adaptive detection in gaussian interference with unknown covariance after reduction by invariance in this paper we address the problem of adaptive detection in homogeneous gaussian interference with unknown covariance matrix after reduction by invariance starting from a maximal invariant statistic which contains all the information for the synthesis of an invariant detector we devise the rao test generalized likelihood ratio test glrt and durbin test moreover we compare their decision statistics with those of the receivers designed according to the same criteria from the raw data i e before reduction by invariance we prove that the glrt in the original data space is statistically equivalent to the glrt designed after reduction by invariance under a very mild assumption and coincide with the conditional uniformly most powerful invariant c umpi test obtained conditioning on an ancillary part of the maximal invariant statistic as to the rao and durbin criteria when they are applied after reduction by invariance lead to detectors different from the counterparts devised in the raw data domain at the analysis stage the performance of the receivers devised in the invariant domain is analyzed in comparison with that of the counterparts synthesized from the original observations

differential space time modulation over frequency selective channels we present herein a new differential space time frequency dstf modulation scheme for systems that are equipped with an arbitrary number of transmit antennas and operate in frequency selective channels the proposed dstf modulator consists of a concatenating spectral encoder and differential encoder that offer full spatio spectral diversity and significant coding gain a unitary structure is imposed on the differential encoder to admit linear decoupled maximum likelihood ml detection in space and time optimum criteria based on pairwise error probability analysis are developed for spectral encoder design we introduce a class of spectral codes namely linear constellation decimation lcd codes which are nonbinary block codes obtained by decimating a phase shift keying psk constellation with a group of decimation factors that are co prime with the constellation size since lcd codes encode across a minimally necessary set of subchannels for full diversity they incur modest decoding complexity among all full diversity codes numerical results are presented to illustrate the performance of the proposed dstf modulation and coding scheme which compares favorably with several existing differential space time schemes in frequency selective channels

time varying autoregressive moving average models for covariance estimation we consider large scale covariance estimation using a small number of samples in applications where there is a natural ordering between the random variables the two classical approaches to this problem rely on banded covariance and banded inverse covariance structures corresponding to time varying moving average ma and autoregressive ar models respectively motivated by this analogy to spectral estimation and the well known modeling power of autoregressive moving average arma processes we propose a novel time varying arma covariance structure similarly to known results in the context of ar and ma we address the completion of an arma covariance matrix from its main band and its estimation based on random samples finally we examine the advantages of our proposed methods using numerical experiments

a fast algorithm for apes and capon spectral estimation in this paper we derive a novel implementation of some very computationally demanding matched filter bank based spectral estimators namely the amplitude and phase estimator apes the amplitude spectrum capon asc estimator and the power spectrum capon psc estimator filter bank based spectral estimation methods that adopt data dependent filter banks can provide spectra characterized by a significantly improved resolution compared to classical approaches however the computational complexity of the currently available implementation algorithms is extremely high a novel technique is introduced that provides efficient algorithms for the computation of the apes asc and psc spectra the proposed method is based on suitable displacement representations of all pertinent data matrices that are subsequently utilized for the computation of the associated complex valued polynomials the computational complexity of the proposed algorithms is lower than all relevant existing methods

iaa spectral estimation fast implementation using the gohberg semencul factorization we consider fast implementations of the weighted least squares based iterative adaptive approach iaa for one dimensional 1 d and two dimensional 2 d spectral estimation of uniformly sampled data iaa is a robust user parameter free and nonparametric adaptive algorithm that can work with a single data sequence or snapshot compared to the conventional periodogram iaa can be used to significantly increase the resolution and suppress the sidelobe levels however due to its high computational complexity iaa can only be used in applications involving small sized data we present herein novel fast implementations of iaa using the gohberg semencul g s type factorization of the iaa covariance matrices by exploiting the toeplitz structure of the said matrices we are able to reduce the computational cost by at least two orders of magnitudes even for moderate data sizes

efficient algorithms for adaptive capon and apes spectral estimation in this paper fast algorithms for adaptive capon and amplitude and phase estimation apes methods for spectral analysis of time varying signals are derived fast stable nonrecursive formulae are derived based on time shifting properties of the pertinent variables as a consequence efficient frequency domain recursive least squares rls based as well as fast rls based algorithms for the adaptive estimation of the power spectra are developed stability issues of the frequency domain estimators are considered and stabilization procedures are proposed the computational complexity of the proposed algorithms is lower than relevant existing methods the performance of the proposed algorithms is demonstrated through extensive simulations

adaptive detection of point like targets in spectrally symmetric interference we address adaptive radar detection of targets embedded in ground clutter dominated environments characterized by a symmetrically structured power spectral density at the design stage we leverage on the spectrum symmetry for the interference to come up with decision schemes capable of capitalizing the a priori information on the covariance structure to this end we prove that the detection problem at hand can be formulated in terms of real variables and then we apply design procedures relying on the glrt the rao test and the wald test specifically the estimates of the unknown parameters under the target presence hypothesis are obtained through an iterative optimization algorithm whose convergence and quality guarantee is thoroughly proved the performance analysis both on simulated and on real radar data confirms the superiority of the considered architectures over their conventional counterparts which do not take advantage of the clutter spectral symmetry

cubic forms in 14 variables the result can be rephrased in geometric language to say that any projective cubic hypersurface defined over q of dimension at least 12 has a q point davenport s result was extended to arbitrary number fields by pleasants 9 and it would be interesting to know whether theorem 1 could similarly be extended these results can be seen as an attempt to extend the classical theorem of meyer 1884 from quadratic forms to cubic forms meyer showed that any indefinite quadratic from over z in 5 or more variables must represent zero indeed meyer s result was generalized by minkowski who showed that a quadratic form over z in any number of variables represents zero if and only if it represents zero over every completion of q it is a well known fact that this local condition is automatically satisfied for qp as soon as n 5 thus meyer s result requires only the condition of indefiniteness the analogous fact for cubic forms is that p adic zeros exist whenever n 10 see davenport 6 chapter 18 for example of course the condition for r holds for any n in this case thus it is natural to conjecture that theorem 1 should hold as soon as n 10 indeed for smaller values of n one might expect that it should suffice for the form c x to represent zero in each field qp however when n 3 or 4 it is possible for a cubic form to have zeros in every completion of q without there being a global zero this is shown by the examples

on output error methods for system identification in this paper are derived consistency and asymptotic normality results for the output error method of system identification the output error estimator has the advantage over the prediction error estimator of being more easily computable however it is shown that the output error estimator can never be more efficient than the prediction error estimator the main result of the paper provides necessary and sufficient conditions for the output error estimator and the prediction error estimator to have the same efficiency irrespective of the spectral density of the noise process

efficient least squares fir system identification if finite impulse response fir system identification is performed by minimizing the squared error between the measured system output and an estimate from an fir system output a set of least squares normal equations to be solved for the fir system coefficients is obtained if the assumed fir system is of duration m samples the usual solution for the m least squares simultaneous equations requires a number of computational operations proportional to m3and storage of normal equation coefficients proportional to m2 the set of normal equations has an underlying structure however that can be exploited to yield a solution with computational operations proportional to m2and storage proportional to m such an efficient algorithmic solution is presented here

a fast algorithm for maximum likelihood estimation of harmonic chirp parameters the analysis of approximately periodic signals is an important element in numerous applications one generalization of standard periodic signals often occurring in practice is harmonic chirp signals where the instantaneous frequency increases decreases linearly as a function of time a statistically efficient estimator for extracting the parameters of the harmonic chirp model in additive white gaussian noise is the maximum likelihood ml estimator which recently has been demonstrated to be robust to noise and accurate even when the model order is unknown the main drawback of the ml estimator is that only very computationally demanding algorithms for computing an estimate are known in this paper we give an algorithm for computing an estimate to the ml estimator for a number of candidate model orders with a much lower computational complexity than previously reported in the literature the lower computational complexity is achieved by exploiting recursive matrix structures including a block toeplitz plus hankel structure the fast fourier transform and using a two step approach composed of a grid and refinement step to reduce the number of required function evaluations the proposed algorithms are assessed via monte carlo and timing studies the timing studies show that the proposed algorithm is orders of magnitude faster than a recently proposed algorithm for practical sizes of the number of harmonics and the length of the signal

exploring super gaussianity toward robust information theoretical time delay estimation time delay estimation tde is a fundamental component of speaker localization and tracking algorithms most of the existing systems are based on the generalized cross correlation method assuming gaussianity of the source it has been shown that the distribution of speech captured with far field microphones is highly varying depending on the noise and reverberation conditions thus the performance of tde is expected to fluctuate depending on the underlying assumption for the speech distribution being also subject to multi path reflections and competitive background noise this paper investigates the effect upon tde when modeling the source signal with different speech based distributions an information theoretical tde method indirectly encapsulating higher order statistics hos formed the basis of this work the underlying assumption of gaussian distributed source has been replaced by that of generalized gaussian distribution that allows evaluating the problem under a larger set of speech shaped distributions ranging from gaussian to laplacian and gamma closed forms of the univariate and multivariate entropy expressions of the generalized gaussian distribution are derived to evaluate the tde the results indicate that tde based on the specific criterion is independent of the underlying assumption for the distribution of the source for the same covariance matrix

counting rational points on hypersurfaces for any n geq 2 let f in mathbb z x 1 ldots x n be a form of degree d geq 2 which produces a geometrically irreducible hypersurface in mathbb p n 1 this paper is concerned with the number n f b of rational points on f 0 which have height at most b for any varepsilon 0 we establish the estimate n f b o b n 2 varepsilon whenever either n leq 5 or the hypersurface is not a union of lines here the implied constant depends at most upon d n and varepsilon

computationally efficient capon and apes based coherence spectrum estimation the coherence spectrum is of notable interest as a bivariate spectral measure in a variety of application and the topic has lately attracted interest with the recent formulation of several high resolution data adaptive estimators in this work we further this development with the presentation of computationally efficient implementations of the capon and apes based msc estimators the presented implementations furthers the recent development of exploiting the estimators inherently low displacement rank of the necessary products of toeplitz like matrices to include also the required cross correlation covariance matrices for the mentioned coherence algorithms numerical simulations together with theoretical complexity measures illustrate the performance of the proposed implementations

computationally efficient two dimensional capon spectrum analysis we present a computationally efficient algorithm for computing the 2 d capon 1969 spectral estimator the implementation is based on the fact that the 2 d data covariance matrix will have a toeplitz block toeplitz structure with the result that the inverse covariance matrix can be expressed in closed form by using a special case of the gohberg heinig 1974 formula that is a function of strictly the forward 2 d prediction matrix polynomials furthermore we present a novel method based on a 2 d lattice algorithm to compute the needed forward prediction matrix polynomials and discuss the difference in the so obtained 2 d spectral estimate as compared with the one obtained by using the prediction matrix polynomials given by the whittle wiggins robinson 1963 1965 algorithm numerical simulations illustrate the improved resolution as well as the clear computational gain in comparison to both the well known classical implementation and the method published by liu et al see ieee trans aerosp electron syst vol 34 p 1314 19 1998

a fast algorithm for the inversion of block toeplitz matrices a technique to solve toeplitz matrices has been generalized to solve block toeplitz matrices this method is more efficient than the levinson recursion for moderately large matrix and block sizes this algorithm is in many cases preferable to cyclically preconditioned conjugate gradient algorithms

superfast approximative implementation of the iaa spectral estimate in this correspondence we develop superfast approximative one dimensional algorithms for the computationally efficient implementation of the recent iterative adaptive approach iaa spectral estimate the proposed methods are based on rewriting the iaa algorithm with suitable gohberg semencul representations solving the resulting linear systems of equations using the preconditioned conjugate gradient method where a novel preconditioning is applied using an incomplete factorization of the toeplitz matrix numerical simulations illustrate the efficiency of both the proposed preconditioning as well as the overall algorithm offering a computational reduction of up to two orders of magnitude as compared to our recently proposed efficient and exact iaa implementation

on the statistical invariance for adaptive radar detection in partially homogeneous disturbance plus structured interference this paper deals with the problem of adaptive vector subspace signal detection in partially homogeneous gaussian disturbance and structured unknown deterministic interference within the framework of invariance theory it is first proved that the maximal invariant statistic mis for the problem at hand is scalar valued and coincides with the well known adaptive normalized matched filter evaluated after data projection in the complementary subspace of the interfering signal second the statistical characterization of the mis under both hypotheses is derived then it is shown the statistical equivalence of two step generalized likelihood ratio test rao and wald tests as well as the more recently considered durbin and gradient test to the above statistic finally simulation results are provided to confirm our findings and analyze the performance trend of the mis with the relevant parameters

modified rao test for multichannel adaptive signal detection the problem of detecting a subspace signal is studied in colored gaussian noise with an unknown covariance matrix in the subspace model the target signal belongs to a known subspace but with unknown coordinates we first present a new derivation of the rao test based on the subspace model and then propose a modified rao test mrt by introducing a tunable parameter the mrt is more general which includes the rao test and the generalized likelihood ratio test as special cases moreover closed form expressions for the probabilities of false alarm and detection of the mrt are derived which show that the mrt bears a constant false alarm rate property against the noise covariance matrix numerical results demonstrate that the mrt can offer the flexibility of being adjustable in the mismatched case where the target signal deviates from the presumed signal subspace in particular the mrt provides better mismatch rejection capacities as the tunable parameter increases

on the invariance coincidence and statistical equivalence of the glrt rao test and wald test three common techniques to discriminate between alternatives in a binary hypothesis testing problem are the generalized likelihood ratio test glrt the rao test and the wald test in this paper we investigate some characteristics of the corresponding decision statistics and provide their expressions for some problems of particular interest in statistical signal processing first of all we focus on the invariance of the rao and wald tests with respect to transformations leaving the testing problem unaltered then we introduce necessary and sufficient conditions in order for their decision statistics to coincide with twice the logarithm of the glrt statistic finally we present some detection problems usually encountered in practical signal processing applications where the decision variables of the three quoted tests are equivalent namely related by strictly monotonic transformations

a diagonal growth curve model and some signal processing applications we consider a variation of the growth curve gc model referred to as the diagonal growth curve dgc model where the steering vectors and waveforms are both known and the complex amplitude matrix is constrained to be diagonal a closed form approximate maximum likelihood aml estimator for this model is derived based on the maximum likelihood principle we analyze the statistical properties of this method theoretically and show that the aml estimate is unbiased and asymptotically statistically efficient for a large snapshot number via several numerical examples in array signal processing and spectral analysis we also show that the proposed aml estimator can achieve better estimation accuracy and exhibit greater robustness than the best existing methods

invariant adaptive detection of range spread targets under structured noise covariance the invariance principle is adopted to develop an exhaustive study for adaptive detection of range spread targets in gaussian noise sharing a block diagonal covariance structure for this problem the usual generalized likelihood ratio principle is intractable in this paper we first determine the largest group of affine transformations that does not alter the decision problem then a maximal invariant identified by this group is derived which can characterize the totality of the invariant detectors and extends the existing results for the point target case a theoretical performance analysis of the maximal invariant is also given finally we propose two classes of invariant detectors which are distinguished by whether a constant false alarm rate cfar is maintained numerical experiments are provided for a comparison of the proposed detectors where a tradeoff between the cfarness and the improved performance has been observed and studied

structured matrices and newton s iteration unified approach abstract recent progress in the study of structured matrices shows advantages of unifying the treatment of various classes of such matrices we recall some fundamental techniques for such a unification and then specify it in full details for newton s iteration which rapidly improves an initial approximation to the inverse matrix by performing two matrix multiplications per recursive step the iteration is particularly suitable for n n structured matrices represented with o n entries of their short generators rather than with their own n 2 entries based on such a representation matrix operations are performed much more rapidly and use much less memory space a major problem is to control the length of the generators which tends to grow quite rapidly in the iterative process two known methods solve this problem for toeplitz like and cauchy like matrices we extend both methods to a more general class of structured matrices and estimate the convergence rate as well as the computational complexity some novel techniques are introduced in this study in particular for the estimation of the norms of the inverse displacement operators

analyticity properties of the feigenbaum function analyticity properties of the feigenbaum function a solution ofg x 1g g x withg 0 1 g 0 0 g 0 0 are investigated by studying its inverse function which turns out to be herglotz or anti herglotz on all its sheets it is found thatg is analytic and uniform in a domain with a natural boundary

robust adaptive parameter estimation of sinusoidal signals a novel two step adaptive identification framework is proposed for sinusoidal signals to estimate the unknown offset amplitude frequency and phase where only the output measurements are used after representing the sinusoidal signal as a linearly parameterized form several adaptive laws are developed the proposed adaptive laws are driven by parameter estimation error information that is derived by applying filter operations on the output measurements so that globally exponential convergence of the parameter estimation is proved by using the sliding mode technique we further improve the design of adaptations to achieve finite time ft parameter estimation the proposed approaches are independent of any observer predictor design and robust to bounded measurement noises the developed estimation methods are finally extended to the full parameter estimation of multi sinusoids with only output measurements comparative simulation results are provided to illustrate the efficacy of the proposed methods

adaptive radar detectors based on the observed fim modified versions of rao wald and durbin tests are considered exploiting an estimator of the fisher information matrix fim in place of the exact one they are asymptotically equivalent under some technical conditions to the standard counterparts and rely on the use of the observed fim ofim which is proportional to the negative hessian of the log likelihood the developed framework is applied to the problem of adaptive radar detection of a point like target in homogeneous or partially homogeneous interference remarkably for both the scenarios it is shown that rao wald and durbin tests with ofim are statistically equivalent to the generalized likelihood ratio test glrt for the specific detection problem namely kelly s detector for the homogeneous environment and the adaptive coherence estimator ace s kraut and l l scharf the cfar adaptive subspace detector is a scale invariant glrt ieee trans signal process vol 47 no 9 pp 2538 2541 sep 1999 also known as adaptive normalized matched filter anmf e conte m lops and g ricci asymptotically optimum radar detection in compound gaussian clutter ieee trans aerosp electron syst vol 31 no 2 pp 617 625 apr 1995 for the partially homogeneous scenario this provides a new interpretation of the mentioned glrts laying the foundations for a better understanding of their theoretical validity

efficient order recursive algorithms for multichannel least squares filtering four efficient order recursive algorithms for least squares ls multichannel fir filtering and multivariable system identification are developed the need for such algorithms arises when the system model assigns an unequal number of delay elements to each input channel all proposed schemes provide considerable improvements over overparametrization or the zero padding approach first a block structures algorithm is derived it operates on boxes or blocks whose dimensions successively increase until their size equals the number of input channels as a result it requires linear system solvers and matrix multiplications the second algorithm manages to get free of block operations by proper decomposition of each block step involved in the first method into a number of scalar steps equal to the size of the block the third and the fourth algorithms provide highly concurrent alternatives that reduce processing time by an order magnitude an illustrative example from multichannel autoregressive spectral estimation is supplied

on the hasse principle for quartic hypersurfaces we establish the hasse principle for smooth projective quartic hypersurfaces of dimension greater than or equal to 28 defined over q

estimation of sinusoidal frequencies and amplitudes using adaptive identifier and observer a simple method is proposed for estimation of amplitudes of multiple sinusoids the estimation is based on the existing adaptive identifier which offers the globally convergent estimate of sinusoidal frequencies to deal with possible singularities of the amplitude estimation adaptive observers are also proposed for estimation of sinusoidal amplitudes simulations illustrate the results

period doubling in maps with a maximum of order z abstract we consider the scaling behaviour in period doubling systems exemplified by the one dimensional map n 1 1 n z which has a maximum of order z z 1 the feigenbaum scaling factors and are studied as functions of z and more generally the scaling functions 1 and a in particular using the universal functions g x and h x we establish the inequality

displacement structure of generalized inverse matrices abstract it is shown that matrices with a uv displacement structure possess generalized inverses with a vu displacement structure estimations for the displacement rank of the generalized inverses are presented the results apply to matrices of toeplitz vandermonde and cauchy type and provide formulas for generalized inverses which are important for fast algorithms

morphology of the period doubling universal function results are presented concerning the structure on the real line of the universal function which is the fixed point solution of the feigenbaum cvitanovic renormalization group equation associated with period doubling chaos in quadratic maps it is shown that the values which the function takes at its turning points can be algebraically characterized by relation to the infinite cycle associated with the original turning point of the map on the interval these extrema become increasingly numerous as the argument increases and their locations can be found progressively using knowledge of the previously determined extrema as well as providing a simple understanding of the structure of the universal function these simple observations may be of assistance in investigating the convergence or asymptotic behavior of approximations to the universal function and perhaps in providing a corrector step to some of these schemes

invariance theory for adaptive detection in interference with group symmetric covariance matrix this paper describes the adaptive detection of point like targets in a gaussian environment assuming a group symmetric structure for the interference covariance matrix this special configuration enforces block sparsity and permits splitting of the original observation space into lower dimensional subspaces each characterized by its own nuisance parameters hence the principle of invariance is invoked to select decision rules enjoying some symmetry defined by a suitable group of transformations which leave the decision problem unaltered all the invariant tests can be expressed as functions of a maximal invariant statistic whose derivation is among the key results of this paper finally some common design criteria are applied to come up with adaptive architectures that share invariance and their behavior is assessed to highlight the interplay between sample size and detection performance in comparison with conventional decision schemes

efficient recursive in order least squares fir filtering and prediction this paper is concerned with the efficient determination of the optimum in the least squares sense fir filter on the basis of data samples of the input and desired response signals by procedures recursive in the filter order this situation typically arises when no a priori statistics are available and the system order is not known the general multiinput multioutput multichannel case is considered here and a fast algorithm is presented requiring for single channel signals approximately 2s 15m multiplications mps per order m s being the number of samples in the special case of linear prediction it calls for about s 12m mps hence it offers a computational reduction of 5m and 2m mps in comparison to the methods of marple 1 and morf et al 2 respectively additionally the proposed scheme is inherently symmetric and is suited very well to initialization of fast sequential algorithms as well as algorithms searching for the optimum lag filter

metrics for vector quantization based parametric speech enhancement and separation speech enhancement and separation algorithms sometimes employ a two stage processing scheme wherein the signal is first mapped to an intermediate low dimensional parametric description after which the parameters are mapped to vectors in codebooks trained on for example individual noise free sources using a vector quantizer to obtain accurate parameters one must employ a good estimator in finding the parameters of the intermediate representation like a maximum likelihood estimator this leaves some unanswered questions however like what metrics to use in the subsequent vector quantization process and how to systematically derive them this paper aims at answering these questions metrics for this are presented and derived and their use is exemplified on a number of different signal models by deriving closed form expressions the metrics essentially take into account in the vector quantization process that some parameters may have been estimated more accurately than others and that there may be dependencies between the estimation errors

matched subspace detectors we formulate a general class of problems for detecting subspace signals in subspace interference and broadband noise we derive the generalized likelihood ratio glr for each problem in the class we then establish the invariances for the glr and argue that these are the natural invariances for the problem in each case the glr is a maximal invariant statistic and the distribution of the maximal invariant statistic is monotone this means that the glr test glrt is the uniformly most powerful invariant detector we illustrate the utility of this finding by solving a number of problems for detecting subspace signals in subspace interference and broadband noise in each case we give the distribution for the detector and compute performance curves

least zero of a cubic form an effective search bound is established for the least non trivial integer zero of an arbitrary cubic form c in mathbb z x 1 ldots x n provided that n 17

amplitude and frequency estimator of a sinusoid a dynamic estimator with global convergence property is proposed to simultaneously reconstruct the unknown values of the amplitude frequency and offset of a sinusoidal signal which is measured several versions of the estimator along with illustrative simulations are presented

a maximal invariant framework for adaptive detection with structured and unstructured covariance matrices we introduce a framework for exploring array detection problems in a reduced dimensional space by exploiting the theory of invariance in hypothesis testing this involves calculating a low dimensional basis set of functions called the maximal invariant the statistics of which are often tractable to obtain thereby making analysis feasible and facilitating the search for tests with some optimality property using this approach we obtain a locally most powerful invariant test for the unstructured covariance case and show that all invariant tests can be expressed in terms of the previously published kelly s generalized likelihood ratio glrt and robey s adaptive matched filter amf test statistics applying this framework to structured covariance matrices corresponding to stochastic interferers in a known subspace for which the glrt is unavailable we obtain the maximal invariant and propose several new invariant detectors that are shown to perform as well or better than existing ad hoc detectors these invariant tests are unaffected by most nuisance parameters hence the variation in the level of performance is sharply reduced this framework facilitates the search for such tests even when the usual glrt is unavailable

optimal invariant detection of a sinusoid with unknown parameters a uniformly most powerful test does not exist for the problem of detecting a sinusoid of unknown amplitude phase and frequency in complex white gaussian noise of unknown variance the problem is shown to be invariant to scale and modulation transformations using invariance principles it is found that the uniformly most powerful invariant umpi test does not exist a test is derived that is umpi if it is given that the signal to noise ratio snr is known for snr unknown the derived test can be used as an upper performance bound for any invariant test suboptimal tests are typically implemented such as the generalized likelihood ratio and locally most powerful tests which are also scale and modulation invariant it is shown analytically that as snr approaches zero the locally most powerful invariant test performance approaches the umpi performance bound and as the snr becomes large the generalized likelihood ratio test performance approaches the performance of the umpi bound computer simulation examples indicate that the generalized likelihood ratio test performance is close to the umpi bound particularly in the low probability of false alarm region of the receiver operating characteristic curve

model order selection rules for covariance structure classification in radar the adaptive classification of the interference covariance matrix structure for radar signal processing applications is addressed in this paper this represents a key issue because many detection architectures are synthesized assuming a specific covariance structure which may not necessarily coincide with the actual one due to the joint action of the system and environment uncertainties the considered classification problem is cast in terms of a multiple hypotheses test with some nested alternatives and the theory of model order selection mos is exploited to devise suitable decision rules several mos techniques such as the akaike takeuchi and bayesian information criteria are adopted and the corresponding merits and drawbacks are discussed at the analysis stage illustrating examples for the probability of correct model selection are presented showing the effectiveness of the proposed rules

efficient implementation of iterative adaptive approach spectral estimation techniques this paper presents computationally efficient implementations for several recent algorithms based on the iterative adaptive approach iaa for uniformly sampled one and two dimensional data sets considering both the complete data case and the cases when the data sets are missing samples either lacking arbitrary locations or having gaps or periodically reoccurring gaps by exploiting the method s inherent low displacement rank together with the development of suitable gohberg semencul representations and the use of data dependent trigonometric polynomials the proposed implementations are shown to offer a reduction of the necessary computational complexity by at least one order of magnitude numerical simulations together with theoretical complexity measures illustrate the achieved performance gain

new levinson schur and lattice type algorithms for linear phase filtering the ls estimate of the impulse response of fir filters with linear phase is known to be given by a structured linear set of equations three new algorithms are presented for the efficient solution of such systems one is the unwindowed extension of a previously derived levinson type structurally symmetric algorithm the second is a novel exact least squares lattice algorithm for time recursive processing the third is a schur type structurally symmetric algorithm with high degree of parallelism the new structures have considerably lower computational complexity and more parsimonious parametrization compared to previously derived algorithms this is because in contrast to the previously derived algorithms the new algorithms are developed so as to respect the symmetry which is intrinsic in the linear phase problem

reply to comment on parseval relationship of nonuniform samples of one and two dimensional signals the author presents a reply to the paper by zhu and leung ieee trans signal processing vol 42 no 3 p 2183 1994 which commented on the paper by marvasti and liu ieee trans acoust speech signal processing vol 38 no 6 p 1061 3 1990 the present author discusses a typographical error in the paper by marvasti and liu it is concluded that the theorem proofs are still correct and the conclusion still valid

amplitude estimation of sinusoidal signals survey new results and an application this paper considers the problem of amplitude estimation of sinusoidal signals from observations corrupted by colored noise a relatively large number of amplitude estimators which encompass least squares ls and weighted least squares wls methods are described additionally filterbank approaches which are widely used for spectral analysis are extended to amplitude estimation more exactly we consider the matched filterbank mafi approach and show that by appropriately designing the prefilters the mafi approach to amplitude estimation includes the wls approach the amplitude estimation techniques discussed in this paper do not model the observation noise and yet they are all asymptotically statistically efficient it is however their different finite sample properties that are of particular interest to this study numerical examples are provided to illustrate the differences among the various amplitude estimators although amplitude estimation applications are numerous we focus herein on the problem of system identification using sinusoidal probing signals for which we provide a computationally simple and statistically accurate solution

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

persymmetric adaptive detectors in homogeneous and partially homogeneous environments this paper addresses the problem of detecting a signal in partially homogeneous and homogeneous environments in partially homogeneous environments i e both the test data and training data share the same noise covariance matrix structure up to an unknown scaling factor a persymmetric adaptive coherence estimator per ace detector is proposed by exploiting the persymmetric structure of the covariance matrix the per ace can reduce training data requirements furthermore the expressions for the probabilities of false alarm and detection are derived along with the distribution of the loss factor in homogeneous environments a persymmetric adaptive matched filter per amf detector has been presented however its probability of detection has not been obtained yet thus we derive the expression for the probability of detection for both the per ace and per amf numerical results of these proposed expressions are confirmed with those of monte carlo trials in addition simulation results show that the proposed per ace outperforms the conventional ace in training limited scenarios

on circularity circularity is an assumption that was originally introduced for the definition of the probability distribution function of complex normal vectors however this concept can be extended in various ways for nonnormal vectors the first purpose of the paper is to introduce and compare some possible definitions of circularity from these definitions it is also possible to introduce the concept of circular signals and to study whether or not the spectral representation of stationary signals introduces circular components therefore the relationships between circularity and stationarity are analyzed in detail finally the theory of linear mean square estimation for complex signals exhibits some connections with circularity and it is shown that without this assumption the estimation theory must be reformulated

amplitude spectrum estimation for two dimensional gapped data the amplitude and phase estimation apes approach to nonparametric spectrum estimation of uniformly sampled data has received considerable interest we consider the extension of apes to gapped data i e uniformly sampled data with missing samples it has been shown that the apes estimate of the spectrum is the minimizer of a certain least squares ls criterion and our extension of apes is based on minimizing this ls criterion with respect to the missing data as well a computationally efficient method for doing this based on cyclic minimization and the conjugate gradient algorithm is proposed the new algorithm is called gapped data apes gapes and is developed for the two dimensional 2 d case with the one dimensional 1 d case as a special instance numerical examples are provided to demonstrate the performance of the algorithm and to show the advantages of 2 d data processing over 1 d row or column wise data processing as well as to show the applicability of the algorithm to synthetic aperture radar sar imaging

a superfast method for solving toeplitz linear least squares problems in this paper we develop a superfast o m n log 2 m n complexity algorithm to solve a linear least squares problem with an m n toeplitz coefficient matrix the algorithm is based on the augmented matrix approach the augmented matrix is further extended to a block circulant matrix and dft is applied this leads to an equivalent tangential interpolation problem where the nodes are roots of unity this interpolation problem can be solved by a divide and conquer strategy in a superfast way to avoid breakdowns and to stabilize the algorithm pivoting is used and a technique is applied that selects difficult points and treats them separately the effectiveness of the approach is demonstrated by several numerical examples 2003 elsevier science inc all rights reserved

testing parametric assumptions on band or time limited signals under noise this paper considers the problem of testing parametric assumptions on signals f from which only noisy observations y sub k f spl tau k spl epsi sub k are available and where the signal is assumed to be either band limited or time limited to this end the signal is reconstructed by an estimator based on the whittaker shannon ws sampling theorem with oversampling as test statistic the minimal l sub 2 distance between the estimated signal and the parametric model is used to construct appropriate tests the asymptotic distribution of the test statistic is derived both under the hypothesis of the validity of the parametric model and under fixed local alternatives as a byproduct the asymptotic distribution of the integrated square error of the estimator is computed which is of interest by itself e g for the analysis of a cross validated bandwidth selector

approximate subspace based iterative adaptive approach for fast two dimensional spectral estimation in this paper we devise a new approach for fast implementation of two dimensional 2 d iterative adaptive approach iaa using single or multiple snapshots our underlying idea is to apply the subspace methodology in this nonparametric technique by performing the iaa on the dominant singular vectors extracted from the singular value decomposition svd or higher order svd of the multidimensional observations in doing so 2 d iaa is approximately realized by multiple steps of 1 d iaa implying that computational attractiveness is achieved particularly for large data size number of grid points and or snapshot number algorithms based on matrix and tensor operations are developed and their implementation complexities are analyzed computer simulations are also included to compare the proposed approach with the state of the art techniques in terms of resolution probability spectral estimation performance and computational requirement

maximal invariants and performance of some invariant hypothesis tests for an adaptive detection problem maximal invariants for adaptive detection of a signal in unknown interference from multiple observations is derived given coherent samples from p sets of observations it is shown that a maximal invariant statistic for the detection problem is a 2p 1 dimensional vector comprising the eigenvalues of two hermitian positive definite matrices obtained from the data set two invariant detectors well known for p 1 are generalized for the case of multiple observations and closed form expressions for the probability of detection and probability of false alarm are derived along with the distributions of the signal to interference plus noise loss factors several novel invariant detectors are constructed from the maximal invariants and the receiver operating characteristics of the detectors compared

performance analysis of the total least squares esprit algorithm the asymptotic distribution of the estimation error for the total least squares tls version of esprit is derived the application to a uniform linear array is treated in some detail and a generalization of esprit to include row weighting is discussed the cramer rao bound crb for the esprit problem formulation is derived and found to coincide with the asymptotic variance of the tls esprit estimates through numerical examples a comparison of this method to least squares esprit music and root music as well as to the crb for a calibrated array is also presented tls esprit is found to be competitive with the other methods and the performance is close to the calibrated crb for many cases of practical interest for highly correlated signals however the performance deviates significantly from the calibrated crb simulations are included to illustrate the applicability of the theoretical results to a finite number of data

fast lcmv based methods for fundamental frequency estimation recently optimal linearly constrained minimum variance lcmv filtering methods have been applied to fundamental frequency estimation such estimators often yield preferable performance but suffer from being computationally cumbersome as the resulting cost functions are multimodal with narrow peaks and require matrix inversions for each point in the search grid in this paper we therefore consider fast implementations of lcmv based fundamental frequency estimators exploiting the estimators inherently low displacement rank of the used toeplitz like data covariance matrices using as such either the classic time domain averaging covariance matrix estimator or if aiming for an increased spectral resolution the covariance matrix resulting from the application of the recent iterative adaptive approach iaa the proposed exact implementations reduce the required computational complexity with several orders of magnitude but as we show further computational savings can be obtained by the adoption of an approximative iaa based data covariance matrix estimator reminiscent of the recently proposed quasi newton iaa technique furthermore it is shown how the considered pitch estimators can be efficiently updated when new observations become available the resulting time recursive updating can reduce the computational complexity even further the experimental results show that the performances of the proposed methods are comparable or better than that of other competing methods in terms of spectral resolution finally it is shown that the time recursive implementations are able to track pitch fluctuations of synthetic as well as real life signals

glrt based detection estimation for undersampled training conditions for sensors where the number of available independent identically distributed training samples t is less than the number of antenna array elements m we propose nondegenerate properly normalized likelihood ratio lr tests both standard and scale invariant to support detection estimation of m point sources m t in white noise based on a generalized likelihood ratio test glrt approach we demonstrate that these tests can detect music specific ldquooutliersrdquo in the direction of arrival doa estimation of closely spaced independent sources caused by insufficient training volume and or signal to noise ratio snr we then compare the performance of the introduced lrs to other test statistics available in this undersampled regime we show that a search for solutions that increase the introduced lr allows us to replace the detected outliers by proper doa estimates this ldquopredict and curerdquo process leverages the snr ldquogaprdquo between music breakdown and breakdown of maximum likelihood estimation itself the resultant lr maximization makes the associated covariance model statistically ldquoas likelyrdquo as the true covariance matrix and removes the vast percentage of outliers in certain scenarios

locally most powerful invariant tests for the properness of quaternion gaussian vectors previous works have addressed the second order statistical characterization of quaternion random vectors introducing different properness definitions and presenting the generalized likelihood ratio tests glrts for determining the kind of quaternion properness this paper considers the more challenging problem of deriving the locally most powerful invariant tests lmpits which can be obtained even without an explicit expression for the maximal invariants thanks to the wijsman s theorem specifically we consider three binary hypothesis testing problems involving the two main kinds of quaternion properness and show that the lmpit statistics are given by the frobenius norm of three previously defined sample coherence matrices the proposed detectors exhibit interesting connections with the problem of testing for the properness of a complex vector and with the problems of testing for the sphericity of a four dimensional real or two dimensional complex proper vector additionally some numerical examples show that in general the proposed lmpits outperform their glrt counterparts and in some cases the performance gap is very noticeable

polarimetric detection in compound gaussian clutter with kronecker structured covariance matrix in this paper we consider polarimetric adaptive detection in compound gaussian clutter whose covariance matrix cm has a kronecker structure we derive the cramer rao bound of a kronecker structured cm estimate and analyze the constant false alarm rate property of the adaptive subspace matched filter detector that uses kronecker structured estimates we provide a general expression for the average signal to clutter ratio loss scrl as a function of the mean square error of the covariance estimate the aforementioned expression is helpful in determining how many samples are required in order to achieve a desired average scrl level in practical scenarios based on that expression we show that the required sample size can be effectively reduced by exploiting the kronecker structure of the clutter cm we also derive the asymptotic detection performance of the adaptive subspace matched filter the analysis of scrl and detection performance can be extended to more general scenarios especially when the maximum likelihood estimate of the structured cm involves solving fixed point equations numerical simulations validate the merits of the proposed methods

adaptive detection of a signal known only to lie on a line in a known subspace when primary and secondary data are partially homogeneous this paper deals with the problem of detecting a signal known only to lie on a line in a subspace in the presence of unknown noise using multiple snapshots in the primary data to account for uncertainties about a signal s signature we assume that the steering vector belongs to a known linear subspace furthermore we consider the partially homogeneous case for which the covariance matrix of the primary and the secondary data have the same structure but possibly different levels this provides an extension to the framework considered by bose and steinhardt the natural invariances of the detection problem are studied which leads to the derivation of the maximal invariant then a detector is proposed that proceeds in two steps first assuming that the noise covariance matrix is known the generalized likelihood ratio test glrt is formulated then the noise covariance matrix is replaced by its sample estimate based on the secondary data to yield the final detector the latter is compared with a similar detector that assumes the steering vector to be known

generalized forward backward subaperture smoothing techniques for sample starved stap a major issue in space time adaptive processing stap for moving target indicator mti radar is the so called sample support problem often the available sample support for estimating the requisite interference covariance matrix is inadequate thereby precluding stap beamforming utilizing many adaptive degrees of freedom dofs although deterministic rank reduction methods can reduce sample support requirements they are invariably suboptimal from a signal to interference plus noise ratio sinr standpoint a new generalized subspatial and subtemporal aperture smoothing method employing forward and backward data vectors is introduced to overcome the data deficiency problem it is shown that multiplicative improvement in data samples can be obtained at the expense of negligible loss in space time aperture of the steering vector

esprit like estimation of real valued sinusoidal frequencies subspace based estimation of multiple real valued sine wave frequencies is considered in this paper a novel data covariance model is proposed in the proposed model the dimension of the signal subspace equals the number of frequencies present in the data which is half of the signal subspace dimension for the conventional model consequently an esprit like algorithm using the proposed data model is presented the proposed algorithm is then extended for the case of complex valued sine waves performance analysis of the proposed algorithms are also carried out the algorithms are tested in numerical simulations when compared with esprit the newly proposed algorithm results in a significant reduction in computational burden without any compromise in the accuracy

rao test for adaptive detection in gaussian interference with unknown covariance matrix this paper deals with the problem of detecting a signal known up to a scaling factor in the presence of gaussian disturbance with unknown covariance matrix a new detector based on the rao test criterion is introduced and its invariance properties and constant false alarm rate cfar behavior are studied at the analysis stage the performance of the new receiver is assessed also in comparison with some classical adaptive radar detectors both in the matched as well as in the mismatched signal case remarkably the rao test may achieve a matched detection performance which is commensurate with that of the generalized likelihood ratio test glrt based detectors if a sufficient number of training data is available moreover it also exhibits better rejection capabilities of mismatched signals than the counterparts in the last part of the work a two stage detector whose second stage coincides with the rao test is devised it represents a suitable means to restore the detection performance of the plain rao test in the presence of a small number of training data finally the performance of the aforementioned two stage processor is analyzed in closed form and the cfar behavior is proved

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

fast mlm power spectrum estimation from uniformly spaced correlations we derive a fast algorithm for calculating the capon maximum likelihood method mlm power spectrum estimate when given uniformly spaced samples of the correlation function this algorithm computes a weighted correlation of the predictor coefficients found by running levinson recursion the fourier transform of the result gives the mlm spectrum this approach also suggests an interesting new comparison between mlm and mem

rational points on quartic hypersurfaces let x be a projective non singular quartic hypersurface of dimension 39 or more which is defined over we show that x is non empty provided that x is non empty and x has p adic points for every prime p 2009 walter de gruyter

displacement structure theory and applications in this survey paper we describe how strands of work that are important in two different fields matrix theory and complex function theory have come together in some work on fast computational algorithms for matrices with what we call displacement structure in particular a fast triangularization procedure can be developed for such matrices generalizing in a striking way an algorithm presented by schur 1917 j reine angew math 147 1917 pp 205 232 in a paper on checking when a power series is bounded in the unit disc this factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra we mention among others inverse scattering analytic and unconstrained rational interpolation theory digital filter design adaptive filtering and state space least squares estimation

glrt based adaptive detection algorithms for range spread targets we address adaptive detection of a range spread target or targets embedded in gaussian noise with unknown covariance matrix to this end we assume that cells referred to in the following as secondary data that are free of signal components are available those secondary data are supposed to possess either the same covariance matrix or the same structure of the covariance matrix of the cells under test in this context we design detectors relying on the generalized likelihood ratio test glrt and on a two step glrt based design procedure remarkably both criteria lead to receivers ensuring the constant false alarm rate cfar property with respect to the unknown quantities a thorough performance assessment of the proposed detection strategies together with the evaluation of their processing cost highlights that the two step design procedure is to be preferred with respect to the plain glrt in fact the former leads to detectors that achieve satisfactory performance under several situations of practical interest and are simpler to implement than those designed resorting to the latter

high resolution spectral analysis of mixtures of complex exponentials modulated by polynomials high resolution methods such as the esprit algorithm are of major interest for estimating discrete spectra since they overcome the resolution limit of the fourier transform and provide very accurate estimates of the signal parameters in signal processing literature most contributions focus on the estimation of exponentially modulated sinusoids in a noisy signal this paper introduces a more general class of signals involving both amplitude and frequency modulations it shows that this polynomial amplitude complex exponentials pace model is the most general model tractable by high resolution methods a generalized esprit algorithm is developed for estimating the signal parameters and it is shown that this model can be characterized by means of a geometrical criterion

performance analysis of forward backward matched filterbank spectral estimators the problem of complex spectral estimation is of great interest in many applications this paper studies the general class of the forward backward matched filterbank mafi spectral estimators including the widely used capon as well as the more recently introduced amplitude and phase estimation of a sinusoid apes methods in particular we show by means of a higher order expansion technique that the one dimensional 1 d capon estimator underestimates the true spectrum whereas the 1 d apes method is unbiased we also show that the bias of the forward backward capon is half that of the forward only capon to within a second order approximation furthermore we show that these results can be extended to the two dimensional 2 d capon and apes estimators numerical examples are also presented to demonstrate quantitatively the properties of and the relation between these mafi estimators

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

design of asymptotically convergent frequency estimator using contraction theory the problem of estimating unknown frequencies of a sinusoidal signal simultaneously is a classical problem in signal and system theory many approaches and algorithms are proposed in literature to develop estimators for a measurable sinusoidal signal having multiple sinusoids with unknown amplitudes frequencies and phases in this work an asymptotically convergent frequency estimator is given for estimation of n unknown frequencies of a measurable sinusoidal signal the contraction theory approach is adopted to show the asymptotic convergence of the proposed estimator in quite a simplified manner approach given here exploits the results of contraction theory related to semi contracting systems the nonlinear estimator based on dynamic system approach guarantees global boundedness and convergence of the state and frequency estimation for all initial conditions and frequency values it further ensures simultaneous globally convergent estimation of states and frequencies of a sinusoid involving multiple frequencies numerical simulations are presented for different combinations of frequencies to justify the claim

iterative generalized likelihood ratio test for mimo radar we consider a multiple input multiple output mimo radar system where both the transmitter and receiver have multiple well separated subarrays with each subarray containing closely spaced antennas because of this general antenna configuration both the coherent processing gain and the spatial diversity gain can be simultaneously achieved we compare several spatial spectral estimators including capon and apes for target detection and parameter estimation we introduce a generalized likelihood ratio test glrt and a conditional generalized likelihood ratio test cglrt for the general antenna configuration based on glrt and cglrt we then propose an iterative glrt iglrt procedure for target detection and parameter estimation via several numerical examples we show that iglrt can provide excellent detection and estimation performance at a low computational cost

cfar detection of multidimensional signals an invariant approach the paper deals with constant false alarm rate cfar detection of multidimensional signals embedded in gaussian noise with unknown covariance we attack the problem by resorting to the principle of invariance which proves a valuable statistical tool for ensuring a priori namely at the design stage the cfar property in this context we determine a maximal invariant statistic with respect to a proper group of transformations that leave unaltered the hypothesis testing problem under study devise the optimum invariant detector and show that no uniformly most powerful invariant umpi test exists thus we establish the conditions an invariant detector must fulfill in order to ensure the cfar property finally we discuss several suboptimal implementable invariant receivers and remarkably show that the generalized likelihood ratio test glrt detector is a member of this class the performance analysis which has been carried out in the presence of a gaussian signal array shows that the proposed detectors exhibit a quite acceptable loss with respect to the optimum neyman pearson detector

an invariant approach to adaptive radar detection under covariance persymmetry in this paper we propose a systematic and unifying framework to deal with adaptive radar detection in the presence of gaussian interference sharing a persymmetric covariance structure first we determine the group of transformations which leaves the considered hypothesis testing problem unaltered then after reduction by sufficiency we determine a maximal invariant statistic which is a four dimensional vector and significantly compresses the original observation space its first two components are the one step and the two step generalized likelihood ratio test decision statistics respectively whereas its last two entries represent an ancillary statistic we provide also the exact statistical characterization of the maximal invariant which is exploited to synthesize both the optimum and the locally optimum in the low signal to interference plus noise ratio regime invariant receivers finally some sub optimum decision rules based on theoretically solid design criteria are discussed and their performances are analyzed in comparison with the benchmark invariant test