conversion and evaluation for two types of parametric surfaces constructed by ntp bases in this paper a new generalized ball basis normalized totally positive ntp basis given by delgado and pena is investigated the conversion formulae between the basis and the bernstein basis are derived we also prove that these formulae not only are valuable for studying the geometric properties such as subdivision of the curves and surfaces constructed by this generalized ball basis but also can improve the computational speed of the bezier curves and surfaces after the bezier surface curve is converted into the generalized ball surface curve the time complexity for evaluation can be reduced from cubic to quadratic of the degree of the surface curve however the intrinsic property such as shape preserving property is not changed so the generalized ball surface and curve have a great future in application of geometric design

efficient computation of the discrete pseudo wigner distribution a description is given of a novel algorithm the fast fourier transform in part fftp for the computation of the discrete pseudo wigner distribution dpwd the fftp computes the cosine and sine parts of the discrete fourier transform dft separately by employing real inverse sinusoidal twiddle factors unlike the conventional methods which directly utilize the complex dft the fftp yields real output since the dpwd is always real in addition the new method reduces the computational cost by making full use of symmetries and removing redundancies in the fftp computation the authors also describe a simple algorithm for computing the discrete hilbert transform dht to produce the nonaliased dpwd a pipeline structure for real time and a bulk processing technique for offline implementations of the method are presented

a flexible and approximate computing approach for time frequency distributions eigen decomposition of the kernel allows fast computation of time frequency distributions this paper proposes a scheme which exploits further approximation by adapting dynamically the accuracy of spectrograms to the eigenvalue magnitudes the approach employs the wavelet transform followed by block recursive fourier transform and a compensation network it provides a set of parameters for the users to trade accuracy against computational complexity so that the implementation can be tailored to a specific application

vandermonde systems on gauss lobatto chebyshev nodes abstract this paper deals with vandermonde matrices v n whose nodes are the gauss lobatto chebyshev nodes also called extrema chebyshev nodes we give an analytic factorization and explicit formula for the entries of their inverse and explore its computational issues we also give asymptotic estimates of the frobenius norm of both v n and its inverse and present an explicit formula for the determinant of v n

kernel decomposition of time frequency distributions bilinear time frequency distributions tfds offer improved time frequency resolution over linear representations but suffer from difficult interpretation higher implementation cost and the lack of associated low cost signal synthesis algorithms in the paper the authors introduce some new tools for the interpretation and quantitative comparison of high resolution tfds these tools are used in related work to define low cost high resolution tfds and to define linear low cost signal synthesis algorithms associated with high resolution tfds first each real valued tfd is associated with a self adjoint linear operator spl psi the spectral representation of spl psi expresses the tfd as a weighted sum of spectrograms sps it is shown that the sp decomposition and weyl correspondence do not yield useful interpretations for high resolution tfds due to the fact that spl psi is not positive

wigner ville spectral analysis of nonstationary processes the wigner ville spectrum has been recently introduced as the unique generalized spectrum for time varying spectral analysis its properties are revised with emphasis on its central role in the analysis of second order properties of nonstationary random signals we propose here a general class of spectral estimators of the wigner ville spectrum this class is based on arbitrarily weighted covariance estimators and its formal description corresponds to the general class of conjoint time frequency representations of deterministic signals with finite energy classical estimators like short time periodograms and the recently introduced pseudo wigner estimators are shown to be special cases of the general class the generalized framework allows the calculation of the moments of general spectral estimators and comparing the results emphasizes the versatility of the new pseudo wigner estimators the effective numerical implementation by an n point fft of pseudo wigner estimators of 2n points is indicated and various examples are given

vector valued time frequency representations cohen s 1989 class of time frequency distributions tfds which includes the spectrogram sp wigner distribution wd and reduced interference distributions rids has become widely known as a useful signal analysis tool it has been shown that every real valued tfd can be written as a weighted sum of sps the sp decomposition has been used to construct fast approximations to desirable tfds using the sp building block for which there exist accessible and efficient hardware and software implementations we introduce a class of linear vector valued time frequency representations tfrs that are easily related to associated bilinear tfds through the sp decomposition we solve a least squares signal synthesis problem on modified vector valued tfrs that are associated with nonnegative tfds as a weighted sum of least squares short time fourier transform stft signal synthesis schemes we extend the solution to vector valued tfrs associated with high resolution tfds in order to define a high resolution alternative to stft signal synthesis as demonstrated by desirable properties and examples the resulting signal synthesis methods can be realized as a weighted sum of stft synthesis schemes for which there exist accessible and efficient hardware and software implementations

a test based on conversion to the bernstein polynomial basis for an interval to be free of zeros applicable to polynomials in chebyshev form and to transcendental functions approximated by chebyshev series abstract polynomials of high degree are much less vulnerable to roundoff error when expressed as truncated chebyshev series rather than the usual power series form recent articles have developed subdivision methods in which all real roots on the canonical chebyshev interval x 1 1 are found by subdividing the interval and finding the roots of separate chebyshev series of moderate degree on each subdomain this strategy can be applied either to polynomials or to transcendental functions if the latter are analytic on the search interval and thus have rapidly convergent chebyshev polynomial approximations the last step is to compute the eigenvalues of the chebyshev frobenius companion matrix for each local polynomial here we propose a simple strategy for flagging some subdomains as zero free so that eigensolving can be omitted the test requires conversion of the polynomial from chebyshev form to a bernstein polynomial basis the interval is zero free if all coefficients in the bernstein basis are of the same sign we give the conversion matrices for various small and moderate n and quote rababah s formulas for general n we show how to exploit parity so as to halve the cost of the conversion for large n to about n 2 floating point operations we show that the conversion matrices have condition numbers that are approximately 5 8 2 n

the weyl correspondence and time frequency analysis describes the weyl correspondence and its properties showing how it gives a window independent definition of time frequency concentration for use in models in signal detection the definition of concentration is justified by showing that it gives reasonable answers in certain intuitive cases the weyl correspondence expresses a linear transformation as a weighted superposition of time frequency shifts of the signal and then authors explain why this is not the same as transforming a signal into the time frequency domain multiplying by a weight in the transform domain and taking the inverse the investigation into time frequency concentration and the weyl correspondence is justified by a new result the authors show that convolving the wigner distribution with a general smoothing function is equivalent to evaluating a weighted sum of spectrograms this is a new interpretation of the process of smoothing the wigner distribution to reduce cross terms it relates smoothing of the wigner distribution to the multiple window technique pioneered by thomson 1982

nonlinear resonance for quasilinear hyperbolic equation the purpose of this paper is to study the wave behavior of hyperbolic conservation laws with a moving source resonance occurs when the speed of the source is too close to one of the characteristic speeds of the system for the nonlinear system characteristic speeds depend on the basic dependence variables and resonance gives rise to nonlinear interactions which lead to rich wave phenomena motivated by physical examples a scalar model is proposed and analyzed to describe the qualitative behavior of waves for a general system in resonance with the source analytical understanding is used to design a numerical scheme based on the random choice method an important physical example is transonic gas flow through a nozzle this analysis provides a transparent and revealing qualitative understanding of wave behavior of gas flow including such phenomena as nonlinear stability instability and changing types of waves

robustness analysis of polynomials with polynomial parameter dependency using bernstein expansion this paper considers the robust stability verification of polynomials with coefficients depending polynomially on parameters varying in given intervals two algorithms are presented both rely on the expansion of a multivariate polynomial into bernstein polynomials the first one is an improvement of the so called bernstein algorithm and checks the hurwitz determinant for positivity over the parameter set the second one is based on the analysis of the value set of the family of polynomials and profits from the convex hull property of the bernstein polynomials numerical results to real world control problems are presented showing the efficiency of both algorithms

kernel design for reduced interference distributions the authors present a class of time frequency signal representations tfrs called the reduced interference distribution rid an overview of commonly used tfrs is given and desirable distribution properties are introduced particular attention is paid to the interpretation of cohen s class of time frequency distributions of tfrs in the ambiguity temporal correlation spectral correlation and time frequency domains based on the desirable kernel requirements the rid is discussed and further defined a systematic procedure to create rid kernels or equivalently compute rids is proposed some aspects and properties of the rid are discussed the authors estimate design considerations for rids and compare various selections of the primitive window some experimental results demonstrating the performance of the rid are presented

a resolution comparison of several time frequency representations two signal components are considered resolved in a time frequency representation when two distinct peaks can be observed the time frequency resolution limit of two gaussian components alike except for their time and frequency centers is determined for the wigner distribution the pseudo wigner distribution the smoother wigner distribution the squared magnitude of the short time fourier transform and the choi williams distribution the relative performance of the various distributions depends on the signal the pseudo wigner distribution is best for signals of this class with only one frequency component at any one time the choi williams distribution is most attractive for signals in which all components have constant frequency content and the matched filter short time fourier transform is best for signal components with significant frequency modulation a relationship between the short time fourier transform and the cross wigner distribution is used to argue that with a properly chosen window the short time fourier transform of the cross wigner distribution must provide better signal component separation that the wigner distribution

robust wigner distribution with application to the instantaneous frequency estimation the wigner distribution wd produces highly concentrated time frequency tf representation of nonstationary signals it may be used as an efficient signal analysis tool including the cases of frequency modulated signals corrupted with the gaussian noise in some applications a significant amount of impulse noise is present then the wd fails to produce satisfactory results the robust periodogram has been introduced for spectral estimation of this kind of noisy signals it can produce good concentration for pure harmonic signals however it is not so efficient in the cases of signals with rapidly varying frequency this is the motivation for introducing the robust wd it is a reliable tf representation tool for wide class of nonstationary signals corrupted with impulse noise this distribution produces good accuracy of the instantaneous frequency if estimation using the huber 1981 loss function a generalization of the wd is presented it includes both the standard and the robust wd as special cases this distribution can be used for tf analysis of signals corrupted with a mixture of impulse and gaussian noise the presented theory is illustrated on examples including applications on the if estimation and time varying filtering of signals corrupted with a mixture of the gaussian and impulse noise the case study analysis of the if estimators accuracy based on the standard and the robust wd forms is performed in order to improve the if estimation a median filter is applied on the obtained if estimate

viscous limits for piecewise smooth solutions to systems of conservation laws in this paper we study the zero dissipation problem for a general system of conservation laws with positive viscosity it is shown that if the solution of the problem with zero viscosity is piecewise smooth with a finite number of noninteracting shocks satisfying the entropy condition then there exist solutions to the corresponding system with viscosity that converge to the solutions of the system without viscosity away from shock discontinuities at a rate of order e as the viscosity coefficient e goes to zero the proof uses a matched asymptotic analysis and an energy estimate related to the stability theory for viscous shock profiles

transition kernels for bilinear time frequency distributions the author proposes a sequence of time frequency signal representations which offer a continuous transition from time smoothed pseudo wigner ville distributions pwvd to a specified class of positive representations this allows the selection of a representation which offers the desired tradeoff between the inherent localization properties of the pwvd and the lack of interference terms present in the class of positive representations a cause of some criticism of the method the sequence converges uniformly to the specified positive distribution

shift covariant time frequency distributions of discrete signals many commonly used time frequency distributions are members of the cohen 1989 class this class is defined for continuous signals and since time frequency distributions in the cohen class are quadratic the formulation for discrete signals is not straightforward the cohen class can be derived as the class of all quadratic time frequency distributions that are covariant to time shifts and frequency shifts we extend this method to three types of discrete signals to derive what we call the discrete cohen classes the properties of the discrete cohen classes differ from those of the original cohen class to illustrate these properties we also provide explicit relationships between the classical wigner distribution and the discrete cohen classes

convergence and round off errors in a two dimensional eigenvalue problem using spectral methods and arnoldi chebyshev algorithm an efficient way of solving 2d stability problems in fluid mechanics is to use after discretization of the equations that cast the problem in the form of a generalized eigenvalue problem the incomplete arnoldi chebyshev method this method preserves the banded structure sparsity of matrices of the algebraic eigenvalue problem and thus decreases memory use and cpu time consumption the errors that affect computed eigenvalues and eigenvectors are due to the truncation in the discretization and to finite precision in the computation of the discretized problem in this paper we analyze those two errors and the interplay between them we use as a test case the 2d eigenvalue problem yielded by the computation of inertial modes in a spherical shell this problem contains many difficulties that make it a very good test case it turns out that single modes especially most damped modes i e with high spatial frequency can be very sensitive to roundoff errors even when apparently good spectral convergence is achieved the influence of roundoff errors is analyzed using the spectral portrait technique and by comparison of double precision and extended precision computations through the analysis we give practical recipes to control the truncation and roundoff errors on eigenvalues and eigenvectors

a note on the connection between bezier curves and linear optimal control we show that bezier curves can in fact be thought of as the solutions to linear optimal control problems using results from hermite interpolation in combination with traditional linear optimal control this provides us with a computational view of bezier curves that differs from the standard decasteljau algorithm and it furthermore points out the close relationship between bezier curves and interpolating dynamic splines

the use of cone shaped kernels for generalized time frequency representations of nonstationary signals generalized time frequency representations gtfrs which use cone shaped kernels for nonstationary signal analysis are presented the cone shaped kernels are formulated for the gtfrs to produce good resolution simultaneously in time and frequency specifically for a gtfr with a cone shaped kernel finite time support is maintained in the time dimension along with an enhanced spectrum in the frequency dimension and the cross terms are smoothed out experimental results on simulated data and real speech show the advantages of the gtfrs with the cone shaped kernels through comparisons to the spectrogram and the pseudo wigner distribution

spectral decomposition of time frequency distribution kernels this paper addresses the general problem of approximating a given time frequency distribution tfd in terms of other distributions with desired properties it relates the approximation of two time frequency distributions to their corresponding kernel approximation it is shown that the singular value decomposition svd of the time frequency t f kernels allows the expression of the time frequency distributions in terms of weighted sum of smoothed pseudo wigner ville distributions or modified periodograms which are the two basic nonparametric power distributions for stationary and nonstationary signals respectively the windows appearing in the decomposition take zero and or negative values and therefore are different than the time and lag windows commonly employed by these two distributions the centrosymmetry and the time support properties of the kernels along with the fast decay of the singular values lead to computational savings and allow for an efficient reduced rank kernel approximations

alias free generalized discrete time time frequency distributions a definition of generalized discrete time time frequency distribution that utilizes all of the outer product terms from a data sequence so that one can avoid aliasing is introduced the new approach provides 1 proper implementation of the discrete time spectrogram 2 correct evaluation of the instantaneous frequency of the underlying continuous time signal and 3 correct frequency marginal the formulation provides a unified framework for implementing members of cohen s class which was formulated in the continuous time domain some requirements for the discrete time kernel in the new approach are discussed in association with desirable distribution properties some experimental results are provided to illustrate the features of the proposed method

subspace analysis of spatial time frequency distribution matrices spatial time frequency distributions stfds have been previously introduced as the natural means to deal with source signals that are localizable in the time frequency domain previous work in the area has not provided the eigenanalysis of stfd matrices which is key to understanding their role in solving direction finding and blind source separation problems in multisensor array receivers the aim of this paper is to examine the eigenstructure of the stfd matrices we develop the analysis and statistical properties of the subspace estimates based on stfds for frequency modulated fm sources it is shown that improved estimates are achieved by constructing the subspaces from the time frequency signatures of the signal arrivals rather than from the data covariance matrices which are commonly used in conventional subspace estimation methods this improvement is evident in a low signal to noise ratio snr environment and in the cases of closely spaced sources the paper considers the music technique to demonstrate the advantages of stfds and uses it as grounds for comparison between time frequency and conventional subspace estimates

improved time frequency representation of multicomponent signals using exponential kernels the authors introduce a time frequency distribution of l cohen s 1966 class and examines its properties this distribution is called exponential distribution ed after its exponential kernel function first the authors interpret the ed from the spectral density estimation point of view they then show how the exponential kernel controls the cross terms as represented in the generalized ambiguity function domain and they analyze the ed for two specific types of multicomponent signals sinusoidal signals and chirp signals next they define the ed for discrete time signals and the running windowed exponential distribution rwed which is computationally efficient finally the authors present numerical examples of the rwed using the synthetically generated signals it is found that the ed is very effective in diminishing the effects of cross terms while retaining most of the properties which are useful for a time frequency distribution

gauss lobatto to bernstein polynomials transformation the aim of this paper is to transform a polynomial expressed as a weighted sum of discrete orthogonal polynomials on gauss lobatto nodes into bernstein form and vice versa explicit formulas and recursion expressions are derived moreover an efficient algorithm for the transformation from gauss lobatto to bernstein is proposed finally in order to show the robustness of the proposed algorithm experimental results are reported

fast computation of the ambiguity function and the wigner distribution on arbitrary line segments by using the fractional fourier transformation of the time domain signals closed form expressions for the projections of their auto or cross ambiguity functions are derived based on a similar formulation for the projections of the auto and cross wigner distributions and the well known two dimensional 2 d fourier transformation relationship between the ambiguity and wigner domains closed form expressions are obtained for the slices of both the wigner distribution and the ambiguity function by using discretization of the obtained analytical expressions efficient algorithms are proposed to compute uniformly spaced samples of the wigner distribution and the ambiguity function located on arbitrary line segments with repeated use of the proposed algorithms samples in the wigner or ambiguity domains can be computed on non cartesian sampling grids such as polar grids

legendre bernstein basis transformations abstract the bernstein form of a polynomial offers valuable insight into its geometrical behavior and has thus won widespread acceptance as the basis for bezier curves and surfaces for least squares approximation problems on the other hand the use of orthogonal bases such as the legendre polynomials permits simple and efficient constructions for convergent sequences of approximants we derive the transformation matrices that map the bernstein and legendre forms of a degree n polynomial on 0 1 into each other and examine the stability of this linear map in the p 1 and norms the condition number of the legendre bernstein transformation matrix grows at a significantly slower rate with n than in the well studied power bernstein case and at a dramatically slower rate than for other common e g bernstein hermite or power hermite basis conversions the utility of legendre representations in approximation problems and their relatively stable transformation to bernstein bezier form argue for more widespread applications of legendre methods in cagd algorithms

fast implementations of generalized discrete time frequency distributions cohen s class of time frequency distributions tfds have significant potential for the analysis of complex signals in order to evaluate the tfd of a signal using its samples discrete time tfds dtfds have been defined as the fourier transform of a smoothed discrete autocorrelation existing algorithms evaluate real valued dtfds using ffts of the conjugate symmetric autocorrelation although the computation required to smooth the autocorrelation is often greater than that for the fft there are no widely applicable fast algorithms for this part of the processing since the fft is relatively inexpensive downsampling is ineffective for reducing computation if the dtfd needs only to be evaluated at a few frequencies for each time instant the cost per time frequency sample can be extremely high the authors introduce two approaches for reducing the computation time of dtfds first they define approximations to real valued dtfds using spectrograms that admit fast space saving evaluations frequency downsampling reduces the computation time of these approximations next they define dtfds that admit fast evaluations over sparse sets of time frequency samples a single short time fourier transform is calculated in order for dtpd time frequency samples to be evaluated at an additional fixed cost per sample

point based methods for estimating the length of a parametric curve this paper studies a general method for estimating the length of a parametric curve using only samples of points we show that by making a special choice of points namely the gauss lobatto nodes we get higher orders of approximation similar to the behaviour of gauss quadrature and we derive some explicit examples

explicit solution of the finite time l2 norm polynomial approximation problem abstract the aim of this paper is to present a new approach to the finite time l 2 norm polynomial approximation problem a new formulation of this problem leads to an equivalent linear system whose solution can be investigated analytically such a solution is then specialized for a polynomial expressed in terms of laguerre and bernstein basis

explicit solution of the polynomial least squares approximation problem on chebyshev extrema nodes in this paper we propose an explicit solution to the polynomial least squares approximation problem on chebyshev extrema nodes we also show that the inverse of the normal matrix on this set of nodes can be represented as the sum of two symmetric matrices a full rank matrix which admits a cholesky factorization and a 2 rank matrix finally we discuss the numerical properties of the proposed formulas