of the subspace of atoms representing the pth period itself unlike in traditional fourier dictionaries it is shown that nonuniform and compact grids are crucial for period estimation interestingly it will be seen that the euler totient function from number theory plays an important role in providing the answers to all such questions

minimum resource commitment for reachability specifications in a discrete time linear setting this paper addresses control input design for a discrete time linear system the goal is to satisfy a reachability specification and at the same time minimize the number of inputs that need to be set influential inputs to this purpose we introduce an appropriate input parametrization so that depending on the parameter values some of the inputs act as control variables while the others are treated as disturbances and can take an arbitrary value in their range we then enforce the specification while maximizing the number of disturbance inputs two approaches are developed one based on an open loop scheme and one based on a compensation scheme in the former we end up solving a linear program in the latter the parametrization is extended so as to allow the influential inputs to depend on the non influential ones and the problem is reduced to a mixed integer linear program a comparison between the two approaches is carried out showing the superiority of the latter possible applications to system design and security of networked control systems are briefly discussed in the introduction

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

the regular module problem i in the study of induced representations the following problem arises let h ag be a finite solvable group and k a field with chark a let v be an irreducible faithful primitive k 4g module suppose h contains a normal extraspecial r subgroup r with z r z h and that a acts faith fully on r under what conditions does a have a regular direct summand in vi in this paper we consider this question under the hypotheses that g mr where 1 m is normal abelian in am a is nilpotent a mr m r 1 and r z r is a faithful irreducible am modu e we show that a has at least three regular direct summands in v unless a exp m and r satisfy certain very restrictive conditions let h ag be a finite solvable group where a is nilpotent g is normal and y4 g 1 let k be a field of characteristic not dividing a and v a zc module assume that v is faithful and irreducible the study of induced representations often leads to the following two questions 1 regular orbit problem under what conditions does a have a regular orbit on f

ramanujan sums in the context of signal processing part i fundamentals the famous mathematician s ramanujan introduced a summation in 1918 now known as the ramanujan sum c q n for any fixed integer q this is a sequence in n with periodicity q ramanujan showed that many standard arithmetic functions in the theory of numbers such as euler s totient function n and the mobius function n can be expressed as linear combinations of c q n 1 q in the last ten years ramanujan sums have aroused some interest in signal processing there is evidence that these sums can be used to extract periodic components in discrete time signals the purpose of this paper and the companion paper part ii is to develop this theory in detail after a brief review of the properties of ramanujan sums the paper introduces a subspace called the ramanujan subspace s q and studies its properties in detail for fixed q the subspace s q includes an entire family of signals with properties similar to c q n these subspaces have a simple integer basis defined in terms of the ramanujan sum c q n and its circular shifts the projection of arbitrary signals onto these subspaces can be calculated using only integer operations linear combinations of signals belonging to two or more such subspaces follows certain specific periodicity patterns which makes it easy to identify periods in the companion paper part ii it is shown that arbitrary finite duration signals can be decomposed into a finite sum of orthogonal projections onto ramanujan subspaces