generalized entropic uncertainty relations a new class of uncertainty relations is derived for pairs of observables in a finite dimensional hilbert space which do not have any common eigenvector this class contains an entropic uncertainty relation which improves a previous result of deutsch and confirms a recent conjecture by kraus some comments are made on the extension of these relations to the case where the hilbert space is infinite dimensional

a new approach to frequency analysis with amplified harmonics summary for stationary time series having a mixed spectrum we introduce a new method for estimating the discrete frequencies based on amplification of the sine waves in the time series and on minimization of least squares regressions on amplified waves under the assumption that the continuous part of the spectrum arises from a stationary autoregressive moving average process we show that the estimators for the discrete frequencies obtained with this method are strongly consistent more precisely as the serial length n tends to infinity their bias almost surely converges to zero with the rate n 3 2 log n 5 6 2 we also establish a central limit theorem which shows that these estimators have exactly the same asymptotic variance as that given by whittle s method which is known to yield the most accurate frequency estimates however the new technique has two advantages over whittle s method first it yields an algorithm that is both robust against improper starting estimates and computationally efficient secondly it gives rise to a simple tool based on its amplified harmonics for identifying the number of sine waves in a time series a simulation study is also reported that gives statistical results on the properties of the new method on simulated series as well as comparative results with whittle s method

rank deficient submatrices of fourier matrices we consider the maximal rank deficient submatrices of fourier matrices with order a power of a prime number we do this by considering a hierarchical subdivision of these matrices into low rank blocks we also explore some connections with the fast fourier transform fft and with an uncertainty principle for fourier transforms over finite abelian groups

threshold effects in parameter estimation from compressed data in this paper we investigate threshold effects associated with the swapping of signal and noise subspaces in estimating signal parameters from compressed noisy data the term threshold effect refers to a sharp departure of mean squared error from the cramer rao bound when the signal to noise ratio falls below a threshold snr in many cases the threshold effect is caused by a subspace swap event when the measured data or its sample covariance is better approximated by a subset of components of an orthogonal subspace than by the components of a signal subspace we derive analytical lower bounds on the probability of a subspace swap in compressively measured noisy data in two canonical models a first order model and a second order model in the first order model the parameters to be estimated modulate the mean of a complex multivariate normal set of measurements in the second order model the parameters modulate the covariance of complex multivariate measurements in both cases the probability bounds are tail probabilities of f distributions and they apply to any linear compression scheme these lower bounds guide our understanding of threshold effects and performance breakdowns for parameter estimation using compression in particular they can be used to quantify the increase in threshold snr as a function of a compression ratio c we demonstrate numerically that this increase in threshold snr is roughly 10 log 10 c db which is consistent with the performance loss that one would expect when measurements in gaussian noise are compressed by a factor c

uncertainty principles for kac algebras in this paper we introduce the notation of bi shift of biprojections in subfactor theory to unimodular kac algebras we characterize the minimizers of the hirschman beckner uncertainty principle and the donoho stark uncertainty principle for unimodular kac algebras with biprojections and prove hardy s uncertainty principle in terms of the minimizers

analysis in a finite time frequency plane we revise our conjecture concerning signals realizing the lower bound on compaction in the phase plane to include a forgotten case of periodization and indicate that this case can be used to develop an orthonormal basis for data lengths n that are the square of an integer

on classes of non gaussian asymptotic minimizers in entropic uncertainty principles abstract in this paper we revisit the bialynicki birula and mycielski uncertainty principle and its cases of equality this shannon entropic version of the well known heisenberg uncertainty principle can be used when dealing with variables that admit no variance in this paper we extend this uncertainty principle to renyi entropies we recall that in both shannon and renyi cases and for a given dimension n the only case of equality occurs for gaussian random vectors we show that as n grows however the bound is also asymptotically attained in the cases of n dimensional student t and student r distributions a complete analytical study is performed in a special case of a student t distribution we also show numerically that this effect exists for the particular case of a n dimensional cauchy variable whatever the renyi entropy considered extending the results of abe and illustrating the analytical asymptotic study of the student t case in the student r case we show numerically that the same behavior occurs for uniformly distributed vectors these particular cases and other ones investigated in this paper are interesting since they show that this asymptotic behavior cannot be considered as a gaussianization of the vector when the dimension increases

dft spectrum filtering a e kahveci and e l hall see ieee trans comput vol c 23 no 9 p 976 81 sept 1974 introduced the concept of filtering discrete fourier transform dft spectra in the walsh sequency domain this is accomplished by finding a real and block diagonal walsh filter matrix g sub w in the walsh domain that performs the sample filtering operation as the prototype complex diagonal fourier filter matrix g sub f in the fourier domain the present authors provide additional information on the structure of g sub w and generalize some results by c j zarowski and m yunik see ibid vol assp 33 p 1246 52 oct 1985 they consider a more general class of transforms the t transforms and the structure of the resulting t transform filter matrices g sub t examples of t besides t w are considered such as the harr transform and fourth order chrestenson transform the implementation of the presented dft spectrum filtering techniques using linear systolic arrays is briefly considered

hardy and uncertainty inequalities on stratified lie groups abstract we prove various hardy type and uncertainty inequalities on a stratified lie group g in particular we show that the operators t f l 2 f where is a homogeneous norm 0 q p and l is the sub laplacian are bounded on the lebesgue space l p g as consequences we estimate the norms of these operators sufficiently precisely to be able to differentiate and prove a logarithmic uncertainty inequality we also deduce a general version of the heisenberg pauli weyl inequality relating the l p norm of a function f to the l q norm of f and the l r norm of l 2 f

singular pseudodifferential operators symmetrizers and oscillatory multidimensional shocks abstract we introduce a calculus of singular pseudodifferential operators spos depending on wavelength e and use them to solve three different types of singular quasilinear hyperbolic systems such systems arise in nonlinear geometric optics and also for example in the study of incompressible limits and of nonlinear wave equations with small nonlinear terms or small data the spos act in both slow and fast variables and are singular not only because their symbols have finite regularity and depend on 1 e but also because their derivatives fail to decay in the usual way in the dual variables there is a necessarily crude calculus with large parameter e g residual operators are just bounded on l2 but the calculus admits the proof of garding inequalities and enables us to symmetrize and sometimes even diagonalize the singular systems being considered by microlocalizing simultaneously in both slow and fast variables the paper culminates in a proof of the existence of oscillatory multidimensional shocks on a fixed time interval independent of the wavelength e as e 0 the use of spos allows us to eliminate the small divisor assumptions made in earlier work and also to construct more general oscillatory solutions in which elliptic boundary layers are present on one or both sides of the shock

reply to comments on resolution in time frequency the authors reply that the first sentence of the conjecture numbered 5 as stated by dr knockaert see ibid vol 48 no 12 p 3585 86 2000 in his comments was proven first by dembo et al 1991 this they discovered in march 1999 furthermore dr m doroslovacki of george washington university pointed out in march 1999 that a time sequence used by donoho et al 1989 was a counter example to conjecture 5 in that paper they correspondingly revised their conjecture to the improved conjecture presented

spectrum estimation by wavelet thresholding of multitaper estimators current methods for power spectrum estimation by wavelet thresholding use the empirical wavelet coefficients derived from the log periodogram unfortunately the periodogram is a very poor estimate when the true spectrum has a high dynamic range and or is rapidly varying in addition because the distribution of the log periodogram is markedly non gaussian special wavelet dependent thresholding schemes are needed these difficulties can be bypassed by starting with a multitaper spectrum estimator the logarithm of this estimator is close to gaussian distributed if a moderate number spl ges 5 of tapers are used in contrast to the log periodogram log multitaper estimates are not approximately pairwise uncorrelated at the fourier frequencies but the form of the correlation can be accurately and simply approximated for scale independent thresholding the correlation acts in accordance with the wavelet shrinkage paradigm to suppress small scale noise spikes while leaving informative coarse scale coefficients relatively unattenuated this simple approach to spectrum estimation is demonstrated to work very well in practice additionally the progression of the variance of wavelet coefficients with scale can be accurately calculated allowing the use of scale dependent thresholds this more involved approach also works well in practice but is not uniformly preferable to the scale independent approach

nonlinear geometric optics for reflecting uniformly stable pulses abstract we provide a justification with rigorous error estimates showing that the leading term in weakly nonlinear geometric optics expansions of highly oscillatory reflecting pulses is close to the uniquely determined exact solution for small wavelengths e pulses reflecting off fixed noncharacteristic boundaries are considered under the assumption that the underlying boundary problem is uniformly spectrally stable in the sense of kreiss there are two respects in which these results make rigorous the formal treatment of pulses in majda and artola 1988 16 and hunter majda and rosales 1986 10 first we give a rigorous construction of leading pulse profiles in problems where pulses traveling with many distinct group velocities are unavoidably present and second we provide a rigorous error analysis which yields a rate of convergence of approximate to exact solutions as e 0 unlike wavetrains interacting pulses do not produce resonances that affect leading order profiles however our error analysis shows the importance of estimating pulse interactions in the construction and estimation of correctors our results apply to a general class of systems that includes quasilinear problems like the compressible euler equations moreover the same methods yield a stability result for uniformly stable euler shocks perturbed by highly oscillatory pulses

the mach stem equation and amplification in strongly nonlinear geometric optics we study highly oscillating solutions to a class of weakly well posed hyperbolic initial boundary value problems weak well posedness is associated with an amplification phenomenon of oscillating waves on the boundary in the previous works cgw14 cw14 we have rigorously justified a weakly nonlinear regime for semilinear problems in that case the forcing term on the boundary has amplitude o e 2 and oscillates at a frequency o 1 e the corresponding exact solution which has been shown to exist on a time interval that is independent of e 0 1 has amplitude o e in this paper we deal with the exact same scaling namely o e 2 forcing term on the boundary and o e solution for quasilinear problems in analogy with cgm03 this corresponds to a strongly nonlinear regime and our main result proves solvability for the corresponding wkb cascade of equations which yields existence of approximate solutions on a time interval that is independent of e 0 1 existence of exact solutions close to approximate ones is a stability issue which as shown in cgm03 highly depends on the hyperbolic system and on the boundary conditions we do not address that question here this work encompasses previous formal expansions in the case of weakly stable shock waves mr83 and two dimensional compressible vortex sheets am87 in particular we prove well posedness for the leading amplitude equation the mach stem equation of mr83 and generalize its derivation to a large class of hyperbolic boundary value problems and to periodic forcing terms the latter case is solved under a crucial nonresonant assumption and a small divisor condition

spectral filtering using the fast walsh transform this paper describes a method of doing spectral filtering using the fast walsh transform fwt rather than the fast fourier transform fft rather than using the walsh transform to find fourier coefficients which can then be filtered by ordinary means as was done in 2 we find a new filter function expressed as a matrix that does the same filtering operation in the walsh domain as the filter function matrix in the fourier domain this new filter matrix called the walsh gain matrix g w is block diagonal and real while the fourier gain matrix g f is complex diagonal the block diagonal structure of g w and a condition that causes g w to be real are proven an off line method for finding g w given g f is presented using the block diagonal structure of g w it is proven that spectral filtering via fwt requires fewer multiplications than spectral filtering via fft for n leq 64 where n is the length of the sequence of samples of the input signal n is a power of 2 a special condition on g f gives a g w such that spectral filtering via fwt becomes better in terms of multiplications than spectral filtering via fft for n leq 128

detection estimation of very close emitters performance breakdown ambiguity and general statistical analysis of maximum likelihood estimation we reexamine the well known problem of threshold behavior or performance breakdown in the detection estimation of very closely spaced emitters in this extreme regime we analyze the performance for maximum likelihood estimation mle of directions of arrival doa for two close gaussian sources over the range of sample volumes and signal to noise ratios snrs where the correct number of sources is reliably estimated by information theoretic criteria itc but where one of the doa estimates is severely erroneous outlier we show that random matrix theory rmt applied to the evaluation of theoretical mle performance gives a relatively simple and accurate analytical description of the threshold behavior of mle and itc in particular the introduced single cluster criterion provides accurate ambiguity bounds for the outliers

an entropy based uncertainty principle for a locally compact abelian group abstract we classify all functions on a locally compact abelian group giving equality in an entropy inequality generalizing the heisenberg uncertainty principle in particular for functions on a real line we proof a conjecture of hirschman published in 1957

wavelet thresholding techniques for power spectrum estimation estimation of the power spectrum s f of a stationary random process can be viewed as a nonparametric statistical estimation problem we introduce a nonparametric approach based on a wavelet representation for the logarithm of the unknown s f this approach offers the ability to capture statistically significant components of ln s f at different resolution levels and guarantees nonnegativity of the spectrum estimator the spectrum estimation problem is set up as a problem of inference on the wavelet coefficients of a signal corrupted by additive non gaussian noise we propose a wavelet thresholding technique to solve this problem under specified noise resolution tradeoffs and show that the wavelet coefficients of the additive noise may be treated as independent random variables the thresholds are computed using a saddle point approximation to the distribution of the noise coefficients

entropy based uncertainty measures for l sup 2 spl ropf sup n spl lscr sup 2 spl zopf and spl lscr sup 2 spl zopf n spl zopf with a hirschman optimal transform for spl lscr sup 2 spl zopf n spl zopf the traditional heisenberg weyl measure quantifies the joint localization uncertainty or concentration of a signal in the phase plane based on a product of energies expressed as signal variances in time and in frequency in the image processing literature the term compactness also has been used to refer to this same notion of joint localization in the sense of a signal representation that is efficient simultaneously in time or space and frequency in this paper we consider hirschman uncertainty principles based not on energies and variances directly but rather on entropies computed with respect to normalized energy densities in time and frequency unlike the heisenberg weyl measure this entropic hirschman notion of joint uncertainty extends naturally from the case of infinitely supported continuous time signals to the cases of both finitely and infinitely supported discrete time signals for the first time we consider these three cases together and study them relative to one another in the case of infinitely supported continuous time signals we find that consistent with the energy based heisenberg principle the optimal time frequency concentration with respect to the hirschman uncertainty principle is realized by translated and modulated gaussian functions in the two discrete cases however the entropy based measure yields optimizers that may be generated by applying compositions of operators to the kronecker delta study of the discrete cases yields two interesting results first in the finitely supported case the hirschman optimal functions coincide with the so called picket fence functions that are also optimal with respect to the joint time frequency counting measure of donoho and stark second the hirschman optimal functions in the infinitely supported case can be reconciled with continuous time gaussians through a certain limiting process while a different limiting process can be used to reconcile the finitely and infinitely supported discrete cases there does not appear to be a straightforward limiting process that unifies all three cases the optimizers from the finitely supported discrete case are decidedly non gaussian we perform a very simple experiment that indicates that the hirschman optimal transform hot is superior to the discrete fourier transform dft and discrete cosine transform dct in terms of its ability to separate or resolve two limiting cases of localization in frequency viz pure tones and additive white noise we believe that these differences arise from the use of entropy rather than energy as an optimality criterion and are intimately related to the apparent incongruence between the infinitely supported continuous time case and the finitely supported discrete time case

nonparametric methods for deconvolving multiperiodic functions summary multiperiodic functions or functions that can be represented as finite additive mixtures of periodic functions arise in problems related to stellar radiation there they represent the overall variation in radiation intensity with time the individual periodic components generally correspond to different sources of radiation and have intrinsic physical meaning provided that they can be deconvolved from the mixture we suggest a combination of kernel and orthogonal series methods for performing the deconvolution and we show how to estimate both the sequence of periods and the periodic functions themselves we pay particular attention to the issue of identifiability in a nonparametric sense of the components this aspect of the problem is shown to exhibit particularly unusual features and to have connections to number theory the matter of rates of convergence of estimators also has links there although we show that the rate of convergence problem can be treated from a relatively conventional viewpoint by considering an appropriate prior distribution for the periods

threshold performance analysis of maximum likelihood doa estimation the purpose of this paper is to investigate the nonasymptotic behavior of the maximum likelihood method in the context of lacunar array processing we derive an analytical expression of the maximum likelihood mean square error which is now valid for all signal to noise ratio snr ranges we also provide an analysis of the snr threshold phenomena which allows us to derive closed form formulas for the threshold values computer simulations confirm the validity of the theoretical investigations

a fast efficient technique for the estimation of frequency interpretation and generalisation in quinn fernandes 1991 an iterative technique was presented for the estimation of the frequency of a sinusoid in the presence of noise the technique is more robust to poor initial estimates than is the periodogram maximiser a generalisation to the case of multiple sinusoids is presented in this paper and it is shown that the original estimator may be interpreted as the maximiser of a smoothed version of the periodogram

quadratic forms over c t1 t2 abstract we show that dimension 4 quadratic forms over c t1 t2 ti transcendental do not satisfy what might be naturally taken as a strong hasse principle though a weak hasse principle exists we reduced solvability to a question of determining the quadratic reciprocity group for the discriminant d the classes modulo d of elements x such that d is a quadratic residue modulo x we determine this group in terms of the class group of a function field we prove that the weak hasse principle does not hold for forms over c t1 d therefore witt and brauer groups of 2 dimensional function fields over c do not restrict monomorphically to their localizations

threshold behavior of the maximum likelihood estimator of frequency we present accurate simplifications of the rife and boorstyn 1974 performance equations for the maximum likelihood estimator of frequency the simplicity of the result allows quick calculation of the onset of threshold as a function of sample size and signal to noise ratio snr the accuracy of the new expression is demonstrated via simulation

uncertainty principles for locally compact quantum groups abstract in this paper we prove the donoho stark uncertainty principle for locally compact quantum groups and characterize the minimizer which are bi shifts of group like projections we also prove the hirschman beckner uncertainty principle for compact quantum groups and discrete quantum groups furthermore we show hardy s uncertainty principle for locally compact quantum groups in terms of bi shifts of group like projections

diophantine undecidability of c t1 t2 we give a short proof that the diophantine theory of the complex function field c t 1 t 2 where t i are algebraically independent over the complex numbers in the language t 1 t 2 0 1 is undecidable we construct a model of z represented by isogenies of an elliptic curve

a fast efficient technique for the estimation of frequency summary a technique is presented for the estimation of the frequency of a sinusoid in the presence of noise the technique is based on fitting arma 2 2 models iteratively in a special way the estimator is shown to be strongly consistent and as efficient as the least squares estimator of the frequency or the periodogram maximizer a simple accelerated version of the technique is shown to converge in a small number of iterations the number depending on the accuracy of the initiator of the procedure the results of a number of simulations are reported

music g music and maximum likelihood performance breakdown direction of arrival estimation performance of music and maximum likelihood estimation in the so called ldquothresholdrdquo area is analyzed by means of general statistical analysis gsa also known as random matrix theory both analytic predictions and direct monte carlo simulations demonstrate that the well known music specific ldquoperformance breakdownrdquo is associated with the loss of resolution capability in the music pseudo spectrum while the sample signal subspace is still reliably separated from the actual noise subspace significant distinctions between music g music specific and mle intrinsic causes of ldquoperformance breakdown rdquo as well as the role of ldquosubspace swaprdquo phenomena are specified analytically and supported by simulation

the uncertainty principle on groups the classical uncertainty principle asserts that both a function and its fourier transform cannot be largely concentrated on intervals of small measure donoho and stark siam j app math 49 1989 pp 906 931 have shown recently that both cannot be largely concentrated on any sets of small measure in the case of functions on the line or functions on finite cyclic groups and with concentrations measured in l 2 the purpose of this note is to extend these results to functions on sigma finite locally compact abelian groups with concentrations measured in l p 1 leqq p leqq 2 the first uncertainty principle on groups due to matolcsi and szucs d r acad sci paris 277 1973 pp 841 843 deals with full rather than large concentration asserting that if a function and its fourier transform are supported by sets t and w then the product of the haar measures of t and w must be at least 1 for the case of full concentration in r n benedicks j math anal appl 106 1985 pp 1

entropic uncertainty bound for open pointer based simultaneous measurements of conjugate observables we discuss the information entropy for a general open pointer based simultaneous measurement and show how it is bound from below this entropic uncertainty bound is a direct consequence of the structure of the entropy and can be obtained from the formal solution of the measurement dynamics furthermore the structural properties of the entropy allow us to give an intuitive interpretation of the noisy influence of the pointers and the environmental heat bath on the measurement results

nonlinear geometric optics for short pulses abstract this paper studies the propagation of pulse like solutions of semilinear hyperbolic equations in the limit of short wavelength the pulses are located at a wavefront 0 where satisfies the eikonal equation and d lies on a regular sheet of the characteristic variety the approximate solutions are u e approx u t x t x e where u t x r is a smooth function with compact support in r when u satisfies a familiar nonlinear transport equation from geometric optics it is proved that there is a family of exact solutions u e exact such that u e approx has relative error o e as e 0 while the transport equation is familiar the construction of correctors and justification of the approximation are different from the analogous problems concerning the propagation of wave trains with slowly varying envelope

gaussian kernels have only gaussian maximizers a gaussian integral kernel g x y on rn x rn is the exponential of a quadratic form in x and y the fourier transform kernel is an example the problem addressed here is to find the sharp bound of g as an operator from l p rp to l q rn and to prove that the l p rn functions that saturate the bound are necessarily gaussians this is accomplished generally for 1 q in some special cases besides greatly extending previous results in this area the proof technique is also essentially different from earlier ones a corollary of these results is a fully multidimensional multilinear generalization of young s inequality

problems equivalent to rational diophantine solvability muthemutics research group alubamu state university montgomery alabama 36195 communicated by walter feit received august 5 1987 under the hypothesis that rational diophantine equations are unsolvable we classify a number of other problems as solvable or unsolvable such as gv c h w for g e g h e h algebraic groups v w subspaces existence of monomorphisms and epimorphisms of modules and algebras existence of submodules of various types

aircraft flight parameter estimation based on passive acoustic techniques using the polynomial wigner ville distribution the acoustic signal from an overflying aircraft as heard by a stationary observer is used to estimate an aircraft s constant height ground speed range and acoustic frequency central to the success of this flight parameter estimation scheme is the need for an accurate estimate of the instantaneous frequency of the observed acoustic signal in this paper the polynomial wigner ville distribution is used in this application as the instantaneous frequency estimator its performance and the issue of the optimal time domain window length are addressed

using the periodogram to estimate period in nonparametric regression summary properties of the periodogram are seldom studied in the setting of nonparametric regression although that is the context in which the periodogram is widely applied in astronomy there it is a competitor with more recent least squares methods the periodogram has the advantage of providing significant graphical insight into statistical and numerical aspects of the problem however as we show in the present paper it also has drawbacks the estimator that it produces has somewhat higher variance than its least squares counterpart and a periodogram based approach is more prone to suffer difficulties caused by periodicity of the observation schedule while the periodogram remains a very attractive tool the information provided in this paper allows users to assess more readily the extent to which it can be relied upon in a nonparametric setting this aspect of our contributions is discussed theoretically and illustrated by numerical studies involving a real dataset

the optimal transform for the discrete hirschman uncertainty principle we determine all signals giving equality for the discrete hirschman uncertainty principle we single out the case where the entropies of the time signal and its fourier transform are equal these signals up to scalar multiples form an orthonormal basis giving an orthogonal transform that optimally packs a finite duration discrete time signal the transform may be computed via a fast algorithm due to its relationship to the discrete fourier transform

on parameter estimates of the lossy wave equation the cramer rao bound crb and maximum likelihood estimates in gaussian noise for the parameters of a sampled signal that satisfies the lossy wave equation such as the voltage sampled along a transmission line are derived results for both equally and unequally spaced samples are given it is shown that for certain values of the wave number frequency the estimates are sensitive to the amount of the ldquoreflected waverdquo in the signal while for other values of the wave number the estimates are insensitive it is also shown that for certain special cases it is possible to improve on standard frequency estimation up to 3 6 db by reflecting some of the noise free signal energy into a backward propagating wave and applying the maximum likelihood estimator mle to the new measured signal

resolution in time frequency we introduce a new measure h sub p that is related to the heisenberg uncertainty principle the measure predicts the compactness of discrete time signal descriptions in the sample frequency phase plane we conjecture a lower limit on the compaction in the phase plane and show that discretized gaussians may not provide the most compact basis

effects of sampling and quantization on single tone frequency estimation the effects of sampling and quantization on frequency estimation for a single sinusoid are investigated the cramer rao bound for 1 bit quantization is derived and compared with the limit of infinite quantization it is found that 1 bit quantization gives a slightly worse performance however with a dramatic increase of variance at certain frequencies this can be avoided by using four times oversampling the effect of sampling when using nonideal antialiasing lowpass filters is therefore investigated through derivation of the cramer rao lower bounds finally fast estimators for 1 bit quantization in particular correlation based estimators are derived and their performance is investigated the paper is concluded with simulation results for four times oversampled 1 bit quantization