frequency estimation for closely spaced sinsoids simple approximations to the cramer rao lower bound simple approximate formulas for the evaluation of the cramer ran lower bound crlb of unbiased frequency estimates for the deterministic and stochastic cases of multiple snapshots of two closely spaced cisoids in complex noise are introduced these are based on general expressions recently published by stoica and nehorai also explored along the same lines is the case of a single snapshot com prising either one real sinusoid in real noise or two cisoids in complex noise these formulas though simple are accurate to within a few percent where the number of data points per snapshot is greater than about 6 it is felt that they add insight into the behavior of the crlb and cover many of the situations commonly cited in the literature

analysis of subspace fitting and ml techniques for parameter estimation from sensor array data it is shown that the multidimensional signal subspace method termed weighted subspace fitting wsf is asymptotically efficient this results in a novel compact matrix expression for the cramer rao bound crb on the estimation error variance the asymptotic analysis of the maximum likelihood ml and wsf methods is extended to deterministic emitter signals the asymptotic properties of the estimates for this case are shown to be identical to the gaussian emitter signal case i e independent of the actual signal waveforms conclusions concerning the modeling aspect of the sensor array problem are drawn

concentrated cramer rao bound expressions we present a method to simplify the analytical computation of the cramer rao bound the method circumvents bound calculations for so called nuisance parameters under mild regularity conditions the technique which replaces expectations with almost sure limits can significantly lower the analytical complexity as compared to traditional methods the dimension of a matrix that requires computation and inversion is reduced to the length of the parameter vector of interest we give applications to random variables having densities in the exponential family for normal distributions the resulting expressions take on particularly simple closed forms

detection of signals by information theoretic criteria a new approach is presented to the problem of detecting the number of signals in a multichannel time series based on the application of the information theoretic criteria for model selection introduced by akaike aic and by schwartz and rissanen mdl unlike the conventional hypothesis testing based approach the new approach does not requite any subjective threshold settings the number of signals is obtained merely by minimizing the aic or the mdl criteria simulation results that illustrate the performance of the new method for the detection of the number of signals received by a sensor array are presented

nonmatrix cramer rao bound expressions for high resolution frequency estimators analytical expressions are derived for the cramer rao cr lower bound on the variance of frequency estimates for the two signal time series data models consisting of either one real sinusoid or two complex sinusoids in white gaussian noise the expressions give the bound in terms of the signal to noise ratio snr the number n of data samples and a function dependent on the frequency separation and the initial phase difference between the two signal components of each model the bounds are examined as the phase difference is varied and the largest and smallest bound expressions and the corresponding critical values of the phase difference are obtained the exact expressions are analyzed for the case of small frequency separations spl delta spl omega it is found that the largest bound is proportional to n spl middot spl delta spl omega sup 4 n sup 3 spl middot snr and that the smallest bound is proportional to n spl middot spl delta spl omega sup 2 n sup 3 spl middot snr for small spl delta spl omega examples indicate that the small spl delta spl omega results closely approximate the exact ones whenever the frequency separation is smaller than the fourier resolution limit based on the asymptotic results it is found that the threshold snr at which an unbiased estimator can resolve the two signal frequencies is at least proportional to n spl middot spl delta spl omega sup 6 n for the worst phase difference case and to n spl middot spl delta spl omega sup 4 n for the best phase difference case for small spl delta spl omega the results are applicable to the general case of sampling where the samples are taken at arbitrary instants

integrality of a ratio of petersson norms and level lowering congruences we prove integrality of the ratiof f g g outside an explicit finite set of primes where g is an arithmetically normalized holomorphic newform on a shimura curve f is a normalized hecke eigenform on gl 2 with the same hecke eigenvalues as g and denotes the petersson inner product the primes dividing this ratio are shown to be closely related to certain level lowering con gruences satisfied by f and to the central values of a family of rankin selberg l functions finally we give two applications the first to proving the integral ity of a certain triple product l value and the second to the computation of the faltings height of jacobians of shimura curves

doa estimation of temporally and spatially correlated narrowband noncircular sources in spatially correlated white noise the main contribution of our work consists in developing for the first time a method of estimating the direction of arrival doa parameters assuming noncircular and temporally and spatially correlated signals this new approach based on a significant enhancement of the two sided instrumental variable signal subspace fitting iv ssf method outperforms its classical version in terms of lower bias and error variance moreover it will be shown that our new method is statistically more efficient than the mode method especially in the case of partly and fully coherent signals where only the extended and the classical two sided iv ssf methods are applicable we also derive an explicit expression for the stochastic cramer rao bound crb of the doa estimates from temporally and spatially correlated signals generated from noncircular sources the new crb is compared to those of circular temporally correlated and noncircular independent and identically distributed signals to show that the crb obtained assuming both noncircular sources and temporally correlated signals is lower than the crbs derived considering only one of these two assumptions this illustrates the potential gain that both noncircularity and temporal correlation provide when considered together it will also be proven that the difference between the three crbs increases with the number of snapshots however as the signal to noise ratio snr increases the crbs merge together and decrease linearly moreover at low snr values it will be shown that temporal correlation is more informative about the unknown doa parameters than noncircularity finally the crb derived assuming noncircular and temporally correlated signals depends on the noncircularity rate the circularity phase separation and the doa separation

simple approximations to the cramer rao lower bound on direction of arrival for closely spaced sources it is demonstrated that the cramer rao lower bound crlb on the directions of arrival doas of two closely spaced uncorrelated plane wave sources impinging on a uniform line array can be approximated by simple algebraic formulas which provide insight into the behavior of the bound with respect to signal to noise ratios snrs source separation etc these formulas are sufficiently accurate for general use with the number of sensors between about 6 and 60 for source separations down to about 0 1 of a classical beamwidth and for snrs from 30 to 30 db

selmer groups of quadratic twists of elliptic curves 1 1 e y a1xy a3y x 3 a2x 2 a4x a6 where a1 a2 a3 a4 a6 z let n e denote the conductor of e j e the j invariant of e and l e s n 1 a n n s the hasse weil l function of e if e is modular then let fe z n 1 ae n q n s2 n e 1 be the associated weight 2 cusp form here 1 denotes the trivial dirichlet character throughout d will denote a square free integer and d shall denote the kronecker character for the field q d let h d denote the order of cl d the ideal class group of q d if l is prime then define h d l by

exact maximum likelihood parameter estimation of superimposed exponential signals in noise a unified framework for the exact maximum likelihood estimation of the parameters of superimposed exponential signals in noise encompassing both the time series and the array problems is presented an exact expression for the ml criterion is derived in terms of the linear prediction polynomial of the signal and an iterative algorithm for the maximization of this criterion is presented the algorithm is equally applicable in the case of signal coherence in the array problem simulation shows the estimator to be capable of providing more accurate frequency estimates than currently existing techniques the algorithm is similar to those independently derived by kumaresan et al in addition to its practical value the present formulation is used to interpret previous methods such as prony s pisarenko s and modifications thereof

cramer rao bound for doa estimators under the partial relaxation framework derivation and comparison a class of computationally efficient doa estimators under the partial relaxation pr framework has recently been proposed conceptually different from conventional doa estimation methods in the literature the estimators under the pr framework rely on the non complete relaxation of the array manifold while performing a spectral search in the field of view this particular type of relaxation essentially implies a modified signal model with partial information loss due to the relaxation the information loss and its impact on the doa estimation performance have not yet been analytically quantified in the literature in this paper the information loss induced by the relaxation of the array manifold is investigated through the cramer rao bound crb the closed form expression of the crb for doa estimation under the pr model on the one hand provides insight on the information loss in the asymptotic region where the number of snapshots tends to infinity on the other hand the proposed crb characterizes the lower bound for the doa estimation performance of all pr estimators we prove that under the assumptions of gaussian source signal and noise the crb of the pr signal model is lower bounded by the conventional stochastic crb we also prove that the previously proposed weighted subspace fitting estimator under the pr framework asymptotically achieves the crb of the pr signal model furthermore it is shown that the asymptotic mean squared errors of all weighted subspace fitting estimators under the pr framework for any positive definite weighting matrix are identical

statistical resolution limits and the complexified crame spl acute r rao bound array resolution limits and accuracy bounds on the multitude of signal parameters e g azimuth elevation doppler range cross range depth frequency chirp polarization amplitude phase etc estimated by array processing algorithms are essential tools in the evaluation of system performance the case in which the complex amplitudes of the signals are unknown is of particular practical interest a computationally efficient formulation of these bounds from the perspective of derivations and analysis is presented for the case of deterministic and unknown signal amplitudes a new derivation is given using the unknown complex signal parameters and their complex conjugates the new formula is readily applicable to obtaining either symbolic or numerical solutions to estimation bounds for a very wide class of problems encountered in adaptive sensor array processing this formula is shown to yield several of the standard crame spl acute r rao results for array processing along with new results of fundamental interest specifically a new closed form expression for the statistical resolution limit of an aperture for any asymptotically unbiased superresolution algorithm e g music esprit is provided the statistical resolution limit is defined as the source separation that equals its own crame spl acute r rao bound providing an algorithm independent bound on the resolution of any high resolution method it is shown that the statistical resolution limit of an array or coherent integration window is about 1 2 spl middot snr sup 1 4 relative to the fourier resolution limit of 2 spl pi n radians large number n of array elements that is the highest achievable resolution is proportional to the reciprocal of the fourth root of the signal to noise ratio snr in contrast to the square root snr sup 1 2 dependence of standard accuracy bounds these theoretical results are consistent with previously published bounds for specific superresolution algorithms derived by other methods it is also shown that the potential resolution improvement obtained by separating two collinear arrays synthetic ultra wideband each with a fixed aperture b wavelengths by m wavelengths assumed large is approximately m b sup 1 2 in contrast to the resolution improvement of m b for a full aperture exact closed form results for these problems with their asymptotic approximations are presented

an approximate expression for the cramer rao bound on doa estimates of closely spaced sources in broadband line array beamforming it is demonstrated that the cramer rao lower bound on doa estimation with two uncorrelated but closely spaced wideband sources is well approximated by a straightforward modification of a previously published extremely simple non matrix narrowband formula the approximation is useful in the context of a uniform line array with constant snr across the band for a number of sensors from about 6 to 60 snrs from 30 db to 30 db source separations down to about 0 1 beamwidths and bandwidths approaching an octave this covers many of the situations in the array processing literature variants of the basic formula are also discussed

central critical values of modular l functions and coefficients of half integral weight modular forms modulo l if f z is a newform of weight 2 and d is a fundamental discriminant then let l f d s be the usual twisted l series we study the algebraic parts of the central critical values of these twisted l series modulo primes we show that if there are two d subject to some local conditions for which the algebraic part of l f d is not 0 mod l then there are infinitely many such d these results depend on precise nonvanishing results for the fourier coefficients of half integral weight modular forms modulo l which are of independent interest

further simple approximations to the cramer rao lower bound on frequency estimates for closely spaced sinusoids it is demonstrated that the cramer rao lower bound on frequency estimates for a data record containing two closely spaced cisoids in complex white gaussian noise can be approximated by an extremely simple nonmatrix expression it extends earlier work by explicitly retaining the difference in initial phases as a parameter of interest the approximation to the bound is shown to have a root mean square error of about 10 with occasional peak errors of about spl plusmn 25 over a wide range of data lengths and for frequency separations down to about one tenth of the rayleigh resolution limit further it is demonstrated that the same basic form of the approximation handles the related cases of a frequency estimation of a single real sinusoid in real noise and b frequency estimation for a closely spaced pair of real sinusoids in real noise

the risk unbiased cramer rao bound for non bayesian multivariate parameter estimation how accurately can one estimate a deterministic parameter subject to other unknown deterministic model nuisance parameters the most popular answer to this question is given by the cramer rao bound crb the main assumption behind the derivation of the crb is local unbiased estimation of all model parameters the foundations of this paper rely on doubting this assumption generally in multivariate parameter estimation each parameter in its turn can be treated as a single parameter of interest whereas the other model parameters are treated as nuisance as their misknowledge interferes with the estimation of the parameter of interest this approach is utilized in this paper to provide a fresh look at deterministic parameter estimation a new cramer rao cr type bound is derived without assuming unbiased estimation of the nuisance parameters rather than that we apply lehmann s concept of unbiasedness for a risk that measures the distance between the estimator and the locally best unbiased estimator which assumes perfect knowledge of the nuisance parameters the proposed risk unbiased crb rucrb is proven to be asymptotically attainable by the maximum likelihood estimator while being tighter than the conventional crb furthermore simulations verify the asymptotic achievability of the rucrb by the maximum likelihood estimator for an array processing problem

an algorithm for pole zero modeling and spectral analysis an explicit connection between fitting exponential models and pole zero models to observed data is made the fitting problem is formulated as a constrained nonlinear minimization problem this problem is then solved using a simplified iterative algorithm the algorithm is applied to simulated data and the performance of the algorithm is compared to previous results

a gtd based parametric model for radar scattering this paper presents a new approach to scattering center extraction based on a scattering model derived from the geometrical theory of diffraction gtd for stepped frequency measurements at high frequencies the model is better matched to the physical scattering process than the damped exponential model and conventional fourier analysis in addition to determining downrange distance energy and polarization the gtd based model extracts frequency dependent scattering information allowing partial identification of scattering center geometry we derive expressions for the cramer rao bound of this model using these expressions we analyze the behavior of the new model as a function of scatterer separation bandwidth number of data points and noise level additionally a maximum likelihood algorithm is developed for estimation of the model parameters we present estimation results using data measured on a compact range to validate the proposed modeling procedure

parameter estimation of superimposed signals using the em algorithm a computationally efficient algorithm for parameter estimation of superimposed signals based on the two step iterative em estimate and maximize with an e step and an m step algorithm is developed the idea is to decompose the observed data into their signal components and then to estimate the parameters of each signal component separately the algorithm iterates back and forth using the current parameter estimates to decompose the observed data better and thus increase the likelihood of the next parameter estimates the application of the algorithm to the multipath time delay and multiple source location estimation problems is considered

computations of homogeneous equilibrium two phase flows with accurate and efficient shock stable schemes for accurate and efficient computations of compressible gas liquid two phase mixture flows the ausmpw and roem schemes for which the accuracy efficiency and robustness have been successfully demonstrated in gas dynamics are extended to two phase flows at all speeds from the mixture equations of state a new shock discontinuity sensing term suitably scaled for two phase flows is derived and its performance is validated in addition several numerical difficulties appearing in the development of the two phase ausmpw and roem schemes are analyzed and successfully cured the two phase ausmpw and roem schemes are then efficiently preconditioned for the simulation of all mach number flows by employing the existing ausm or harten lax van leer with contact restoration types of preconditioning strategies various gas liquid two phase flows from highly compressible to nearly incompressible flow conditions are tested the numerical results show the accurate and robust behavior of the proposed schemes for all speeds of two phase flows

coefficients of half integral weight modular forms modulo lj suppose that l 5 is prime that j 0 is an integer and that f z is a half integral weight modular form with integral fourier coefficients we give some general conditions under which the coefficients of f are well distributed modulo lj as a consequence we settle many cases of a classical conjecture of newman by proving for each prime power lj with l 5 that the ordinary partition function p n takes each value modulo lj infinitely often

optimal and suboptimal broad band source location estimation two maximum likelihood ml estimators are considered for direction of arrival doa estimation of broadband sources with unknown spectral parameters one is based on the assumption that the sources radiate stochastic gaussian signals and therefore is called the stochastic gaussian ml sgml estimator the other using estimates of the actual signals not their assumed distribution is called the conditional ml cml estimator neither is efficient if the source spectral parameters are completely arbitrary and unknown but the problem can be avoided for a version of the sgml estimation if the signal and noise spectra are known to satisfy certain smoothness conditions while this version of the sgml is formally superior to the cml it is demonstrated that the performance difference is small with underconditions not infrequently encountered in practice when these are satisfied the computationally simpler cml can be used without significant loss the required conditions become more stringent as the source separation decreases or correlation between sources increases a closed form analytic expression is obtained for the small error variance of the cml estimator of the doa of the nth source in the presence of n 1 other sources

music maximum likelihood and cramer rao bound further results and comparisons the problem of determining the direction of arrival of narrowband plane waves using sensor arrays and the related problem of estimating the parameters of superimposed signals from noisy measurements are studied a number of results have been recently presented by the authors on the statistical performance of the multiple signal characterization music and the maximum likelihood ml estimators for the above problems this work extends those results in several directions first it establishes that in the class of weighted music estimators the unweighted music achieves the best performance i e the minimum variance of estimation errors in large samples next it derives the covariance matrix of the ml estimator and presents detailed analytic studies of the statistical efficiency of music and ml estimators these studies include performance comparisons of music and mle with each other as well as with the ultimate performance corresponding to the cramer rao bound finally some numerical examples are given which provide a more quantitative study of performance for the problem of finding two directions with uniform linear sensor arrays

bounds for estimation of complex exponentials in unknown colored noise we consider the problem of estimating the parameters of complex exponentials in the presence of complex additive gaussian noise with unknown covariance bounds are derived for the accuracy of jointly estimating the parameters of the exponentials and the noise we first present an exact cramer rao bound crb for this problem and specialize it for the cases of circular gaussian processes and autoregressive processes we also derive an approximate expression for the crb which is related to the conditional likelihood function numerical evaluation of these bounds provides some insights on the effect of various signal and noise parameters on the achievable estimation accuracy

maximum likelihood array processing in non gaussian noise with gaussian mixtures many approaches have been studied for the array processing problem when the additive noise is modeled with a gaussian distribution but these schemes typically perform poorly when the noise is non gaussian and or impulsive this paper is concerned with maximum likelihood array processing in non gaussian noise we present the cramer rao bound on the variance of angle of arrival estimates for arbitrary additive independent identically distributed iid symmetric non gaussian noise then we focus on non gaussian noise modeling with a finite gaussian mixture distribution which is capable of representing a broad class of non gaussian distributions that include heavy tailed impulsive cases arising in wireless communications and other applications based on the gaussian mixture model we develop an expectation maximization em algorithm for estimating the source locations the signal waveforms and the noise distribution parameters the important problems of detecting the number of sources and obtaining initial parameter estimates for the iterative em algorithm are discussed in detail the initialization procedure by itself is an effective algorithm for array processing in impulsive noise novel features of the em algorithm and the associated maximum likelihood formulation include a nonlinear beamformer that separates multiple source signals in non gaussian noise and a robust covariance matrix estimate that suppresses impulsive noise while also performing a model based interpolation to restore the low rank signal subspace the em approach yields improvement over initial robust estimates and is valid for a wide snr range the results are also robust to pdf model mismatch and work well with infinite variance cases such as the symmetric stable distributions simulations confirm the optimality of the em estimation procedure in a variety of cases including a multiuser communications scenario we also compare with existing array processing algorithms for non gaussian noise

maximum likelihood localization of multiple sources by alternating projection an algorithm referred to as apm for computing the maximum likelihood estimator of the locations of simple sources in passive sensor arrays is presented the algorithm is equally applicable to the case of coherent signals and to the case of a single snapshot the algorithm is iterative the maximum of the likelihood function is computed by successive approximations the convergence of the algorithm to the global maximum is demonstrated for a variety of scenarios the key to this global convergence is the initialization scheme

a multiphase godunov method for compressible multifluid and multiphase flows we propose a new model and a solution method for two phase compressible flows the model involves six equations obtained from conservation principles applied to each phase completed by a seventh equation for the evolution of the volume fraction this equation is necessary to close the overall system the model is valid for fluid mixtures as well as for pure fluids the system of partial differential equations is hyperbolic hyperbolicity is obtained because each phase is considered to be compressible two difficulties arise for the solution one of the equations is written in non conservative form non conservative terms exist in the momentum and energy equations we propose robust and accurate discretisation of these terms the method solves the same system at each mesh point with the same algorithm it allows the simulation of interface problems between pure fluids as well as multiphase mixtures several test cases where fluids have compressible behavior are shown as well as some other test problems where one of the phases is incompressible the method provides reliable results is able to compute strong shock waves and deals with complex equations of state

optimal utilization of signal free samples for array processing in unknown colored noise fields many algorithms for direction of arrival doa estimation require the noise covariance matrix to be known or to possess a known structure in many cases the noise covariance is in fact estimated from separate measurements this paper addresses the combined effects of finite sample sizes both in the estimated noise covariance matrix and in the data with signals present it is assumed that a batch of signal free samples is available in addition to the signal containing samples no assumption is made on the structure of the noise covariance in this paper the asymptotic covariance of the weighted subspace fitting wsf algorithm is derived for the case in which the data are whitened using an estimated noise covariance the expression obtained suggests an optimal weighting that improves performance compared to the standard choice in addition a new method based on covariance matching is proposed both methods are asymptotically statistically efficient the cramer rao lower bound crb on the covariance of the estimate for the data model is also derived monte carlo simulations show promising small sample performance for the two new methods and confirm the asymptotic results

trace formulae and imaginary quadratic fields for the proof of this result we utilize the trace formulae of hecke operators for certain modular groups which have been established in 3 15 and are naturally related to the p adic galois representations associated with the jacobian varieties of the corresponding modular curves the hypothesis that p is sufficiently large can be omitted or made effective in some cases for instance holds the following cf also i 5 6 theorem 2 let n o denote the product of all primes in po and assume that 4 2 4 n 3 log x o p 7r2

on the performance of signal subspace processing part i narrow band systems this paper presents an analytical evaluation of detection determination of the number of sources and estimation performances of narrow band signal subspace processing for multiple source direction finding the probabilities of underestimating and overestimating the number of sources are derived under asymptotic conditions and around the threshold regions in terms of the choice of a penalty function and signal noise and array parameters for the cases of at most two closely spaced sources in the spatially white noise a scalar measure is introduced for the evaluation of the quality of the estimated signal subspace based on the statistics of this measure performance thresholds are demonstrated for the signal to noise ratio angle separation and correlation between two equipowered sources

low diffusion flux splitting methods for flows at all speeds methods for extending the advective upwind splitting method ausm family of low diffusion flux splitting schemes to operate effectively at all flow speeds are developed the extensions developed are designed for use with time derivative preconditioning and are based on the idea that the speed of sound should cease to be an important scaling parameter for the diffusive contributions to the interface flux as the mach number becomes small using this criterion alternative definitions for the interface mach numbers are developed and methods for ensuring pressure velocity coupling at low speeds are formulated results are presented for inviscid flows through a channel at various mach numbers developing viscous flow in a two dimensional duct driven cavity flows at various mach and reynolds numbers flow over a backward facing step and hydrogen nitrogen mixing layers

computation of the exact information matrix of gaussian time series with stationary random components the paper presents an algorithm for efficient recursive computation of the fisher information matrix of gaussian time series whose random components are stationary and whose means and covariances are functions of a parameter vector the algorithm is first developed in a general framework and then specialized to the case of autoregressive moving average processes with possible additive white noise the asymptotic behavior of the algorithm is explored and a termination criterion is derived finally the algorithm is used to demonstrate the behavior of the exact cramer rao bound for unbiased estimates for some arma processes as a function of the number of data points it is shown that for processes with zeros near the unit circle and short data records the exact cramer rao bound differs dramatically from its common approximation based on asymptotic theory

temporal and spatial sampling influence on the estimates of superimposed narrowband signals when less can mean more this paper addresses the influence that the sampling locations have on the estimated frequencies of superimposed sinusoids this problem has application in harmonic time series analysis or direction finding phased array systems generalized mathematical bounds are developed in terms that are independent of the array locations and have an intuitively appealing physical interpretation they establish the influence of the sampling locations on the variance of the frequency estimate and the limit at which two sources can be resolved using signal subspace estimators for the resolution criteria an expression dominated by the fourth central moment of the sensor locations expresses the resolving ability of the sensing array irrespective of the array aperture or number of sensors increasing the fourth central moment increases an array s resolution ability the commonly accepted notion that resolution necessarily depends on the array aperture is misleading and indeed that fewer snapshots from a reduced aperture array can outperform a larger array of more elements for the estimator variance criteria it is found that the product of the number of sensors and the second central moment array variance characterizes the estimator variance lower bound the metrics developed demonstrate that the sampling topology is itself an important factor in determining the performance of the sampling system and not the covariance lags sampled or the aperture spanned simulations are used to describe the results

a detailed derivation of arrays free of higher rank ambiguities since linear dependence of steering vectors would lead to ambiguous direction of arrival doa estimates it is crucial to employ array with linearly independent steering vectors in doa estimation applications in fact it is well known that in order to determine uniquely the doas of spl nu uncorrelated signals one requires an array that is free of up to rank spl nu ambiguities i e an array whose every spl nu 1 steering vectors with distinct doas are linearly independent however the question as to the sensor arrangement that gives rise to an array that is free of up to rank spl nu ambiguities where spl nu spl isin 3 remains unanswered we construct a class of cross arrays that are free of up to rank spl nu ambiguities where spl nu spl isin 1 through this study we derive a theorem for characterizing inherent ambiguities associated with cross arrays we also provide counterexamples to the conjecture for characterizing higher rank ambiguity proposed by lo and marple see ibid vol 40 p 2641 50 1992 and that proposed by wang et al see icassp 9 vol iv p 509 12 1994

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

blind frequency synchronization for multiuser ofdm uplink with large number of receive antennas in this paper we propose a new blind frequency synchronization scheme for multiuser orthogonal frequency division multiplexing ofdm uplink transmissions the spatial multiplexing is supported in the considered model that allows the subcarriers to be simultaneously occupied by multiple users while the orthogonal frequency division multiple access ofdma can serve as a special case of our discussion we propose to assign different null subcarriers to different users and design algorithm that can perform blind carrier frequency offset cfo estimation for each individual user with the aid of large number of receive antennas which then removes the necessity of multidimensional searching after estimating cfo for one user the orthogonal complement space of the other users can be obtained which could be further utilized for multiuser interference cancelation hence the multiuser ofdm uplink model reduces to an equivalent single user signal model where the blind channel estimation with scalar ambiguity can be designed and the conventional single user data detection can be peformed to make the proposed study complete the closed form performance analysis for mean square error mse of the cfo estimation is derived based on which the discussion about optimal null subcarrier assignment scheme is further provided finally the cramer rao bound crb is derived and the numerical results are provided to corroborate the proposed studies

performance study of conditional and unconditional direction of arrival estimation a numerical and analytical study of conditional and unconditional direction of arrival doa estimation is presented explicit expressions for the unconditional cramer rao bounds on the doa estimation accuracy and the covariance matrix of the conditional maximum likelihood method are given it is shown that many doa estimation methods have the same asymptotic statistical properties under conditional and unconditional models the situation of two narrowband plane signals impinging on a uniformly spaced linear array is discussed

on the cramer rao bound for direction finding of correlated signals in this paper we present compact closed form formulas for the cramer rao bound corresponding to the joint estimation of the directions of arrival the signal covariance matrix and the noise variance using these formulas we investigate the effect of signal correlation on the achievable accuracy of direction finding system in a correlated signal environment as expected estimation accuracy decreases with increasing correlation magnitude we observe that under certain conditions small aperture high correlation magnitude correlation phase has a strong effect on doa estimation accuracy

weakly holomorphic modular forms of half integral weight with nonvanishing constant terms modulo l let l be a prime and j 0 be an integer suppose that f z n a n q n is a weakly holomorphic modular form of weight 1 2 and that a 0 0 mod l we prove that if the coefficients of f z are not well distributed modulo l j then 0 or 1 mod l 1 2 this implies that under the additional restriction 0 0 mod l the following conjecture of balog darmon and ono is true if the coefficients of a modular form of weight 1 2 are almost but not all divisible by l then either 0 mod l 1 2 or 1 mod l 1 2 we also prove that if 0 and 1 mod l 1 2 then there does not exist an integer 0 l such that ln 0 mod l for every nonnegative integer n as an application we study congruences for the values of the overpartition function

on the expectation of the product of four matrix valued gaussian random variables the formula for the expectation of the product of four scalar real gaussian random variables is generalized to matrix valued real or complex gaussian random variables as an application of the extended formula a simple derivation is presented of the covariance matrix of instrumental variable estimates of parameters in multivariable regression models

extension of harten lax van leer scheme for flows at all speeds the harten lax and van leer with contact restoration hllc scheme has been modified and extended in conjunction with time derivative preconditioning to compute flow problems at all speeds it is found that a simple modification of signal velocities in the hllc scheme based on the eigenvalues of the preconditioned system is only needed to reduce excessive numerical diffusion at the low mach number the modified scheme has been implemented and used to compute a variety of flow problems in both two and three dimensions on unstructured grids numerical results obtained indicate that the modified hllc scheme is accurate robust and efficient for flow calculations across the mach number range istorically numerical algorithms for the solution of the euler and navier stokes equations are classified as either pressure based or density based solution methods the pressurebased methods originally developed and well suited for incompressible flows are typically based on the pressure correction techniques they usually use a staggered grid and solve the governing equations in a segregated manner the density based methods originally developed and robust for compressible flows use time arching procedures to solve the hyperbolic system of governing equations in a coupled manner in general density based methods are not suitable for efficiently solving low mach number or incompressible flow problems because of large ratio of acoustic and convective timescales at the low speed flow regimes to alleviate this stiffness and associated convergence problems time derivative preconditioning techniques have been developed and used successfully for solving low machnumber and incompressible flows by many investigators including chorin 1 choi and merkle 2 turkel 3 weiss and smith 4 and dailey and pletcher 5 among others such methods seek to modify the time component of the governing equations so that the convergence can be made independent of mach number this is accomplished by altering the acoustic speeds of the system so that all eigenvalues become of the same order and thus condition number remains bounded independent of the mach number of the flows over the last two decades characteristic based upwind methods have established themselves as the methods of choice for prescribing the numerical fluxes for compressible euler equations when these upwind methods are used to compute the numerical fluxes for the preconditioned euler equations solution accuracy at low speeds can be compromised unless the numerical flux formulation is modified to take into account the eigensystem of the precondi

maximum likelihood methods for direction of arrival estimation five methods of direction of arrival doa estimation which can be derived from the maximum likelihood ml principle are considered the ml method mlm results from the application of the ml principle to the statistics of the observed raw data the standard multiple signal classification music procedure called music 1 is obtained as a brute force approximation of the mlm an improved music procedure named music 2 is obtained by applying the ml principle to the statistics of certain linear combinations of the sample noise space eigenvectors a procedure which compromises between the good performance of the mlm and the computational simplicity of music is a method of direction estimation mode 1 which is derived as a large sample realization of the mlm a fifth method called mode 2 is obtained by using the ml principle on the statistics of certain linear combinations of the sample eigenvectors mode 2 is computationally less demanding than the mlm it is of the same complexity as mode 1 and statistically more efficient a numerical comparison of these five doa estimation methods is presented it confirms the analytic results on their theoretical performance levels

optimality of high resolution array processing using the eigensystem approach in the classical approach to underwater passive listening the medium is sampled in a convenient number of look directions from which the signals are estimated in order to build an image of the noise field in contrast a modern trend is to consider the noise field as a global entity depending on few parameters to be estimated simultaneously in a gaussian context it is worthwhile to consider the application of likelihood methods in order to derive a detection test for the number of sources and estimators for their locations and spectral levels this paper aims to compute such estimators when the wavefront shapes are not assumed known a priori this justifies results previously found using the asymptotical properties of the eigenvalue eigenvector decomposition of the estimated spectral density matrix of the sensor signals they have led to a variety of high resolution array processing methods more specifically a covariance matrix test for equality of the smallest eigenvalues is presented for source detection for source localization a best fit method and a test of orthogonality between the smallest eigenvectors and the source vectors are discussed

frequency estimation performance of several weighted burg algorithms the frequency estimation performance of a set of weighted burg algorithms is investigated for single complex and real sinusoids in noise it is demonstrated that in the complex case an extension of ibrahim s 1987 1989 modification of kaveh and lipert s 1983 optimum tapered burg algorithm meets the cramer rao lower bound for a sufficiently high signal to noise ratio and a sufficiently large model order this is in contrast with the original burg method in the case of a real sinusoid the same algorithm most closely approaches this bound making it the technique of choice for both applications spectral resolution is addressed briefly

new results on deterministic cramer rao bounds for real and complex parameters the cramer rao bounds crb is a lower bound of great interest for system analysis and design in the asymptotic region high signal to noise ratio snr and or large number of snapshots as it is simple to calculate and it is usually possible to obtain closed form expressions the first part of the paper is a generalization to complex parameters of the barankin rationale for deriving mse lower bounds that is the minimization of a norm under a set of linear constraints with the norm minimization approach the study of fisher information matrix fim singularity constrained crb and regularity conditions become straightforward corollaries of the derivation the second part provides new results useful for system analysis and design a general reparameterization inequality the equivalence between reparameterization and equality constraints and an explicit relationship between parameters unidentifiability and fim singularity

subspace leakage analysis and improved doa estimation with small sample size classical methods of doa estimation such as the music algorithm are based on estimating the signal and noise subspaces from the sample covariance matrix for a small number of samples such methods are exposed to performance breakdown as the sample covariance matrix can largely deviate from the true covariance matrix in this paper the problem of doa estimation performance breakdown is investigated we consider the structure of the sample covariance matrix and the dynamics of the root music algorithm the performance breakdown in the threshold region is associated with the subspace leakage where some portion of the true signal subspace resides in the estimated noise subspace in this paper the subspace leakage is theoretically derived we also propose a two step method that improves the performance by modifying the sample covariance matrix such that the amount of the subspace leakage is reduced furthermore we introduce a phenomenon named as root swap which occurs in the root music algorithm in the low sample size region and degrades the performance of the doa estimation a new method is then proposed to alleviate this problem numerical examples and simulation results are given for uncorrelated and correlated sources to illustrate the improvement achieved by the proposed methods moreover the proposed algorithms are combined with the pseudo noise resampling method to further improve the performance