wavelets and filter banks theory and design the wavelet transform is compared with the more classical short time fourier transform approach to signal analysis then the relations between wavelets filter banks and multiresolution signal processing are explored a brief review is given of perfect reconstruction filter banks which can be used both for computing the discrete wavelet transform and for deriving continuous wavelet bases provided that the filters meet a constraint known as regularity given a low pass filter necessary and sufficient conditions for the existence of a complementary high pass filter that will permit perfect reconstruction are derived the perfect reconstruction condition is posed as a bezout identity and it is shown how it is possible to find all higher degree complementary filters based on an analogy with the theory of diophantine equations an alternative approach based on the theory of continued fractions is also given these results are used to design highly regular filter banks which generate biorthogonal continuous wavelet bases with symmetries

the scale representation the authors considers scale a physical attribute of a signal and develop its properties he presents an operator which represents scale and study its characteristics and representation this allows one to define the scale transform and the energy scale density spectrum which is an indication of the intensity of scale values in a signal he obtains explicit expressions for the mean scale scale bandwidth instantaneous scale and scale group delay furthermore he derives expressions for mean time mean frequency duration frequency bandwidth in terms of the scale variable the short time transform is defined and used to obtain the conditional value of scale for a given time he shows that as the windows narrows one obtains instantaneous scale convolution and correlation theorems for scale are derived a formulation is devised for studying linear scale invariant systems he derives joint representations of time scale and frequency scale general classes for each are presented using the same methodology as for the time frequency case as special cases the joint distributions of marinovich altes 1978 1986 and bertrand bertrand 1984 are recovered also joint representations of the three quantities time frequency scale are devised a general expression for the local scale autocorrelation function is given uncertainty principles for scale and time and scale and frequency are derived

optimal quadratic detection and estimation using generalized joint signal representations time frequency analysis has significant advances in two main directions statistically optimized methods that extend the scope of time frequency based techniques from merely exploratory data analysis to more quantitative application and generalized joint signal representations that extend time frequency based methods to a richer class of nonstationary signals this paper fuses the two advances by developing optimal detection and estimation techniques based on generalized joint signal representations by generalizing the statistical methods developed for time frequency representations to arbitrary joint signal representations this paper develops a unified theory applicable to a wide variety of problems in nonstationary statistical signal processing

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

leak location in gas pipelines using cross time frequency spectrum of leakage induced acoustic vibrations abstract the correlation based location methods are widely used in leak detection of the pipelines assuming that the acoustic speed has been known and constant in practice the acoustic speed is frequency varying due to the dispersions of gas leak induced acoustic waves and thus the assumption is not supported in this work a location scheme based on cross time frequency spectrum ctfs is intended for the gas leak induced acoustic waves with frequency varying acoustic speed in the scheme the ctfs is obtained by the one dimensional fourier transform of the time domain convolution between the kernel function in correlation domain and the instantaneous cross correlation of the two spatially separately collected acoustic signals on either sides of a leakage then the time difference of arrival tdoa and the corresponding frequency information are extracted simultaneously when the ctfs reaches the maximum value the resulting peak frequency is used to online determine the frequency dependant acoustic speed in combination with the known dispersive curve of gas leak induced dominated mode finally the gas leakage is located by the tdoa and the frequency dependant acoustic speed of real time determination instead of constant acoustic speed consequently for the proposed scheme the constant acoustic speed is no longer a prerequisite the proposed scheme has been experimentally validated in leak detection of gas pipelines and results demonstrate that the average relative location errors are reduced by six times compared with the commonly used correlation based location method

optimal data based kernel estimation of evolutionary spectra complex demodulation of evolutionary spectra is formulated as a two dimensional kernel smoother in the time frequency domain first a tapered fourier transform y sub v f t is calculated then the log spectral estimate is smoothed as the characteristic widths of the kernel smoother increase the bias from the temporal and frequency averaging increases while the variance decreases the demodulation parameters such as the order length and bandwidth of spectral taper and the kernel smoother are determined by minimizing the expected error for well resolved evolutionary spectra the optimal taper length is a small fraction of the optimal kernel halfwidth the optimal frequency bandwidth w for the spectral window scales as w sup 2 approximately lambda tau where tau is the characteristic time and lambda sub f is the characteristic frequency scalelength in contrast the optimal halfwidths for the second stage kernel smoother scales as h approximately 1 tau lambda sub f sup 1 p 2 where p is the order of the kernel smoother the ratio of the optimal frequency halfwidth to the optimal time halfwidth is determined

scheme of lines on a family of 2 dimensional quadrics geometry and derived category given a generic family q of 2 dimensional quadrics over a smooth 3 dimensional base y we consider the relative fano scheme m of lines of it the scheme m has a structure of a generically conic bundle m rightarrow x over a double covering x rightarrow y ramified in the degeneration locus of q rightarrow y the double covering x rightarrow y is singular in a finite number of points corresponding to the points y in y such that the quadric q y degenerates to a union of two planes the fibers of m over such points are unions of two planes intersecting in a point the main result of the paper is a construction of a semiorthogonal decomposition for the derived category of coherent sheaves on m this decomposition has three components the first is the derived category of a small resolution x of singularities of the double covering x rightarrow y the second is a twisted resolution of singularities of x given by the sheaf of even parts of clifford algebras on y and the third is generated by a completely orthogonal exceptional collection

optimal kernels for nonstationary spectral estimation current theories of a time varying spectrum of a nonstationary process all involve either by definition or by difficulties in estimation an assumption that the signal statistics vary slowly over time this restrictive quasistationarity assumption limits the use of existing estimation techniques to a small class of nonstationary processes we overcome this limitation by deriving a statistically optimal kernel within cohen s 1989 class of time frequency representations tfr s for estimating the wigner ville spectrum of a nonstationary process we also solve the related problem of minimum mean squared error estimation of an arbitrary bilinear tfr of a realization of a process from a correlated observation both optimal time frequency invariant and time frequency varying kernels are derived it is shown that in the presence of any additive independent noise optimal performance requires a nontrivial kernel and that optimal estimation may require smoothing filters that are very different from those based on a quasistationarity assumption examples confirm that the optimal estimators often yield tremendous improvements in performance over existing methods in particular the ability of the optimal kernel to suppress interference is quite remarkable thus making the proposed framework potentially useful for interference suppression via time frequency filtering

anechoic blind source separation using wigner marginals blind source separation problems emerge in many applications where signals can be modeled as superpositions of multiple sources many popular applications of blind source separation are based on linear instantaneous mixture models if specific invariance properties are known about the sources for example translation or rotation invariance the simple linear model can be extended by inclusion of the corresponding transformations when the sources are invariant against translations spatial displacements or time shifts the resulting model is called an anechoic mixing model we present a new algorithmic framework for the solution of anechoic problems in arbitrary dimensions this framework is derived from stochastic time frequency analysis in general and the marginal properties of the wigner ville spectrum in particular the method reduces the general anechoic problem to a set of anechoic problems with non negativity constraints and a phase retrieval problem the first type of subproblem can be solved by existing algorithms for example by an appropriate modification of non negative matrix factorization nmf the second subproblem is solved by established phase retrieval methods we discuss and compare implementations of this new algorithmic framework for several example problems with synthetic and real world data including music streams natural 2d images human motion trajectories and two dimensional shapes

kernels and multiple windows for estimation of the wigner ville spectrum of gaussian locally stationary processes this paper treats estimation of the wigner ville spectrum wvs of gaussian continuous time stochastic processes using cohen s class of time frequency representations of random signals we study the minimum mean square error estimation kernel for locally stationary processes in silverman s sense and two modifications where we first allow chirp multiplication and then allow nonnegative linear combinations of covariances of the first kind we also treat the equivalent multitaper estimation formulation and the associated problem of eigenvalue eigenfunction decomposition of a certain hermitian function for a certain family of locally stationary processes which parametrizes the transition from stationarity to nonstationarity the optimal windows are approximately dilated hermite functions we determine the optimal coefficients and the dilation factor for these functions as a function of the process family parameter

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

hybrid linear bilinear time scale analysis the introduce a new method for the time scale analysis of nonstationary signals our work leverages the success of the time frequency distribution series cross term deleted representations into the time scale domain to match wideband signals that are better modeled in terms of time shifts and scale changes than in terms of time and frequency shifts using a wavelet decomposition and the bertrand time scale distribution we locally balance linearity and bilinearity in order to provide good resolution while suppressing troublesome interference components the theory of frames provides a unifying perspective for cross term deleted representations in general

derived categories of quadric fibrations and intersections of quadrics abstract we construct a semiorthogonal decomposition of the derived category of coherent sheaves on a quadric fibration consisting of several copies of the derived category of the base of the fibration and the derived category of coherent sheaves of modules over the sheaf of even parts of the clifford algebras on the base corresponding to this quadric fibration generalizing the kapranov s description of the derived category of a single quadric as an application we verify that the noncommutative algebraic variety p s 2 w b 0 where b 0 is the universal sheaf of even parts of clifford algebras is homologically projectively dual to the projective space p w in the double veronese embedding p w p s 2 w using the properties of the homological projective duality we obtain a description of the derived category of coherent sheaves on a complete intersection of any number of quadrics

time scale energy density functions scale like frequency is a physical characteristic of a signal to measure the scale content of a signal the signal must be appropriately transformed a theory for joint time scale energy density functions is presented and a method for generating such functions for any signal is given examples for synthetic signals and real data are presented the theory and method can be extended to arbitrary joint densities of any variables for example frequency and scale

the power classes quadratic time frequency representations with scale covariance and dispersive time shift covariance we consider scale covariant quadratic time frequency representations qtfrs specifically suited for the analysis of signals passing through dispersive systems these qtfrs satisfy a scale covariance property that is equal to the scale covariance property satisfied by the continuous wavelet transform and a covariance property with respect to generalized time shifts we derive an existence representation theorem that shows the exceptional role of time shifts corresponding to group delay functions that are proportional to powers of frequency this motivates the definition of the power classes pcs of qtfr s the pcs contain the affine qtfr class as a special case and thus they extend the affine class we show that the pcs can be defined axiomatically by the two covariance properties they satisfy or they can be obtained from the affine class through a warping transformation we discuss signal transformations related to the pcs the description of the pcs by kernel functions desirable properties and kernel constraints and specific pc members furthermore we consider three important pc subclasses one of which contains the bertrand 1992 p sub k distributions finally we comment on the discrete time implementation of pc qtfrs and we present simulation results that demonstrate the potential advantage of pc qtfrs

nonstationary spectral analysis based on time frequency operator symbols and underspread approximations we present a unified framework for time varying or time frequency tf spectra of nonstationary random processes in terms of tf operator symbols we provide axiomatic definitions and tf operator symbol formulations for two broad classes of tf spectra one of which is new these classes contain all major existing tf spectra such as the wigner ville evolutionary instantaneous power and physical spectrum our subsequent analysis focuses on the practically important case of nonstationary processes with negligible high lag tf correlations so called underspread processes we demonstrate that for underspread processes all tf spectra yield effectively identical results and satisfy several desirable properties at least approximately we also show that gabor frames provide approximate karhunen loeve kl functions of underspread processes and tf spectra provide a corresponding approximate kl spectrum finally we formulate simple approximate input output relations for the tf spectra of underspread processes that are passed through underspread linear time varying systems all approximations are substantiated mathematically by upper bounds on the associated approximation errors our results establish a tf calculus for the second order analysis and time varying filtering of underspread processes that is as simple as the conventional spectral calculus for stationary processes

group delay shift covariant quadratic time frequency representations we propose classes of quadratic time frequency representations qtfrs that are covariant to group delay shifts gdss the gds covariance qtfr property is important for analyzing signals propagating through dispersive systems with frequency dependent characteristics this is because a qtfr satisfying this property provides a succinct representation whenever the time shift is selected to match the frequency dependent changes in the signal s group delay that may occur in dispersive systems we obtain the gds covariant classes from known qtfr classes such as cohen s 1995 class the affine class the hyperbolic class and the power classes using warping transformations that depend on the relevant group delay change we provide the formulation of the gds covariant classes using two dimensional 2 d kernel functions and we list desirable qtfr properties and kernel constraints as well as specific class members we present known examples of the gds covariant classes and we provide a new class the power exponential qtfr class we study the localized kernel subclasses of the gds covariant classes that simplify the theoretical development as the kernels reduce from 2 d to one dimensional 1 d functions and we develop various intersections between the qtfr classes finally we present simulation results to demonstrate the advantage of using qtfrs that are matched to changes in the group delay of a signal

the transient spectrum of a random system the frequency spectrum is a fundamental quantity for the analysis and design of a system when a system is turned on or off the frequency spectrum of its output changes with time we define the transient spectrum as the time frequency spectrum of the system output during the transient phase we obtain the exact transient spectrum for a wide class of random systems and we formulate it with respect to the classical power spectrum which is reached for large times we apply the developed method to the case of a harmonic oscillator and of a system with two resonances

estimation of ambiguity functions with limited spread in this correspondence we propose a class of estimators of the ambiguity function of a discrete time process using thresholding to this purpose the distributional and moment properties of the empirical ambiguity function for a realization of an analytic random process are derived and the moments for the special cases of stationary uniformly modulated and deterministic signals immersed in noise processes are determined suitable thresholds are chosen based on the distributional properties of the empirical ambiguity function and the procedure is tested on simulated data

an iterative numerical method for singularly perturbed reaction diffusion equations with negative shift in this paper a numerical method based on an iterative scheme is proposed for a singularly perturbed delay differential equation of reaction diffusion type in this method the solution of the delay problem is obtained as the limit of the solutions to a sequence of the non delay problems the numerical solutions of the non delay problems are obtained by applying existing finite difference schemes and finite element method for the non delay singularly perturbed equations error estimates are derived in the supremum norm numerical results illustrating the theory are also included

the chirplet transform physical considerations we consider a multidimensional parameter space formed by inner products of a parameterizable family of chirp functions with a signal under analysis we propose the use of quadratic chirp functions which we will call q chirps for short giving rise to a parameter space that includes both the time frequency plane and the time scale plane as 2 d subspaces the parameter space contains a time frequency scale volume and thus encompasses both the short time fourier transform as a slice along the time and frequency axes and the wavelet transform as a slice along the time and scale axes in addition to time frequency and scale there are two other coordinate axes within this transform space shear in time obtained through convolution with a q chirp and shear in frequency obtained through multiplication by a q chirp signals in this multidimensional space can be obtained by a new transform which we call the q chirplet transform or simply the chirplet transform the proposed chirplets are generalizations of wavelets related to each other by 2 d affine coordinate transformations translations dilations rotations and shears in the time frequency plane as opposed to wavelets which are related to each other by 1 d affine coordinate transformations translations and dilations in the time domain only

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

wideband weyl symbols for dispersive time varying processing of systems and random signals we extend the narrowband weyl symbol ws and the wideband p sub o weyl symbol pows for dispersive time frequency tf analysis of nonstationary random processes and time varying systems we obtain the new tf symbols using unitary transformations on the ws and the p sub o ws for example whereas the ws is matched to systems with constant or linear tf characteristics the new symbols are better matched to systems with dispersive nonlinear tf structures this results from matching the geometry of the unitary transformation to the specific tf characteristics of a system we also develop new classes of smoothed weyl symbols that are covariant to tf shifts or time shift and scaling system transformations these classes of symbols are also extended via unitary warpings to obtain classes of tf symbols covariant to dispersive shifts we provide examples of the new symbols and symbol classes and we list some of their desirable properties using simulation examples we demonstrate the advantage of using tf symbols that are matched to the changes in the tf characteristics of a system or random process we also provide new tf formulations for matched detection applications

derived equivalent calabi yau threefolds from cubic fourfolds we describe pretty examples of derived equivalences and autoequivalences of calabi yau threefolds arising from pencils of cubic fourfolds the cubic fourfolds are chosen to be special so they each have an associated k3 surface thus a pencil gives rise to two different calabi yau threefolds the associated pencil of k3 surfaces and the baselocus of the original pencil the intersection of two cubic fourfolds they both have crepant resolutions which are derived equivalent

singular perturbation analysis of boundary value problems for differential difference equations a study is made of a class of boundary value problems for linear second order differential difference equations in which the highest order derivative is multiplied by a small parameter consideration of the exact solution of simple model equations provides insight into the appropriate use of singular perturbation techniques for more general problems the resulting analysis leads to several novel features which are not present in problems without differences

time frequency representation of mimo dynamical systems many deterministic and random physical signals can be modeled as the output of a multi input multi output mimo dynamical system since physical signals are typically nonstationary their frequency content changes with time to understand this time variation we transform the mimo system to the time frequency domain the result is a time frequency mimo dynamical system whose input and output are the time frequency spectra of the original input and output signals in the time domain the time frequency system reveals the spectral mechanisms involved in the generation of nonstationary signals we apply our method to the case of a mimo system with two vibrational modes and a nonstationary noise at the input we obtain the time frequency spectrum of the output which shows how the spectrum of the modes changes with time this result cannot be achieved with classical spectral techniques because they require the input random process to be wide sense stationary

on the birationality of complete intersections associated to nef partitions we prove that generic complete intersections associated to multiple mirror nef partitions are all birational this result solves a conjecture by batyrev and nill in 6 under some mild assumptions

on the equivalence of the operator and kernel methods for joint distributions of arbitrary variables generalizing the concept of time frequency representations cohen see englewood cliffs nj prentice hall 1995 has proposed a method based on operator correspondence rules for generating joint distributions of arbitrary variables as an alternative to considering all such rules which is a practical impossibility in general cohen has proposed the kernel method in which different distributions are generated from a fixed rule via an arbitrary kernel we derive a simple but rather stringent necessary condition on the underlying operators for the kernel method with the kernel functionally independent of the variables to generate all bilinear distributions of the specific pairs of variables that have been studied essentially only time and frequency satisfy the condition in particular the important variables of time and scale do not the results warrant further study for a systematic characterization of bilinear distributions in cohen s method

on clifford double mirrors of toric complete intersections abstract we present a construction of noncommutative double mirrors to complete intersections in toric varieties this construction unifies existing sporadic examples and explains the underlying combinatorial and physical reasons for their existence

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

adaptive covariance estimation of locally stationary processes it is shown that the covariance operator of a locally stationary process has approximate eigenvectors that are local cosine functions we model locally stationary processes with pseudo differential operators that are time varying convolutions an adaptive covariance estimation is calculated by searching first for a best local cosine basis which approximates the covariance by a band or a diagonal matrix the estimation is obtained from regularized versions of the diagonal coefficients in the best basis

a numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equations abstract a boundary value problem for second order singularly perturbed delay differential equation is considered when the delay argument is sufficiently small to tackle the delay term the researchers m k kadalbajoo k k sharma numerical analysis of singularly perturbed delay differential equations with layer behavior appl math comput 157 2004 11 28 r e o malley jr singular perturbation methods for ordinary differential equations springer verlag new york 1991 used taylor s series expansion and presented an asymptotic as well as numerical approach to solve such type boundary value problem but the existing methods in the literature fail in the case when the delay argument is bigger one because in this case the use of taylor s series expansion for the term containing delay may lead to a bad approximation in this paper to short out this problem we present a numerical scheme for solving such type of boundary value problems which works nicely in both the cases i e when delay argument is bigger one as well as smaller one to handle the delay argument we construct a special type of mesh so that the term containing delay lies on nodal points after discretization the proposed method is analyzed for stability and convergence to demonstrate the efficiency of the method and how the size of the delay argument and the coefficient of the delay term affects the layer behavior of the solution several test examples are considered

time scale energy distributions a general class extending wavelet transforms the theory of a new general class of signal energy representations depending on time and scale is developed time scale analysis has been introduced recently as a powerful tool through linear representations called continuous wavelet transforms wts a concept for which an exhaustive bilinear generalization is given although time scale is presented as an alternative method to time frequency strong links relating the two are emphasized thus combining both descriptions into a unified perspective the authors provide a full characterization of the new class the result is expressed as an affine smoothing of the wigner ville distribution on which interesting properties may be further imposed through proper choices of the smoothing function parameters not only do specific choices allow recovery of known definitions but they also provide via separable smoothing a continuous transition from wigner ville to either spectrograms or scalograms squared modulus of the wt this property makes time scale representations a very flexible tool for nonstationary signal analysis

a comparison between different discrete ambiguity domain definitions in stochastic time frequency analysis the ambiguity domain plays a central role in estimating the time varying spectrum and in estimating the covariance function of nonstationary random processes in continuous time for processes in discrete time there exist different definitions of the ambiguity domain but it is well known that neither of these definitions perfectly resembles the usefulness of the continuous ambiguity domain in this paper we present some of the most frequently used definitions of the ambiguity domain in discrete time the claasen mecklenbrauker the jeong williams and the nuttall definitions for the first time we prove their equivalence within some necessary conditions and we present theorems that justify their usage

multiple window time varying spectral analysis a multiwindow method for generating a time varying spectrum of nonstationary signals is presented the time varying spectrum is computed from an optimally weighted average of multiple orthogonal windowed spectrograms the weights are determined using linear least squares estimation with respect to a reference time frequency distribution examples are provided with performance criteria measures to demonstrate and quantify the effectiveness of the method

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed