an iterative algorithm for single frequency estimation an algorithm for the estimation of the frequency of a complex sinusoid in noise is proposed the estimator consists of multiple applications of lowpass filtering and decimation frequency estimation by linear prediction and digital heterodyning the estimator has a significantly reduced threshold relative to existing phase based algorithms and performance close to that of maximum likelihood estimation in addition the mean squared error performance is within 0 7 db of the cramer rao bound crb at signal to noise ratios snrs above threshold unlike many autocorrelation and phase based methods the proposed algorithm s performance is uniform across a frequency range of spl pi to spl pi the computational complexity of the algorithm is shown to be favorable compared with maximum likelihood estimation via the fast fourier transform fft algorithm when significant zero padding is required

analysis of polynomial phase signals by the integrated generalized ambiguity function the aim of this work is the performance analysis of a method for the detection and parameter estimation of mono or multicomponent polynomial phase signals pps embedded in white gaussian noise and based on a generalized ambiguity function the proposed method is shown to be asymptotically efficient for second order pps and nearly asymptotically efficient for third order ppss the method presents some advantages with respect to similar techniques like the polynomial phase transform for example in terms of i a closer approach to the cramer rao lower bounds ii a lower snr threshold iii a better capability of discriminating multicomponent signals

a simple estimator for frequency and decay rate a simple computationally inexpensive algorithm is developed for estimating the frequency and decay rate of a complex exponential two iterations of the algorithm attains the cramer rao bound on the variance of the frequency and decay rate estimate for a complex exponential in white gaussian noise the algorithm uses an adaptive window that changes its shape with the estimate of the decay rate formulas are derived for the bias and variance of the estimator and its performance is demonstrated in simulations

harmonics in gaussian multiplicative and additive noise cramer rao bounds the concern here is retrieval of single and multiple tone harmonics observed in white gaussian multiplicative and additive noise computable cramer rao bound crb expressions are derived on the frequency and phase estimates as well as on the sample mean or variance of the multiplicative noise processes the zero and nonzero mean multiplicative noise cases are addressed separately and are shown to yield distinct crbs on the frequency and phase estimates tight lower and upper bounds on the crbs themselves are developed which relative to the crbs are intuitively more appealing and easier to implement well established formulas on the achievable accuracy for estimates of constant amplitude harmonics turn put to be special cases of our results numerical studies support our claims

subspace based algorithm for parameter estimation of polynomial phase signals in this correspondence parameter estimation of a polynomial phase signal pps in additive white gaussian noise is addressed assuming that the order of the pps is at least 3 the basic idea is first to separate its phase parameters into two sets by a novel signal transformation procedure and then the multiple signal classification music method is utilized for joint estimating the phase parameters with second order and above in doing so the parameter search dimension is reduced by a half as compared to the maximum likelihood and nonlinear least squares approaches in particular the problem of cubic phase signal estimation is studied in detail and its simplification for a chirp signal is given the effectiveness of the proposed approach is also demonstrated by comparing with several conventional techniques via computer simulations

parameter estimation of chirp signals the problem of the parameter estimation of chirp signals is addressed several closely related estimators are proposed whose main characteristics are simplicity accuracy and ease of online or offline implementation for moderately high signal to noise ratios they are unbiased and attain the cramer rao bound monte carlo simulations verify the expected performance of the estimators it should be easy to extend this approach to signals having polynomials of any degree in the exponent all the derivations will be done under the assumption that the signal to noise ratio is sufficiently high

low complexity frequency estimation using multiple disjoint pilot blocks in burst mode communications two low complexity data aided frequency estimators suitable for use in burst mode communications are described and analyzed the estimators are based on pilot symbols organized into disjoint blocks embedded in the burst the first estimator is a generalization of the phase increment frequency estimation technique and the second estimator is a generalization of the autocorrelation frequency estimation technique the generalizations are needed to account for the spacings between the pilot blocks it is shown that the frequency estimators exhibit good accuracy while maintaining useful operating ranges

a new class of bayesian cyclic bounds for periodic parameter estimation many practical signal processing applications involve estimation of parameters with periodic nature such as phase frequency and direction of arrival estimation the commonly used mean squared error mse risk does not take periodicity into account and thus is inappropriate for periodic parameter estimation in which one is interested in the modulo t error rather than the plain error value as a result mse lower bounds are not valid for periodic estimation in addition conventional bayesian mse lower bounds such as the bayesian cramer rao bound bcrb and the bobrovsky zakai bound bzb require restrictive regularity conditions and usually do not exist in periodic settings an alternative risk which is commonly used in periodic parameter estimation problems is the mean cyclic error mce in this paper we establish a new class of bayesian lower bounds on the mce of any estimator the new class includes cyclic versions of the bcrb and the bzb that have less restrictive regularity conditions than those of the conventional bcrb and bzb respectively the tightest bound in the proposed class is derived and its tightness is discussed in addition the proposed class is extended to mixed vector parameter estimation with both periodic and nonperiodic parameters the new cyclic lower bounds are compared with the mce performance of the minimum mce and maximum a posteriori probability estimators for von mises parameter estimation and for frequency estimation

two algorithms for adaptive retrieval of slowly time varying multiple cisoids in noise two algorithms for tracking parameters of slowly varying multiple complex sine waves cisoids in noise the multiple frequency tracker and the adaptive notch filter are described for high signal to noise ratio snr the properties of the algorithms i e stability noise rejection and tracking speed are studied analytically using a linear filter approximation technique the tradeoff between noise rejection and tracking error for both algorithms is shown to be similar different choices of the design variables are discussed namely i minimal mean square estimation error for random walk modeled frequency variations and ii minimal stationary estimation variance subject to a given tracking delay

product high order ambiguity function for multicomponent polynomial phase signal modeling parameter estimation and performance analysis issues are studied for multicomponent polynomial phase signals ppss embedded in white gaussian noise identifiability issues arising with existing approaches are described first when dealing with multicomponent pps having the same highest order phase coefficients this situation is encountered in applications such as synthetic aperture radar imaging or propagation of polynomial phase signals through channels affected by multipath and is thus worthy of a careful analysis a new approach is proposed based on a transformation called product high order ambiguity function phaf the use of the phaf offers a number of advantages with respect to the high order ambiguity function haf more specifically it removes the identifiability problem and improves noise rejection capabilities performance analysis is carried out using the perturbation method and verified by simulation results

the statistical performance of some instantaneous frequency estimators the authors examine the class of smoothed central finite difference scfd instantaneous frequency if estimators which are based on finite differencing of the phase of the analytic signal these estimators are closely related to if estimation via the periodic first moment with respect to frequency of discrete time frequency representations tfrs in l cohen s 1966 class the authors determine the distribution of this class of estimators and establish a framework which allows the comparison of several other estimators such as the zero crossing estimator and one based on linear regression on the signal phase it is found that the regression if estimator is biased and exhibits a large threshold for much of the frequency range by replacing the linear convolution operation in the regression estimator with the appropriate convolution operation for circular data the authors obtain the parabolic scfd pscfd estimator which is unbiased and has a frequency independent variance yet retains the optimal performance and simplicity of the original estimator

discrete time instantaneous frequency and its computation classical results indicate that the means of the fourier and instantaneous frequencies are exactly equal and the weighted variance of the fourier frequency is greater than or equal to that of the instantaneous frequency these results are investigated for the discrete case in which the signal and its spectrum are both assumed to be periodic specifically the cases in which the discrete time instantaneous frequency is defined as the forward backward and central differences are discussed a simple algorithm for computing the discrete time instantaneous frequency is presented whereby descriptors for the characterization of the signal are constructed the application of this algorithm is discussed through examples

nonminimum phase identification based on higher order spectrum slices a new family of methods named phase estimation using polyspectrum slices pep for the reconstruction of the fourier phase of a complex linear time invariant lti system excited by a white non gaussian input is proposed more precisely we propose two subfamilies of methods the q pep q ges 3 and q1 q2 pep q2 q1 ges 3 algorithms the q pep methods exploit the best two dimensional 2 d slice of the data qth order spectrum the originality of the q1 q2 pep methods consists of exploiting simultaneously one 1 d slice of the q1th order spectrum and one 2 d slice of the q2th order spectrum these new algorithms are easy both to implement and to use moreover the asymptotic unbiasedness and consistency of these methods are demonstrated eventually computer simulations show that the pep algorithms exhibit in general better performances than classical methods especially for band limited systems

an evaluation of the use of magnetic field maps to undistort echo planar images when a head is placed in an mri scanner differences between the magnetic susceptibility of tissue bone and air distort the magnetic field while some of the resulting inhomogeneity can be corrected by the shimming process much of it cannot and this causes distortion sometimes referred to as geometric distortion of echo planar images epis one strategy for the correction of distortion is to acquire a map of the magnetic field achieved in each subject and then to use this to undistort their epis after reconstruction jezzard and balaban 1995 here we present five experiments to evaluate the application of such a strategy on data from a 3 t scanner we show that after undistortion the shape of epis is more similar to the true shape of the brain and we investigate the effect of head movement on the efficacy of undistortion if undistortion was applied first it was found that less nonlinear warping was required to transform epis into a standard space particularly in the phaseencode direction we show that if spm 99 normalization is used to perform a nonlinear warp to standard space the prior application of undistortion increases the statistical power of group studies with motor and auditory tasks we show that this increase in power is due to an increase in the overlap of activation of different subjects finally we evaluate where in the brain undistorting epis might be expected to have the greatest effect in terms both of mislocalization of activation and of a reduction in power overall undistorting epis using field maps has proved extremely successful improving the anatomical localization of activation and increasing statistical power 2002 elsevier science usa

blind high resolution localization and tracking of multiple frequency hopped signals this paper considers the problem of blind localization and tracking of multiple frequency hopped spread spectrum signals using a uniform linear antenna array without knowledge of hopping patterns or directions of arrival as a preprocessing step we propose to identify a hop free subset of data by discarding high entropy spectral slices from the spectrogram high resolution localization is then achieved via either quadrilinear regression of four way data generated by capitalizing on both spatial and temporal shift invariance or a new maximum likelihood ml based two dimensional 2 d harmonic retrieval algorithm the latter option achieves the best known model identifiability bound while remaining close to the cramer rao bound even at low signal to noise ratios snrs following beamforming using the recovered directions a dynamic programming approach is developed for joint ml estimation of signal frequencies and hop instants in single user tracking the efficacy of the proposed algorithms is illustrated in pertinent simulations

approximate maximum likelihood algorithms for two dimensional frequency estimation of a complex sinusoid starting with the maximum likelihood ml formulation three iterative algorithms for approximate ml frequency estimation of a two dimensional 2 d complex sinusoid in white gaussian noise are developed mean and variance analyses of the proposed methods are provided which show that they are approximately unbiased and their performance achieves cramer rao lower bound crlb at sufficiently high signal to noise ratio snr conditions computer simulation results are included to corroborate the theoretical development as well as to contrast the performance of the proposed algorithms with kay s estimators and the crlb

on periodic pulse interval analysis with outliers and missing observations analysis of periodic pulse trains based on time of arrival is considered with perhaps very many missing observations and contaminated data a period estimator is developed based on a modified euclidean algorithm this algorithm is a computationally simple robust method for estimating the greatest common divisor of a noisy contaminated data set the resulting estimate although it is not maximum likelihood is used as initialization in a three step algorithm that achieves the cramer rao bound crb for moderate noise levels as shown by comparing monte carlo results with the crbs this approach solves linear regression problems with missing observations and outliers comparisons with a periodogram approach based on a point process model are shown an extension using multiple independent data records is also developed that overcomes high levels of contamination

joint 2 d doa estimation and phase calibration for uniform rectangular arrays a precise model of the array response is required to maintain the performance of direction of arrival doa estimation when modeling errors are present or the sensor environment is time varying autocalibration becomes necessary in this paper the problem of phase autocalibration for uniform rectangular array ura geometries is considered for the case with a single source a simple and robust least squares algorithm for joint 2 d doa estimation and phase calibration is presented when performing phase autocalibration with a ura the phase and doa parameters cannot be identified together without ambiguity this problem is discussed and a suitable remedy is suggested an approximate cramer rao bound and analytical expressions for the mean squared error performance of the proposed estimator are presented the proposed algorithm for phase autocalibration is extended for the case with multiple sources the results are evaluated using a representative body of simulations

polynomial phase estimation by least squares phase unwrapping estimating the coefficients of a noisy polynomial phase signal is important in fields including radar biology and radio communications one approach attempts to perform polynomial regression on the phase of the signal this is complicated by the fact that the phase is wrapped modulo 2 and must be unwrapped before regression can be performed in this paper we consider an estimator that performs phase unwrapping in a least squares manner we call this the least squares unwrapping lsu estimator the lsu estimator can be computed in a reasonable amount of time for data sets of moderate size using existing general purpose algorithms from algebraic number theory under mild conditions on the distribution of the noise we describe the asymptotic properties of this estimator showing that it is strongly consistent and asymptotically normally distributed a key feature is that the lsu estimator is accurate over a far wider range of parameters than many popular existing estimators monte carlo simulations support our theoretical results and demonstrate the excellent statistical performance of the lsu estimator when compared with existing state of the art estimators

bootstrap modeling of a class of nonstationary signals the problem of modeling polynomial phase signals is considered techniques based on the bootstrap and multiple hypotheses tests for optimal model selection of constant amplitude polynomial phase signals embedded in stationary noise are developed phase parameter estimators based on both the least squares method and the polynomial phase transform are used the proposed techniques are compared with existing ones including akaike s 1978 1974 information criterion and rissanen s 1978 minimum description length criterion and are shown to outperform these procedures for the considered small sample sizes

limitations on snr estimator accuracy we consider the samples of a pure tone in additive white gaussian noise awgn for which we wish to determine the signal to noise ratio snr defined here to be spl alpha a sup 2 2 spl sigma sup 2 where a is the tone amplitude and spl sigma sup 2 is the noise variance a and spl sigma sup 2 are assumed to be deterministic but unknown a priori if the variance of an unbiased estimator of spl alpha is spl sigma sub spl alpha spl circ sup 2 we show that at high snr the normalized standard a deviation satisfies the cramer rao lower bound crlb according to spl sigma sub spl alpha spl circ spl alpha spl ges spl radic 2 n where n is the number of independent observables used to obtain the snr estimate spl sigma spl circ

a generalized weighted linear predictor frequency estimation approach for a complex sinusoid based on linear prediction and weighted least squares three simple iterative algorithms for frequency estimation of a complex sinusoid in additive white noise are devised the proposed approach which utilizes the first order as well as higher order linear prediction terms simultaneously but does not require phase unwrapping can be considered as a generalized version of the weighted linear predictor frequency estimator in particular convergence as well as mean and variance analysis of the most computationally efficient frequency estimator namely gwlp 2 are provided computer simulations are included to contrast the performance of the proposed algorithms with several conventional computationally attractive frequency estimators and crame spl acute r rao lower bound for different frequencies observation lengths and signal to noise ratios

analysis of the variance threshold of kay s weighted linear predictor frequency estimator a theoretical approximation for the variance of kay s weighted linear predictor frequency estimator is derived from this expression an inequality describing the variance threshold of the estimator is found the window weights are then optimized to improve the variance numerical simulations demonstrate that the variance approximations are valid for medium to high signal to noise ratios or for large numbers of samples

modifications of the euclidean algorithm for isolating periodicities from a sparse set of noisy measurements modifications of the euclidean algorithm are presented for determining the period from a sparse set of noisy measurements the elements of the set are the noisy occurrence times of a periodic event with perhaps very many missing measurements this problem arises in radar pulse repetition interval pri analysis in bit synchronization in communications and in other scenarios the proposed algorithms are computationally straightforward and converge quickly a robust version is developed that is stable despite the presence of arbitrary outliers the euclidean algorithm approach is justified by a theorem that shows that for a set of randomly chosen positive integers the probability that they do not all share a common prime factor approaches one quickly as the cardinality of the set increases in the noise free case this implies that the algorithm produces the correct answer with only 10 data samples independent of the percentage of missing measurements in the case of noisy data simulation results show for example good estimation of the period from 100 data samples with 50 of the measurements missing and 25 of the data samples being arbitrary outliers

frequency estimation from proper sets of correlations as a complement to the periodogram low complexity frequency estimators are of interest one such estimator is based on prony s method and rely on phase information of the auto correlations without prior knowledge of the frequency e g a given frequency interval the frequency cannot be unambiguously estimated from a single correlation only we introduce a new method of phase unwrapping using an arbitrary number more than one of correlations from this arbitrary set of correlations we propose a weighted average estimator we derive the asymptotic performance and show how the correlation lags should be properly chosen from a design aspect there is often a restriction of using a fixed number of computations in addition we therefore propose a strategy to find a proper set of correlation lags subject to a given computational complexity finally simulation results that lend support to the theoretical findings are included

cyclic barankin type bounds for non bayesian periodic parameter estimation in many practical periodic parameter estimation problems the appropriate performance criteria are periodic in the parameter space the existing mean square error mse lower bounds such as cramer rao bound crb and barankin type bounds do not provide valid lower bounds in such problems in this paper cyclic versions of the crb and the barankin type bounds hammersley chapman robbins and mcaulay seidman are derived for non bayesian parameter estimation the proposed bounds are lower bounds on the mean cyclic error mce of any cyclic unbiased estimator where the cyclic unbiasedness is defined by using lehmann unbiasedness these mce lower bounds can be readily obtained from existing mse lower bounds and thus can be easily calculated the cyclic barankin type bounds and the performance of the maximum likelihood ml estimator are compared in terms of mce in von mises distributed measurements problem and for frequency and amplitude estimation with gaussian noise in these problems the ml estimator is found to be cyclic unbiased

estimating the frequencies of two sinusoids using only the phase angles of complex valued data the problem of estimating the frequencies of two complex sinusoids using only the phase data is addressed under the assumption that the amplitudes of the two signals are not equal a computationally efficient approximate maximum likelihood estimation amle is obtained this amle uses only the phase angles of the complex valued data to estimate frequencies and uses the envelope data to resolve an inherent algebraic sign ambiguity in the difference frequency at moderately high snrs satisfactory results are obtained from computer simulations the mean square errors of the estimates are compared with the corresponding values of the cramer rao lower bounds crlbs

an inverse problem approach to the correction of distortion in epi images magnetic resonance imaging using the echo planar imaging epi technique is particularly sensitive to main b sub 0 field inhomogeneities the primary effect is geometrical distortion in the phase encoding direction here the authors present a method based on the conjugate gradient algorithm to correct for this geometrical distortion by solving the epi imaging equation two versions are presented one that attempts to solve the full four dimensional 4 d imaging equation and one that independently solves for each profile along the blip encoding direction results are presented for both phantom and in vivo brain epi images and compared with other proposed correction methods

relationship between matrix pencil and state space based harmonic retrieval methods the author studies the relationship between a state space approach and the matrix pencil method for the problem of estimating the parameters of damped and undamped sinusoids it is shown that the methods are very similar in the exact data case and have minor difference in implementation in the noisy data case

an improved feedforward frequency offset estimator in this correspondence we propose a new feedforward frequency offset or single frequency estimator which is based on a single weighted sample correlation with lag k computed from n noisy observations the new estimator is a simple generalization of the unweighted lank reed pollon lrp estimator introduced in g w lank i s reed and g e pollon a semicoherent detection and doppler estimation statistic ieee transactions on aerospace and electronic systems vol 9 no 2 pp 151 165 march 1973 it is shown that by relaxing the requirement for frequency estimation range an estimator with improved accuracy and lower threshold may be designed in particular the estimator is able to provide a 20 log10 k db threshold reduction when n is an integer multiple of k closed form expressions are derived for the weighting coefficients and also for the minimum variance analytical and simulation results are presented which assess and demonstrate the estimator performance

esprit a subspace rotation approach to estimation of parameters of cisoids in noise the application of a subspace invariance approach esprit to the estimation of parameters frequencies and powers of cisoids in noise is described esprit exploits an underlying rotational invariance of signal subspaces spanned by two temporally displaced data sets the new approach has several advantages including improved resolution over pisarenko s technique for harmonic retrieval

estimation of multicomponent polynomial phase signals impinging on a multisensor array using state space modeling this contribution addresses the problem of estimating the parameters of multicomponent polynomial phase signals when impinging on a multisensor array an original approach is proposed based on a state space modelization of the signal and the application of an extended kalman filter for the state estimation the use of a multisensor array allows the exploitation of a spatial information and leads to the consideration of multiple filters with different observation equations computer simulations are used to demonstrate the effectiveness of the proposed algorithm

on the performance of the weighted linear predictor frequency estimator a statistical analysis of kay s 1989 weighted linear predictor frequency estimator was carried out using a different analysis technique to that employed in that paper the asymptotic variance of the weighted linear predictor frequency estimator is derived

comments on on the performance of the weighted linear predictor frequency estimator in a recent correspondence handel see ibid vol 43 p 3070 1 dec 1995 provides an alternate analysis to that of clarkson et al see ibid vol 42 p 2370 9 sept 1994 of kay s linear predictor frequency estimator we take this opportunity to clarify our result and emphasize the care needed when analyzing the performance of a nonlinear estimator

parameter estimation of phase modulated signals using bayesian unwrapping parametric estimation of phase modulated signals pms in additive white gaussian noise is considered the prohibitive computational expense of maximum likelihood estimation for this problem has led to the development of many suboptimal estimators which are relatively inaccurate and cannot operate at low signal to noise ratios snrs in this paper a novel technique based on a probabilistic unwrapping of the phase of the observations is developed the method is capable of more accurate estimation and operates effectively at much lower snrs than existing algorithms this is demonstrated in monte carlo simulations

kalman estimation of single tone parameters and performance comparison with map estimator by incorporating a priori knowledge via the a priori probability density function of the frequency and phase the kalman estimator for linear minimum mean square error estimation of these single tone parameters in additive white gaussian noise is presented first a linear two dimensional state space model for the frequency and phase of a sinusoid is formulated then two kalman filters are proposed one based on tretter s phase noise model s a tretter ldquoestimating the frequency of a noisy sinusoid by linear regression rdquo ieee transactions on information theory vol it 31 no 6 pp 832 835 nov 1985 and the other based on our newly proposed model in h fu and p y kam ldquomap ml estimation of the frequency and phase of a single sinusoid in noise rdquo ieee transactions on signal processing vol 55 no 3 mar 2007 finally their mean square error performances are compared with each other and with that of the maximum a posteriori probability map estimator using computer simulations the results show that the map estimator performs best and the kalman filter based on the improved phase noise model has better performance than that based on tretter s model especially at low signal to noise ratio

variable precision registration via wavelets optimal spatial scales for inter subject registration of functional mri abstract the detection of significantly activated brain regions in multi subject functional magnetic resonance imaging fmri studies almost invariably entails the coregistration of individual subjects data in a standard space here we investigate how sensitivity to detect loci of generic activation in such studies may be conditioned by the precision of anatomical registration we describe a novel algorithm implemented in the wavelet domain for inhomogeneous deformation of individual images to match a template the algorithm matches anatomical features in a coarse to fine fashion first minimising a cost function in terms of relatively coarse spatial features and then proceeding iteratively to match the images in terms of progressively more detailed anatomical features applying the method to data acquired from two groups of 12 healthy volunteers with mean age 27 and 70 years respectively during performance of a paired associate learning task we show that geometrical overlap between template and individual images is monotonically improved compared to an affine transform by additional inhomogeneous deformations informed by more detailed features likewise sensitivity to detect activated voxels can be substantially improved by a factor of 4 or more if wavelet mediated deformations informed by medium sized anatomical features are applied in addition to a preliminary affine transform however sensitivity to detect activated voxels was reduced by over registering data or matching anatomical features at the finest scales of the wavelet transform the benefits of variable precision registration are particularly salient for data acquired in older subjects which showed evidence of greater inter subject anatomic variability and generally required more extensive local deformation to achieve a satisfactory match to the template image we conclude that major benefits in sensitivity to detect functional activation in multi subject fmri studies can be attained with an inhomogeneous deformation applied over appropriate spatial scales

non bayesian periodic cramer rao bound the cramer rao bound crb is one of the most important tools for performance analysis in parameter estimation in many practical periodic parameter estimation problems the appropriate criterion is periodic in the parameter space however the crb does not provide a valid lower bound in such problems in this paper the periodic crb for non bayesian periodic parameter estimation is derived the proposed periodic crb is a lower bound on the mean square periodic error mspe of any periodic unbiased estimator where the periodic unbiasedness is defined by using lehmann unbiasedness it is shown that if there exists a periodic unbiased estimator which achieves the bound then the maximum likelihood produces it the periodic crb and the performance of some periodic unbiased estimators are compared in terms of mspe in a linear gaussian problem with modulo measurements and for phase estimation problems

convergence acceleration for the numerical solution of second order linear recurrence relations the concept of modification used for accelerating the convergence of ordinary continued fractions is adapted to the case of the gautschi aggarwal burgmeier algorithm for the computation of nondominant solutions of nonhomogeneous second order linear recurrence relations

instantaneous frequency estimation of quadratic and cubic fm signals using the cross polynomial wigner ville distribution the cross polynomial wigner ville distribution x pwvd is applied to instantaneous frequency if estimation of polynomial fm signals the x pwvd is used in a simple iterative algorithm similar to the one proposed by boashash and o shea see ibid vol 41 no 3 p 1439 1993 for the cross wigner ville distribution x wvd the advantage of the x pwvd over the polynomial wigner ville distribution is that the former is linear with respect to the observed signal and as a result it attains the cramer rao bound at lower signal to noise ratios snrs

frequency rate estimation at high snr the problem of estimating the frequency rate of change of complex valued frequency modulated signals from noisy observations is considered the performance of four related estimators is studied both analytically and by means of simulations and their relationship to the estimators proposed by djuric and kay 1990 and lang and musicus 1989 is established

improved approximate time domain ml estimators of chirp signal parameters and their performance analysis we derive an improved approximate maximum likelihood ml estimator for the parameters of a single component chirp signal in the time domain which has much lower computational complexity than that of the traditional estimators in the frequency domain we term it the zero order amplitude weighted phase based estimator awpe it is a weighted linear combination of the phases of the received signal samples where the weights are determined by the amplitudes of the received signals it achieves better performance than the only existing time domain estimator in the literature that only exploits the phase information a phase unwrapping pu algorithm is needed to recover the phase information correctly from the received signal samples several pu algorithms are compared and analyzed a new robust pu algorithm based on the first order phase differences of the consecutive received signal samples is proposed this new unwrapping algorithm has smaller pu failure probability than that based directly on the phases of the received signal samples the need for pu can even be removed by using the second order phase differences of the consecutive received signal samples two additional structures for the ml based awpe using the phase differences are derived starting from using the zero order awpe obtained the first is based on the first order phase differences and the second on the second order phase differences when the pu algorithm works perfectly i e no phase unwrapping failure occurs the estimators based on phase differences do not perform as well as the original one based on absolute phase information due to the increased noise variance this is also applicable when the chirp parameters take on very small values however when no knowledge of the ranges of the chirp parameters is available and the possible pu failures are taken into account the estimators based on the phase differences could perform better performance analysis of the estimators is given based on an improved phase noise model it is easy to verify that the estimates obtained are unbiased the cramer rao lower bound crlb on the mean square error mse is derived the mse performance of the estimators approaches the crlb at high signal to noise ratios the variance performance of the estimators based on phase differences is analyzed whose validity is demonstrated by simulation results

a robust wrap reduction algorithm for fringe projection profilometry and applications in magnetic resonance imaging in this paper we present an effective algorithm to reduce the number of wraps in a 2d phase signal provided as input the technique is based on an accurate estimate of the fundamental frequency of a 2d complex signal with the phase given by the input and the removal of a dependent additive term from the phase map unlike existing methods based on the discrete fourier transform dft the frequency is computed by using noise robust estimates that are not restricted to integer values then to deal with the problem of a non integer shift in the frequency domain an equivalent operation is carried out on the original phase signal this consists of the subtraction of a tilted plane whose slope is computed from the frequency followed by a re wrapping operation the technique has been exhaustively tested on fringe projection profilometry fpp and magnetic resonance imaging mri signals in addition the performance of several frequency estimation methods has been compared the proposed methodology is particularly effective on fpp signals showing a higher performance than the state of the art wrap reduction approaches in this context it contributes to canceling the carrier effect at the same time as it eliminates any potential slope that affects the entire signal its effectiveness on other carrier free phase signals e g mri is limited to the case that inherent slopes are present in the phase data

asymptotic cramer rao bounds for estimation of the parameters of damped sine waves in noise the problem of estimating the parameters of a signal composed of several damped sine waves in noise has applications for example in transient signal analysis explicit closed form expressions are derived for the cramer rao bound of this problem in the case of low damping low spectral interference between the signal components and a sufficiently large number of samples among several conclusions which are drawn from the expressions it is found that the estimation problem is much more sensitive to errors in the damping factor and frequency than to errors in amplitude and phase

phase based time domain estimation of the frequency and phase of a single sinusoid in awgn the role and applications of the additive observation phase noise model this paper presents the theoretical foundation for time domain phase based estimation of the frequency and phase of a single sinusoid in additive white gaussian noise awgn analogous to the theoretical foundation provided by rife and boorstyn for frequency domain fourier transform based estimation it is shown from the maximum a posteriori probability map and the maximum likelihood ml estimation principles that with the additive observation phase noise aopn due to the awgn being described by its a posteriori distribution conditioned on the received signal magnitude the received signal phase is a sufficient statistic for estimating the single sinusoid angle parameters using a geometric approach the exact statistical model for the aopn is derived where the a posteriori probability density function pdf and the corresponding a priori pdf are given by explicit closed form expressions that are valid for arbitrary signal to noise ratios snrs the a posteriori pdf is tikhonov and is of particular interest as it establishes the aopn model for phase based frequency phase map ml estimation in the time domain it is further illustrated that the results derived can yield various aopn models as special cases and the underlying physical insights and interconnections that exist among these models are revealed it is shown that the model derived by tretter is an ultimate specialization in the high snr limit of the aopn models developed here for high snr the a posteriori tikhonov pdf can be accurately approximated by a gaussian distribution which leads to the best linearized aopn model the applications of these aopn models to the design of linear estimators including the linear minimum mean square error lmmse estimator the linear minimum variance estimator and the lmmse implementation of the weighted phase averager are presented and their estimation performances are compared through computer simulations with the cramer rao lower bound crlb and the bayesian crlb as the benchmark to facilitate estimator design the a priori statistical models of the frequency and phase are proposed from the information theoretic perspective and an improved phase unwrapping algorithm over that given by fu and kam is presented it is shown that by incorporating all the information available in the aopn the estimation accuracy can be much improved

estimating signal parameters using the nonlinear instantaneous least squares approach a novel method for signal parameter estimation is presented termed the nonlinear instantaneous least squares nils estimator the basic idea is to use the observations in a sliding window to compute an instantaneous short term estimate of the amplitude used in the separated nonlinear least squares nlls criterion the effect is a significant improvement of the numerical properties in the criterion function which becomes well suited for a signal parameter search for small sized sliding windows the global minimum in the nlis criterion function is wide and becomes easy to find for maximum size windows the nils is equivalent to the nlls estimator which implies statistical efficiency for gaussian noise a blind signal parameter search algorithm that does not use any a priori information is proposed the nils estimator can be interpreted as a signal subspace projection based algorithm moreover the nils estimator can be interpreted as an estimator based on the prediction error of a structured linear predictor hereby a link is established between nlls signal subspace fitting and linear prediction based estimation approaches the nils approach is primarily applicable to deterministic signal models specifically polynomial phase signals are studied and the nils approach is evaluated and compared with other approaches simulations show that the signal to noise ratio snr threshold is significantly lower than that of the other methods and it is confirmed that the estimates are statistically efficient just as the nlls approach the nils estimator can be applied to nonuniformly sampled data

subspace approach for fast and accurate single tone frequency estimation a new signal subspace approach for estimating the frequency of a single complex tone in additive white noise is proposed in this correspondence our main ideas are to use a matrix without repeated elements to represent the observed signal and exploit the principal singular vectors of this matrix for frequency estimation it is proved that for small error conditions the frequency estimate is approximately unbiased and its variance is equal to cramer rao lower bound computer simulations are included to compare the proposed approach with the generalized weighted linear predictor periodogram and phase based maximum likelihood estimators in terms of estimation accuracy computational complexity and threshold performance

a fast and accurate single frequency estimator a novel frequency estimator for a single complex sinusoid in complex white gaussian noise is proposed the estimator is more computationally efficient that the optimal maximum likelihood estimator yet attains equally good performance at moderately high signal to noise ratios the estimator is shown to be related to the linear prediction estimator this relationship is used to reveal why the linear prediction estimator does not attain the cramer rao bound even at high signal to noise ratios

high snr asymptotics for signal subspace methods in sinusoidal frequency estimation high snr limit second order properties of multiple signal classification music minimum norm mn and subspace rotation sur signal subspace methods for sinusoidal frequency estimation are discussed an alternative to large sample analysis of the methods is presented the two most important variants of these methods are considered in connection with the choice of the sample covariance matrix the simpler technique follows the principle of a linear prediction and the more complex one is based on the idea of a forward backward prediction explicit expressions for the high snr covariance elements of the estimation errors associated with all the methods are derived the expressions for the covariances are used to analyze and compare the statistical performances of music mn and sur estimation methods in both of the variants to discuss the problem of the optimal dimension of the data covariance matrix and to study the limit statistical efficiency of the methods performances of the large sample and high snr asymptotics derived using monte carlo simulations are presented

how to correct susceptibility distortions in spin echo echo planar images application to diffusion tensor imaging diffusion tensor imaging is often performed by acquiring a series of diffusion weighted spin echo echo planar images with different direction diffusion gradients a problem of echo planar images is the geometrical distortions that obtain near junctions between tissues of differing magnetic susceptibility this results in distorted diffusion tensor maps to resolve this we suggest acquiring two images for each diffusion gradient one with bottom up and one with top down traversal of k space in the phase encode direction this achieves the simultaneous goals of providing information on the underlying displacement field and intensity maps with adequate spatial sampling density even in distorted areas the resulting dt maps exhibit considerably higher geometric fidelity as assessed by comparison to an image volume acquired using a conventional 3d mr technique

frequency estimation of a single complex sinusoid using a generalized kay estimator this letter introduces a generalized version of kay s estimator for the frequency of a single complex sinusoid in complex additive white gaussian noise the kay estimator is a maximum likelihood ml estimator at high signal to noise ratio snr based on differential phase measurements with a delay of one symbol interval in this letter the corresponding ml estimator with an arbitrary delay in the differential phase measurements is derived the proposed estimator reduces the variance at low snr compared with kay s original estimator for certain delay values explicit expressions for the window function and the corresponding high snr variance of the proposed generalized kay gk estimator are presented furthermore for some delay values the window function is nearly uniform and the implementation complexity is reduced compared with the original kay estimator for a delay value of two we show that the variance at asymptotically high snr approaches the cramer rao bound as the sequence length tends to infinity we also explore the effect of exchanging the order of summation and phase extraction for reduced complexity reasons the resulting generalized weighted linear predictor estimator and the gk estimator are compared with both autocorrelation based and periodogram based estimators in terms of computational complexity estimation range and performance at both low and high snrs

spatiotemporal approach for time varying global image motion estimation image motion estimation using the spatiotemporal approach has largely relied on the constant velocity assumption and thus becomes inappropriate when the velocity of the imaged scene or the camera changes during the data acquisition time using a polynomial or a trigonometric polynomial model for the time variation of the image motion spatiotemporal algorithms are developed in this paper to handle time varying but space invariant motion under these models it is shown that time varying image motion estimation is equivalent to parameter estimation of one dimensional 1 d polynomial phase or phase modulated signals which allows one to exploit well established results in radar signal processing when compared with alternative approaches the resulting motion estimation algorithms produce more accurate estimates simulation results are provided to demonstrate the proposed schemes

an exact direct method of sinusoidal parameter estimation derived from finite fourier integral of differential equation in this paper we propose a novel method for estimating the parameters frequency amplitude and phase of real sinusoids to derive the estimator we start from the characteristic differential equation of a sinusoid to remove differentials and obtain an algebraic relation for frequency we introduce finite period weighted integrals of the differential equation which become equivalent to the differential equation when a sufficient number of weight functions are applied as weight functions we show that fourier kernels have excellent properties terms related to integral boundaries are readily eliminated observations are provided by fourier coefficients and the relation becomes independently accurate for multiple sinusoids if they are sufficiently spaced we solve the obtained equations in two ways one is for approaching to the cramer rao lower bound crlb and the other is for enhancing the interference rejection capability also methods are proposed to calculate the weighted integrals from sampled signals with an improved accuracy proposed algorithms are examined under noise and sinusoidal interference error variances are compared with the crlb and other fast fourier transform fft based methods

frequency estimation by phase unwrapping single frequency estimation is a long studied problem with application domains including radar sonar telecommunications astronomy and medicine one method of estimation called phase unwrapping attempts to estimate the frequency by performing linear regression on the phase of the received signal this procedure is complicated by the fact that the received phase is wrapped modulo and therefore must be unwrapped before the regression can be performed in this paper we propose an estimator that performs phase unwrapping in the least squares sense the estimator is shown to be strongly consistent and its asymptotic distribution is derived we then show that the problem of computing the least squares phase unwrapping is related to a problem in algorithmic number theory known as the nearest lattice point problem we derive a polynomial time algorithm that computes the least squares estimator the results of various simulations are described for different values of sample size and snr

estimating the frequency of a noisy sinusoid by linear regression corresp the estimation of the parameters of a sinusoid from observations of signal samples corrupted by additive noise is investigated at high signal to noise ratios the additive noise is viewed as an equivalent phase noise suggesting frequency and phase estimation by linear regression on the signal phase the variances of the regression estimates are shown to achieve the cramer rao bounds a formula for the variance of the regression frequency estimator is derived in terms of the noise power spectrum a simple formula for the variance with 1 f 2 phase noise is presented

approximate maximum likelihood estimators for array processing in multiplicative noise environments we consider the problem of localizing a source by means of a sensor array when the received signal is corrupted by multiplicative noise this scenario is encountered for example in communications owing to the presence of local scatterers in the vicinity of the mobile or due to wavefronts that propagate through random inhomogeneous media since the exact maximum likelihood ml estimator is computationally intensive two approximate solutions are proposed originating from the analysis of the high and low signal to noise ratio snr cases respectively first starting with the no additive noise case a very simple approximate ml aml sub 1 estimator is derived the performance of the aml sub 1 estimator in the presence of additive noise is studied and a theoretical expression for its asymptotic variance is derived its performance is shown to be close to the cramer rao bound crb for moderate to high snr next the low snr case is considered and the corresponding aml sub 2 solution is derived it is shown that the approximate ml criterion can be concentrated with respect to both the multiplicative and additive noise powers leaving out a two dimensional 2 d minimization problem instead of a four dimensional 4 d problem required by the exact ml numerical results illustrate the performance of the estimators and confirm the validity of the theoretical analysis

analytic estimation of subsample spatial shift using the phases of multidimensional analytic signals in this correspondence a method of analytic subsample spatial shift estimation based on an a priori n d signal model is proposed the estimation uses the linear phases of n analytic signals defined with the multidimensional hilbert transform this estimation proposes i an analytic solution to the n d shift estimation and ii an estimation without processing complex cross correlation function or cross spectra between signals contrary to most phase shift estimators the method provides better performance in estimating subsample shifts than two classical estimators one using the maximum of cross correlation function and the other seeking the zero of the complex correlation function phase two delay estimators using the in phase and quadrature phase components of signals are also compared to our estimator like most estimators using the complex signal phases the estimator proposed herein presents the advantage of unaltered accuracy when low sampled signals are used moreover we show that this method can be applied to motion tracking with ultrasound images thus included in a block based motion estimation method and tested with ultrasound data this estimator provides an analytical solution to the translation estimation problem

estimation of complex single tone parameters in the dft domain in this correspondance a new method is presented that uses the model of the complex single tone in the dft domain to set a system of linear equations using 2 dft samples the solution of this linear system gives the value of 2 intermediary parameters from which it is easy to determine the frequency and the complex amplitude this deterministic method is then applied directly for parameter estimation in noisy conditions with appropriate choice of the dft samples the estimators are robust in poor snr conditions

map ml estimation of the frequency and phase of a single sinusoid in noise the problem of estimating the frequency and carrier phase of a single sinusoid observed in additive white gaussian noise is addressed much of the work in the literature considers maximum likelihood ml estimation however the ml estimator given by the location of the peak of a periodogram in the frequency domain shown in d c rife and r r boorstyn single tone parameter estimation from discrete time observations ieee transactions on information theory vol it 20 pp 591 598 sep 1974 has a very high computational complexity this paper derives the explicit structure of the ml estimator for data processing in the time domain assuming only reasonably high signal to noise ratio snr the result of this approximate ml estimator shows that both the phase and the magnitude of the noisy signal samples are utilized in the estimator and the phase data alone as assumed in s a tretter estimating the frequency of a noisy sinusoid by linear regression ieee transactions on information theory vol it 31 pp 832 835 nov 1985 and s kay a fast and accurate single frequency estimator ieee transactions on acoustics speech and signal processing vol 39 pp 1203 1205 may 1991 is not a sufficient statistic the sample by sample iterative processing nature of the estimator enables us to propose a novel recursive phase unwrapping algorithm that allows the estimator to be implemented efficiently to facilitate the performance analysis a new linearized observation model for the instantaneous signal phase that is more accurate than that of s a tretter estimating the frequency of a noisy sinusoid by linear regression ieee transactions on information theory vol it 31 pp 832 835 nov 1985 and of s kay a fast and accurate single frequency estimator vol 39 pp 1203 1205 may 1991 is proposed this new model explains physically why the phase data are weighted by the magnitude information in the ml estimator moreover by incorporating a priori knowledge via the a priori probability density function of the unknown frequency and the carrier phase the explicit structure of the approximate maximum a posteriori probability map estimator is derived and the bayesian cramer rao lower bound bcrlb on the mean square error mse is obtained our analysis shows that the mse performance of the map estimator can approach the bcrlb very closely

an efficient two dimensional frequency estimator the problem of two dimensional frequency estimation of a sinusoid embedded in complex white gaussian noise is addressed a frequency estimator and its statistical performance are derived the estimator is demonstrated to be an efficient estimator it achieves the cramer rao bound for high signal to noise ratios the results of a computer simulator which compares the variance with that of the maximum likelihood estimator and the theoretical two dimensional cramer rao bound are presented

improving a method for computing non dominant solutions of certain second order recurrence relations of poincare type in this paper we present a method of convergence acceleration for the calculation of non dominant solutions of second order linear recurrence relations for which the coefficients satisfy certain asymptotic conditions it represents an improvement of the method recently proposed by jacobsen and waadeland 3 4 for limit periodic continued fractions for continued fractions the method corresponds to a repeated application of the bauer muir transformation some examples and a generalization to non homogeneous recurrence relations are given

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

rapid carrier acquisition from baud rate samples maximum likelihood estimation of phase and frequency offset involves the maximization of a nonlinear likelihood function we develop an estimator for carrier acquisition by linearizing the likelihood function and show that its performance is close to the cramer rao bound crb the estimator is then extended to blind frequency acquisition and combined with testing to eliminate unlikely hypotheses with proper modeling of the likelihood resulting from the blind search simulations show that the algorithm quickly rejects all but the correct frequency hypothesis

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

improved cramer rao lower bounds for phase and frequency estimation with m psk signals we derive new cramer rao lower bounds crbs for the estimation of carrier phase and frequency offset from an unmodulated carrier or from a m psk data signal the new crbs are obtained formally from the exact likelihood function for the carrier phase and frequency offset build from a block of received phase samples and are applicable for a general alphabet size m the bounds are compared with other previously obtained bounds which also take into account the effect of random phase modulation and with the performance of some popular feedforward algorithms for carrier phase and frequency offset estimation

new robust 3 d phase unwrapping algorithms application to magnetic field mapping and undistorting echoplanar images abstract the phase as well as the magnitude of mri images can carry useful information it may be used to encode flow or temperature or to map the magnetic field for the undistorting of epis and automated shimming in all cases we measure the extra spin given to nuclei unfortunately we can only measure the final phase of the spins the rotation is wrapped into the range and to obtain a measure of the parameter of interest the missing multiples of 2 must be replaced a process known as phase unwrapping while simple in principle standard phase unwrapping algorithms fail catastrophically in the presence of even small amounts of noise here we present a new algorithm for robust three dimensional phase unwrapping in which unwrapping is guided so that it initially works on less noisy regions we test the algorithm on simulated phase data and on maps of magnetic field which were then used to successfully undistort epi images the unwrapping algorithm could be directly applied to other kinds of phase data

an algorithm for nondominant solutions of linear second order inhomogeneous difference equations an algorithm is given for computing a weighted sum of a nondominant solution of a linear second order inhomogeneous difference equation to a prescribed accuracy by estimating the truncation error the present method is an extension of both the stable numerical method due to olver and sookne and a summation technique due to deuflhard for computing minimal solutions of a homogeneous difference equation the method is illustrated by numerical examples

quasi ml period estimation from incomplete timing data given a noisy sequence of possibly shifted integer multiples of a certain period it is often of interest to accurately estimate the period with known integer regressors the problem is classical linear regression in many applications however the regressors are unknown integers and only loose bounds on the period are available examples include hop period and timing estimation wherein hops may be missed at the output of the frequency discriminator or the emitter may hop out of band pulse repetition interval pri analysis and passive rotating beam radio scanning we study several pertinent period estimators our emphasis is on a quasi maximum likelihood approach developed herein and an earlier method based on the fourier transform of a dirac delta train representation of the data surprisingly both are capable of attaining the clairvoyant crame spl acute r rao bound at moderate signal to noise ratios snrs even for short e g 10 samples we carefully address parameter identifiability issues and corroborate our findings with extensive simulations

outlier probability of generalized chirpogram based estimators for multicomponent polynomial phase signals outliers are often encountered by estimators that have a nonlinear nature in this paper the probability of outliers is derived concerning the generalized chirpogram gc based estimators for polynomial phase signals pps to derive the approximate theoretical expressions we develop an approach that is based on the extrema theory of complex gaussian envelope and the asymptotic poisson model the semi aliasing effect of pps is also considered in order to predict the outliers more accurately then expressions of the probability of outliers are presented for several typical cases of pps including single tone and chirp signals and their two component versions the gc with the hamming windowing is also incorporated numerical examples of computer simulations are also provided to show the accuracy of the theoretical expressions