on metaplectic modular categories and their applications for non abelian simple objects in a unitary modular category the density of their braid group representations the p hard evaluation of their associated link invariants and the bqp completeness of their anyonic quantum computing models are closely related we systematically study such properties of the non abelian simple objects in the metaplectic modular categories so m 2 for an odd integer m 3 the simple objects with quantum dimensions sqrt m have finite image braid group representations and their link invariants are classically efficient to evaluate we also provide classically efficient simulations of their braid group representations these simulations of the braid group representations can be regarded as qudit generalizations of the knill gottesmann theorem for the qubit case the simple objects of dimension 2 give us a surprising result while their braid group representations have finite images and are efficiently simulable classically after a generalized localization their link invariants are p hard to evaluate exactly we sharpen the p hardness by showing that any sufficiently accurate approximation of their associated link invariants is already p hard

a note on an overview of aliasing errors in discrete time formulations of time frequency representations various realizations asserting to produce the alias free discrete time wigner distribution have been reviewed and analyzed by costa and boudreaux bartels see ibid vol 47 no 5 p 1463 74 may 1999 here a simple alias free form of the pseudo wigner distribution is proposed it is obtained from the fact that the sampled signal can formally be considered as a multicomponent signal

geometric quantisation of the mic kepler problem the geometric quantisation scheme is applied to the compact kaehler orbit manifolds of the modified kepler problem thus the authors obtain the quantisation of the magnetic charge and energy spectrum of the corresponding quantum problem a new regularisation of the standard kepler problem is presented

a modular functor which is universal for quantum computation we show that the topological modular functor from witten chern simons theory is universal for quantum computation in the sense that a quantum circuit computation can be efficiently approximated by an intertwining action of a braid on the functor s state space a computational model based on chern simons theory at a fifth root of unity is defined and shown to be polynomially equivalent to the quantum circuit model the chief technical advance the density of the irreducible sectors of the jones representation has topological implications which will be considered elsewhere

on classification of modular tensor categories we classify all unitary modular tensor categories umtcs of rank 4 there are a total of 35 umtcs of rank 4 up to ribbon tensor equivalence since the distinction between the modular s matrix s and s has both topological and physical significance so in our convention there are a total of 70 umtcs of rank 4 in particular there are two trivial umtcs with s 1 each such umtc can be obtained from 10 non trivial prime umtcs by direct product and some symmetry operations explicit data of the 10 non trivial prime umtcs are given in sect 5 relevance of umtcs to topological quantum computation and various conjectures are given in sect 6

density of the so 3 tqft representation of mapping class groups we show that for an odd prime r 3 and an integer g 1 in the projective representation given by the so 3 witten reshitikhin turaev theory at an rth root of unity the image of the mapping class group of a surface of genus g is dense

micz kepler problems in all dimensions the kepler problem is a physical problem about two bodies which attract each other by a force proportional to the inverse square of the distance the micz kepler problems are its natural cousins and have been previously generalized from dimension 3 to dimension 5 in this paper we construct and analyze the quantum micz kepler problems in all dimensions higher than 2

a general approach for obtaining joint representations in signal analysis i characteristic function operator method a general approach for obtaining joint representations for arbitrary physical quantities is presented the characteristic function operator method of moyal 1949 and ville 1960 and generalized by cohen 1966 1976 and scully and cohen 1987 for the time frequency case is developed for arbitrary variables

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

geometric quantization of the brst charge in the first half of this paper sects 1 4 we generalise the standard geometric quantization procedure to symplectic supermanifolds in the second half sects 5 6 we apply this to two examples that exhibit classical brst symmetry i e we quantize the brst charge and the ghost number more precisely in the first example we consider the reduced symplectic manifold obtained by symplectic reduction from a free group action with ad equivariant moment map in the second example we consider a foliated configuration space whose cotangent bundle admits the construction of a brst charge associated to this foliation we show that the classical brst symmetry can be described in terms of a hamiltonian supergroup action on the extended phase space and that geometric quantization gives us a super unitary representation of this supergroup finally we point out how these results are related to reduction at the quantum level as compared with the reduction at the classical level

the development of a discrete transform for the wigner distribution and ambiguity function a theoretical basis for the development of the discrete wigner distribution and ambiguity function is presented that is based on the temporal and spectral properties of the two dimensional wigner kernel function this two dimensional description is helpful in visualizing the various functions and shows that the increase in the number of points required to compute the discrete wigner distribution is a natural consequence of the definition of the kernel and that it does not contradict the nyquist sampling theorem a generalized set of pseudofunctions is introduced that involves applying a two dimensional weighting function to the wigner kernel or its two dimensional fourier transform resulting in a wigner distribution that is weighted or smoothed in time or frequency two forms of the wigner distribution are derived the first is a pure wigner distribution without any windowing of the kernel that may be computed over the entire data set the second involves the general set of pseudo wigner distribution that may be calculated over a subset of the data set the discrete wigner distribution of the analytic signal is also developed and it is shown that the number of points may be reduced by a factor of 2 in the same way that the number of points required to represent the discrete analytic signal may be reduced the discrete ambiguity function is developed along with the discrete wigner distribution and it is shown that the four two dimensional functions are interrelated by discrete one and two dimensional fourier transforms

profinite completions of burnside type quotients of surface groups using quantum representations of mapping class groups we prove that profinite completions of burnside type surface group quotients are not virtually prosolvable in general further we construct infinitely many finite simple characteristic quotients of surface groups

braid representations from unitary braided vector spaces we investigate braid group representations associated with unitary braided vector spaces focusing on a conjecture that such representations should have virtually abelian images in general and finite image provided the braiding has finite order we verify this conjecture for the two infinite families of gaussian and group type braided vector spaces as well as the generalization to quasi braided vector spaces of group type

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

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

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

image of the braid groups inside the finite iwahori hecke algebras we determine the image of the braid groups inside the iwahori hecke algebras of type a when defined over a finite field in the semisimple case and for suitably large but controlable order of the defining quantum parameter

discrete time implementation of the cone kernel time frequency representation subtle differences exist between the continuous time and discrete time formulations of the cone kernel time frequency representation that should be taken into account prior to implementation the article derives the discrete formulation from equivalence in the ambiguity domain yielding a kernel different from that obtained by directly sampling the continuous time kernel yet preserving the important properties of the continuous time representation

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

an adaptive optimal kernel time frequency representation time frequency representations with fixed windows or kernels figure prominently in many applications but perform well only for limited classes of signals representations with signal dependent kernels can overcome this limitation however while they often perform well most existing schemes are block oriented techniques unsuitable for on line implementation or for tracking signal components with characteristics that change with time the time frequency representation developed in the present paper based on a signal dependent radially gaussian kernel that adapts over time surmounts these difficulties the method employs a short time ambiguity function both for kernel optimization and as an intermediate step in computing constant time slices of the representation careful algorithm design provides reasonably efficient computation and allows on line implementation certain enhancements such as cone kernel constraints and approximate retention of marginals are easily incorporated with little additional computation while somewhat more expensive than fixed kernel representations this new technique often provides much better performance several examples illustrate its behavior on synthetic and real world signals

a centrosymmetric kernel decomposition for time frequency distribution computation time frequency distributions tfds are bilinear transforms of the signal and as such suffer from a high computational complexity previous work has shown that one can decompose any tfd in cohen s class into a weighted sum of spectrograms this is accomplished by decomposing the kernel of the distribution in terms of an orthogonal set of windows in this paper we introduce a mathematical framework for kernel decomposition such that the windows in the decomposition algorithm are not arbitrary and that the resulting decomposition provides a fast algorithm to compute tfds using the centrosymmetric structure of the time frequency kernels we introduce a decomposition algorithm such that any tfd associated with a bounded kernel can be written as a weighted sum of cross spectrograms the decomposition for several different discrete time kernels are given and the performance of the approximation algorithm is illustrated for different types of signals

the discrete time phase derivative as a definition of discrete instantaneous frequency and its relation to discrete time frequency distributions we compare two definitions of the instantaneous frequency of a discrete signal the two point symmetric phase difference and the discrete time phase derivative the phase derivative definition avoids the pitfalls associated with the two point definition and is equal to the first moment w r t frequency of a properly defined alias free time frequency distribution of the signal which is consistent with the continuous time case

covariant schrodinger operator we analyse the schrodinger operator for a quantum scalar particle in a curved spacetime which is fibred over absolute time and is equipped with given spacelike metric gravitational field and electromagnetic field we approach the schrodinger operator in three independent ways in terms of covariant differentials induced by the quantum connection via a quantum lagrangian and directly by the only requirement of general covariance in particular in the flat case our schrodinger operator coincides with the standard one

conformal regularization of the kepler problem the manifoldm of null rays through the origin of 2 n 1 is diffeomorphic tos1 s n and it is a homogeneous space of so 2 n 1 this group therefore acts ont m which we show to be the generating manifold of the extended phase space of the regularized kepler problem a local canonical chart int m is found such that the restriction to the subbundle of the null nonvanishing covectors is given byp0 h q p 0 whereh q p is the hamiltonian of the kepler problem by means of this construction we get some results that clarify and complete the previous approaches to the problem

on alias free formulations of discrete time cohen s class of distributions the transition of the cohen s 1989 class of distributions from the continuous time case to the discrete time case is not straightforward because of aliasing problems we classify the aliasing problems which occur for joint time frequency representations tfrs into two categories type i and type ii aliasings type i aliasing can be avoided by properly defined discrete time versions of some members of cohen s class in particular properly defined kernels whereas type ii aliasing can be reduced and or eliminated by increasing the sampling rate a type i alias free formulation of the discrete time cohen s class af dtcc which is equivalent to the af gdtft of joeng and williams see ibid vol 40 no 2 p 1084 1992 is then introduced based on the fact that the cohen s class can be expressed as the 2 d fourier transform of the generalized ambiguity function af based on this definition two discretization schemes for kernel functions are presented in both the af domain and the time lag domain and are shown to be equivalent under certain conditions we also do the following 1 we show that a discrete time wigner ville distribution dwvd and discrete time spectrogram dspg are type i alias free and members of af dtcc 2 we use all the available correlation information from a given data sequence by using the woodward af instead of the sussman af 3 we give kernel constraints in the af domain for various distribution properties and 4 we provide a type i and type ii alias free formulation for those distributions whose kernel functions satisfy the finite frequency support constraint

pukanszky s condition and symplectic induction pukanszky s condition is a condition used in obtaining representations from coadjoint orbits in order to obtain more geometric insight in this condition we relate it to symplectic induction it turns out to be equivalent to the condition that the orbit in question is a symplectic subbundle of a modified cotangent bundle

quantum morphing and the jones polynomial we will explore the experimental observation that on the set of knots with bounded crossing number algebraically independent vassiliev invariants become correlated as noticed first by s willerton we will see this through the value distribution of the jones polynomial at roots of unit as the degree of the roots of unit is getting larger the higher order fluctuation is diminishing and a more organized shape will emerge from a rather random value distribution of the jones polynomial we call such a phenomenon quantum morphing evaluations of the jones polynomial at roots of unity play a crucial role for example in the volume conjecture

variables separation and superintegrability of the nine dimensional micz kepler problem the nine dimensional micz kepler problem is of recent interest this is a system describing a charged particle moving in the coulomb field plus the field of a so 8 monopole in a nine dimensional space interestingly this problem is equivalent to a 16 dimensional harmonic oscillator via the hurwitz transformation in the present paper we report on the multiseparability a common property of superintegrable systems and the superintegrability of the problem first we show the solvability of the schrodinger equation of the problem by the variables separation method in different coordinates second based on the so 10 symmetry algebra of the system we construct explicitly a set of seventeen invariant operators which are all in the second order of the momentum components satisfying the condition of superintegrability the found number 17 coincides with the prediction of 2n 1 law of maximal superintegrability order in the case n 9 until now this law is accepted to apply only to scalar hamiltonian eigenvalue equations in n dimensional space therefore our results can be treated as evidence that this definition of superintegrability may also apply to some vector equations such as the schrodinger equation for the nine dimensional micz kepler problem the nine dimensional micz kepler problem is of recent interest this is a system describing a charged particle moving in the coulomb field plus the field of a so 8 monopole in a nine dimensional space interestingly this problem is equivalent to a 16 dimensional harmonic oscillator via the hurwitz transformation in the present paper we report on the multiseparability a common property of superintegrable systems and the superintegrability of the problem first we show the solvability of the schrodinger equation of the problem by the variables separation method in different coordinates second based on the so 10 symmetry algebra of the system we construct explicitly a set of seventeen invariant operators which are all in the second order of the momentum components satisfying the condition of superintegrability the found number 17 coincides with the prediction of 2n 1 law of maximal superintegrability order in the case n 9 until now this law is accepted to apply only to scalar hamiltonia

on the regularization of the kepler problem an intimate relationship between moser s regularization 1 and the ks regularization 3 of the 3 dimensional kepler problem is established explicit formulae linking moser s and the ks transformation are obtained in the case of negative as well as in the case of positive energies as a side result it is shown that the ks transformation owes its existence to the local isomorphism of so 2 4 and su 2 2

a unified definition for the discrete time discrete frequency and discrete time frequency wigner distributions various discrete definitions of the wigner distribution wd for discrete time signals have been proposed in previous works the formulation developed in this paper leads to natural and unified definitions of discrete versions of the wd they are directly related to the continuous and preserve most of its properties the discretization is first considered in the time domain dtwd in the frequency domain dfwd and then in both domains simultaneously dtfwd in each case the aliasing problem is studied and generalized interpolation formulas allowing the reconstruction of the continuous wd are derived the dtfwd is particulary relevant for computer implementation of the wd

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

improving the readability of time frequency and time scale representations by the reassignment method in this paper the use of the reassignment method first applied by kodera gendrin and de villedary 1976 to the spectrogram is generalized to any bilinear time frequency or time scale distribution this method creates a modified version of a representation by moving its values away from where they are computed so as to produce a better localization of the signal components we first propose a new formulation of this method followed by a thorough theoretical study of its characteristics its practical use for a large variety of known time frequency and time scale distributions is then addressed finally some experimental results are reported to demonstrate the performance of this method

nonlinear symmetry algebra of the mic kepler problem on the sphere s3 the quantum mechanical problem of motion in the dual charged coulomb field modified by a centrifugal term mic kepler problem is considered in the three dimensional space of constant positive curvature s3 conserved operators in this problem form a cubic algebra similar to that of the kepler problem on s3 the explicit form of invariants of this algebra shows that its representation associated with the mic kepler problem on s3 is a nondegenerate one the cubic symmetry algebra is used to obtain the energy spectrum of the problem

the generalized mic kepler system this paper deals with the dynamical system that generalizes the mic kepler system it is shown that the schrodinger equation for this generalized mic kepler system can be separated in spherical and parabolic coordinates the spectral problem in spherical and parabolic coordinates is solved

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

reduction quantization and nonunimodular groups it is shown that even in relatively nice cases the naive approach to the quantization of constraints is not correct in general i e the procedure that if f 0 is a classical constraint and f is the associated quantum operator then the quantum constraint is f 0 an explicit procedure for the quantization of constraints in the case of a configuration space with a symmetry group is provided and proven where the reduced configuration space is the orbit space it is not thought that the group acts freely merely that all isotropy subgroups are conjugated to each other

high spectral resolution time frequency distribution kernels a new class of time frequency distribution tfd kernels is introduced members in this class satisfy the desirable tfd properties and simultaneously provide local autocorrelation functions laf that are amenable to high frequency resolution modeling techniques it is shown that members of the proposed class are product kernels fast implementation multiplication free kernels recursive kernels and optimum kernels with respect to autoterm localization

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

discrete time discrete frequency time frequency analysis a formulation of a discrete time discrete frequency wigner distribution for analysis of discrete time periodic signals is given using an approach involving group representation theory this approach is motivated by a well known connection between group theory and the continuous wigner distribution the advantage of this approach is that the resulting discrete distribution satisfies mathematical properties analogous to those satisfied by the continuous distribution after outlining the relationship between group representation theory and time frequency analysis we derive the discrete distribution and exhibit many of its mathematical properties these include time and frequency marginals the weyl correspondence and covariance in particular the interpretation of covariance for the discrete distribution is shown to be different than that for the continuous distribution finally we note some unusual features of this discrete distribution which are a consequence of the group theoretic derivation

the aliasing problem in discrete time wigner distributions there is no straightforward way to proceed from the continuous time wigner distribution to a discrete time version of this time frequency signal representation a previously given definition of such a function turned out to yield a distribution that was periodic with period instead of 2 and this caused aliasing to occur various alternative definitions are considered and compared with respect to aliasing and computational complexity from this comparison it appears that no definition leads to a function that is optimum in all respects this is illustrated by an example

the two eigenvalue problem and density of jones representation of braid groups 1 the two eigenvalue problem 179 2 hecke algebra representations of braid groups 183 3 duality of jones wenzl representations 186 4 closed images of jones wenzl sectors 188 5 distribution of evaluations of jones polynomials 192 6 fibonacci representations 196

reduction in stages and complete quantization of the mic kepler problem the one parameter deformation family of the standard kepler problem known as the mic kepler problem is completely quantized using the explicit momentum mapping of the torus actions on some toric manifolds and some equivariant cohomology theory these manifolds appear as symplectic faces of the system during reduction process

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

image of the braid groups inside the finite temperley lieb algebras we determine the image of the braid groups inside the temperley lieb algebras defined over finite field in the semisimple case and for suitably large but controlable order of the defining quantum parameter we also prove that under natural conditions on this parameter the representations of the hecke algebras over a finite field are unitary for the action of the braid groups

the wigner distribution of noisy signals with adaptive time frequency varying window time frequency representations using the wigner distribution wd may be significantly obscured by the noise in the observations the analysis performed for the wd of discrete time noisy signals shows that this time frequency representation can be optimized by the appropriate choice of the window length however the practical value of this analysis is not significant because the optimization requires knowledge of the bias which depends on the unknown derivatives of the wd a simple adaptive algorithm for the efficient time frequency representation of noisy signals is developed in this paper the algorithm uses only the noisy estimate of the wd and the analytical formula for the variance of this estimate the quality of this adaptive algorithm is close to the one that could be achieved by the algorithm with the optimal window length provided that the wd derivatives were known in advance the proposed algorithm is based on the idea that has been developed in our previous work for the instantaneous frequency if estimation here a direct addressing to the wd itself rather than to the instantaneous frequency resulted in a time and frequency varying window length and showed that the assumption of small noise and bias is no longer necessary a simplified version of the algorithm using only two different window lengths is presented it is shown that the procedure developed for the adaptive window length selection can be generalized for application on multicomponent signals with any distribution from the cohen 1989 1990 1992 class simulations show that the developed algorithms are efficient even for a very low value of the signal to noise ratio

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

unitarity in quantization commutes with reduction let m be a compact kahler manifold equipped with a hamiltonian action of a compact lie group g in this paper we study the geometric quantization of the symplectic quotient m g guillemin and sternberg invent math 67 515 538 1982 have shown under suitable regularity assumptions that there is a natural invertible map between the quantum hilbert space over m g and the g invariant subspace of the quantum hilbert space over m

on the quantization of presymplectic dynamical systems via coisotropic imbeddings we study certain aspects of the problem of quantizing a presymplectic dynamical system such a system is quantized by imbedding the presymplectic manifoldm under consideration into a symplectic manifoldx and then geometrically quantizing the latter it is known that this procedure will yield consistent results provided the imbeddingm x is coisotropic we show that such imbeddings always exist and are locally unique furthermore we investigate the extent to which the resulting quantum dynamics is independent of the choice of coisotropic imbedding that is we examine to what extent the presymplectic phase spacem determines the quantum representation space and the quantization of observables the quantization is carried out within the geometric quantization framework of kostant and souriau

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

geometric quantization and the generalized segal bargmann transform for lie groups of compact type let k be a connected lie group of compact type and let t k be its cotangent bundle this paper considers geometric quantization of t k first using the vertical polarization and then using a natural kahler polarization obtained by identifying t k with the complexified group k the first main result is that the hilbert space obtained by using the kahler polarization is naturally identifiable with the generalized segal bargmann space introduced by the author from a different point of view namely that of heat kernels the second main result is that the pairing map of geometric quantization coincides with the generalized segal bargmann transform introduced by the author this means that the pairing map in this case is a constant multiple of a unitary map for both results it is essential that the half form correction be included when using the kahler polarization

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

radon transformation of time frequency distributions for analysis of multicomponent signals the radon transform of a time frequency distribution produces local areas of signal concentration that facilitate interpretation of multicomponent signals the radon wigner transform can be efficiently implemented with dechirping in the time domain however only half of the possible projections through the time frequency plane can be realized because of aliasing we show here that the frequency dual to dechirping exists so that all of the time frequency plane projections can be calculated efficiently both time and frequency dechirping are shown to warp the time frequency plane rather rotating it producing an angle dependent dilation of the radon wigner projection axis we derive the discrete time equations for both time and frequency dechirping and highlight some practical implementation issues discrete dechirping is shown to correspond to line integration through the extended discrete rather than the discrete wigner ville distribution computationally dechirping is o 2n log 2n instead of o n sup 3 for direct projection and the computation is dominated by the fast fourier transform calculation the noise and cross term suppression of the radon wigner transform are demonstrated by several examples using dechirping and using direct radon wigner transformation

covariant geometric quantization of nonrelativistic time dependent mechanics we provide geometric quantization of the vertical cotangent bundle v q q r equipped with the canonical poisson structure and treated as a momentum phase space of nonrelativistic time dependent mechanics we show that this quantization is equivalent to fiberwise quantization of symplectic fibers of v q r and that the quantum algebra of time dependent mechanics is an instantwise algebra quantization of the classical evolution equation defines a connection on this instantwise algebra and describes quantum evolution in time dependent mechanics as a parallel transport

improved discrete definition of quadratic time frequency distributions computation of a time frequency distribution tfd requires a discrete version of the continuous distribution this discrete tfd dtfd should be free from aliasing and conserve all the important mathematical properties of the continuous distribution existing dtfd definitions however poorly approximate this ideal one popular definition the generalized dtfd gdtfd is alias free but does not retain all the desirable properties from the continuous distribution another definition the so called alias free gdtfd af gdtfd retains most properties yet is not always alias free we propose a new dtfd definition based on the gdtfd that retains all desirable properties and is always alias free

constraints reduction and quantization theorems are proved that establish the unitary equivalence of the extended and reduced phase space quantizations of a constrained classical system with symmetry several examples are presented

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

note on the use of the wigner distribution for time frequency signal analysis it is shown that a correct use of the wigner distribution wd for time frequency signal analysis requires use of the analytic signal this version often referred to as the wigner ville distribution wvd is straightforward to compute does not exhibit any aliasing problem and introduces no frequency artifacts the problems introduced by the use of the wigner distribution with a real signal are clarified

an overview of aliasing errors in discrete time formulations of time frequency representations discrete time time frequency representation tfr algorithms claim to provide alias free approximations to their continuous time tfr counterparts without requiring oversampling of the signal by counterexamples we demonstrate that some of these claims are invalid we give new necessary conditions for reducing aliasing errors in these discrete time tfr algorithms

the zak transform and decimated time frequency distributions in this paper the interrelation between the zak transform and the generalized discrete time frequency distribution gdtfd is examined starting with the discrete zak transform its definition is broadened to include arbitrary windows creating the windowed zak transform wzt the wzt is then combined with the spectrogram it is demonstrated that the spectrogram based upon the wzt called the zak spectrogram zs is a generalization of the standard spectrogram next building upon the idea of the weighted spectrogram the weighted zs is used to produce a new class of gdtfd called the decimated gdtfd d gdtfd the decimated gdtfd is similar to the gdtfd except it trades bandwidth for computational speed and requires significantly less storage in order to be implemented the reduction in discrete bandwidth is from 2 spl pi for the gdtfd to 2 spl pi m for the d gdtfd an important attribute of the dgdtfd is that it requires significantly less storage than the gdtfd the d gdtfd requires only 1 m sup 2 of the storage of the gdtfd an example using the binomial distribution is given to illustrate the connection between the d gdtfd and the gdtfd throughout the paper examples of multirate systems are given which could be used to implement the building blocks of the dgdtfd this makes the d gdtfd practical to implement using slower and less expensive elements

the sp 1 kepler problems let n 2 be a positive integer to each irreducible representation of sp 1 an sp 1 kepler problem in dimension 4n 3 is constructed and analyzed this system is superintegrable and when n 2 it is equivalent to a generalized micz kepler problem in dimension of 5 the dynamical symmetry group of this system is o 4n with the hilbert space of bound states h being the unitary highest weight representation of o 4n with highest weight 1 1 1 which occurs at the rightmost nontrivial reduction point in the enright howe wallach classification diagram for the unitary highest weight modules here is the highest weight of furthermore it is shown that the correspondence h is the theta correspondence for dual pair sp 1 o 4n sp 8n r

a high resolution quadratic time frequency distribution for multicomponent signals analysis the paper introduces a new kernel for the design of a high resolution time frequency distribution tfd we show that this distribution can solve problems that the wigner ville distribution wvd or the spectrogram cannot in particular the proposed distribution can resolve two close signals in the time frequency domain that the two other distributions cannot moreover we show that the proposed distribution is more accurate than the wvd and the spectrogram in the estimation of the instantaneous frequency of a stepped fm signal embedded in additive gaussian noise synthetic and real data collected from real world applications are shown to validate the proposed distribution

finite word length effects in implementation of distributions for time frequency signal analysis this correspondence presents an analysis of the finite register length influence on the accuracy of results obtained by the time frequency distributions tfds in order to measure quality of the obtained results the variance of the proposed model is found signal to quantization noise ratio snr is defined and appropriate expressions are derived floating and fixed point arithmetic are considered with the analysis of discrete random and discrete deterministic signals it is shown that commonly used reduced interference distributions rids exhibit similar performance with respect to the snr we have also derived the expressions establishing the relationship between the number of bits and the required quality of representation which is defined by the snr which may be used for register length design in hardware implementation of time frequency algorithms

the time frequency distributions of nonstationary signals based on a bessel kernel a kernel based on the first kind bessel function of order one is proposed to compute the time frequency distributions of nonstationary signals this kernel can suppress the cross terms of the distribution effectively it is shown that the bessel distribution the time frequency distribution using bessel kernel meets most of the desirable properties with high time frequency resolution a numerical alias free implementation of the distribution is presented examples of applications in time frequency analysis of the heart s sound and doppler blood flow signals are given to show that the bessel distribution can be easily adapted to two very different signals for cardiovascular signal processing by controlling a kernel parameter this distribution can be used to compute the time frequency representations of transient deterministic and random signals the study confirms the potentials of the proposed distribution in nonstationary signal analysis