This article derives maximal invariants (MIs) for group invariant hypothesis tests to detect subspace signals in zero-mean complex Gaussian multivariate interference and noise with unknown covariance ...matrix given multiple observations. The signal is assumed to belong to a known <inline-formula><tex-math notation="LaTeX">M</tex-math></inline-formula>-dimensional subspace in <inline-formula><tex-math notation="LaTeX">\mathbb {C}^{N \times 1}</tex-math></inline-formula>. Given <inline-formula><tex-math notation="LaTeX">P</tex-math></inline-formula> independent observations of the test vector, we show that MIs are any two of three <inline-formula><tex-math notation="LaTeX">P \times P</tex-math></inline-formula> matrices constructed from the test matrix and the interference training data. The importance of MI is that the detection statistic of all constant false alarm rate (CFAR) tests are derived from MI and as such MI provides information on how input test matrix data and interference training data must be processed in order for the receiver to construct any one of the numerous CFAR tests.
This article compares the receiver operating characteristics (ROC) of two subspace-based generalized likelihood ratio tests (GLRTs) referred to in this article as the unconstrained and constrained ...GLRTs. The tests are derived for slightly different assumptions and are applied to detect an unknown rank-1 signal that belongs to a known <inline-formula><tex-math notation="LaTeX">M</tex-math></inline-formula>-dimensional subspace in <inline-formula><tex-math notation="LaTeX">\mathbb {C}^{N \times 1}</tex-math></inline-formula>. The receiver is given multiple (say <inline-formula><tex-math notation="LaTeX">P</tex-math></inline-formula>) statistically independent observations of unknown multivariate zero-mean complex Gaussian interference-plus-noise that may contain a rank-1 signal as the test matrix and signal-free training data to estimate the interference-plus-noise covariance matrix. The two detectors are shown to coincide when <inline-formula><tex-math notation="LaTeX">M = 1</tex-math></inline-formula> and/or <inline-formula><tex-math notation="LaTeX">P = 1</tex-math></inline-formula>. New analytical expressions for the probability of false alarm and probability of detection are derived for the unconstrained GLRT and to the best of our knowledge, no such analytical expressions are available for evaluating the performance of the constrained GLRT in the general case, where <inline-formula><tex-math notation="LaTeX">M > 1</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">P > 1</tex-math></inline-formula>. The ROC of the unconstrained GLRT is a lower bound on the performance of the constrained GLRT. For small signal-to-interference-plus-noise ratios (SINRs) and <inline-formula><tex-math notation="LaTeX">M/N \ll 1</tex-math></inline-formula>, the lower bound is tight, which makes the derived analytical results both interesting and useful.
In this paper, an approach for the adaptive detection of a hypothesized signal in unknown multivariate Gaussian interference-plus-noise is considered under conditions where the set of signal space ...eigenvalues of the interference-plus-noise covariance matrix of the training samples and the test vector may be mismatched. The detector is required to have the constant false alarm rate (CFAR) property under these conditions. The proposed approach uses two sets of interference-plus-noise data: First, vectors from a reference set, typically from range cells in the vicinity of the test cell that have the same interference-plus-noise covariance matrix C as the test vector, and second, vectors from a training set that are used to compute the weights for interference suppression in the test vector and the reference vectors. Because the matrices C and Σ are unknown, the average power level of the residual interference in the test cell and reference cells after interference suppression is unknown. The adaptive matched filter statistic at the test cell is normalized by the sample mean of similar statistics for the reference cells to evaluate the detection statistic, which is shown to have the CFAR property. The detection performance of the CFAR detector is analyzed and the effect of mismatches in the eigenvalues of the covariance matrices C and Σ is shown to be characterized by a single random variable ρ, defined as the signal-to-interference-plus-noise ratio loss factor. Sample results are provided for purposes of illustration.
Adaptive signal detection algorithms in unknown interference are generally formulated under the assumption that training sets are available for the estimation of the test cell interference ...characteristics. A significant problem occurs in applications such as radar surveillance when the interference covariance matrices of vectors from the training cells and that of the test cell vector are mismatched. False alarm rates may increase beyond acceptable levels and thus overwhelm a receiver if the interference in the test cell is not adequately cancelled. A number of adaptive detection algorithms with the constant false alarm rate (CFAR) property have been designed assuming the availability of training data that have the same covariance matrix as the interference in the test cell (although the adaptive coherence estimator algorithm requires the two covariance matrices to be related by a positive multiplicative constant that is not required to be known). The detection threshold in these CFAR detectors can be selected without knowledge of the interference-plus-noise covariance matrix. To the best of our knowledge, an analytical characterization of the performance effects of a receiver using a mismatched training set (perhaps inadvertently) in adaptive detection is currently not available. This paper addresses the problem analytically. An exact analysis of the effects of interference covariance matrix mismatch on the performance of the adaptive matched filter test is carried out. Results provide insights to the specific aspects of covariance matrix mismatch that cause the increase in false alarms when compared to the matched training set case. Analytical results are illustrated with an example and verified independently with simulations. These results are useful in developing CFAR algorithms under conditions of mismatched training.
Rational design of small molecular gelators is an elusive and herculean task, despite the rapidly growing body of literature devoted to such gels over the past decade. The process of self-assembly, ...in molecular gels, is intricate and must balance parameters influencing solubility and those contrasting forces that govern epitaxial growth into axially symmetric elongated aggregates. Although the gelator-gelator interactions are of paramount importance in understanding gelation, the solvent-gelator specific (i.e., H-bonding) and nonspecific (dipole-dipole, dipole-induced and instantaneous dipole induced forces) intermolecular interactions are equally important. Solvent properties mediate the self-assembly of molecular gelators into their self-assembled fibrillar networks. Herein, solubility parameters of solvents, ranging from partition coefficients (log P), to Henry's law constants (HLC), to solvatochromic parameters (ET(30)), and Kamlet-Taft parameters (β, α and π), and to Hansen solubility parameters (δp, δd, δh), are correlated with the gelation ability of numerous classes of molecular gelators. Advanced solvent clustering techniques have led to the development of a priori tools that can identify the solvents that will be gelled and not gelled by molecular gelators. These tools will greatly aid in the development of novel gelators without solely relying on serendipitous discoveries. These tools illustrate that the quest for the universal gelator should be left in the hands of Don Quixote and as researchers we must focus on identifying gelators capable of gelling classes of solvents as there is likely no one gelator capable of gelling all solvents.
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 \times 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.
This paper considers the problem of detecting a signal that belongs to an unknown one dimensional subspace of <inline-formula><tex-math notation="LaTeX">\mathbb {C}^{N \times ...1}</tex-math></inline-formula> in additive interference-plus-noise whose covariance matrix is unknown. The interference-plus-noise is assumed to be modeled as a complex multivariate zero-mean random vector whose covariance matrix <inline-formula><tex-math notation="LaTeX">{\bf R}</tex-math></inline-formula>, is estimated from signal-free training vectors. The hypothesis test, labeled the generalized Adaptive Coherence Estimator (GACE) involves two test vectors, both of which contain the unknown signal. The test statistic reduces to the ACE test statistic as the signal-to-interference-plus-noise ratio of any one of the test vectors increases without limit. In the limit of large number of training samples the GACE test statistic reduces to the magnitude square of the inner-product of a signal vector in additive statistically independent white noise vectors. Analytical expressions for the probability of false alarm and the probability of detection of the GACE test are derived and the test is shown to have the constant false alarm rate (CFAR) property. Sample results to illustrate the performance of the detector are provided and compared with the performance of the generalized likelihood ratio test (GLRT) for the specific problem, along with results on the sequential application of the GLRT and GACE.
In this paper, an approach for the design and analysis of coherent constant false alarm rate (CFAR) detectors in clutter and interference with a Kronecker covariance structure is described. In a ...two-dimensional example considered, the interference-plus-noise matrix X ∈ C N×L is modeled by a doubly correlated, zero-mean multivariate complex Gaussian distribution described by two covariance matrices C and R that are unknown to the receiver. The concatenated columns of X has a structured covariance matrix Σ given by Σ = R* ⊗ C. In the approach described, an estimate of R is used to "prewhiten" and match filter all the rows of both the training data matrices and the test data matrix. The processing enables one to reduce the detection problem to a one-dimensional case that can be handled by any one of the several adaptive detection algorithms. The proposed algorithm for the doubly correlated clutter is analyzed to show that the detection performance is determined by two statistically independent signal-to-interference-plus-noise loss factors both of which have complex beta distributions. Sample results show that the proposed approach requires training samples that is a multiple of N + L, while an adaptive detection algorithm that do not explicitly use the Kronecker constraint on the covariance structure requires training samples that is a multiple of N × L for comparable detection performance.
The construction of a virtual array response is integral to the process of target detection for coherent airborne multiple input multiple output (MIMO) radars. Multiple scatterers on a target with ...complex scattering characteristics have an impact on the virtual array response. In this paper, an analytical expression for the virtual array response of a target is derived in terms of the response vectors of uniform linear arrays involving scatterer characteristics at multiple delays, the auto-ambiguity function and cross-ambiguity function of transmitted signals. Clutter at the same Doppler as the target is modeled similarly as signal returns from an ensemble of scatterers distributed in range. Analysis shows that the ideal Vandermonde vector of the virtual array response to a point scatterer must be modified to account for multiple scatterers and can provide a basis to develop detection/classification algorithms for coherent MIMO radar. A likelihood ratio test formulation of the detection problem is shown to result in the adaptive coherence estimator (ACE) test. The detection performance of the ACE test is derived in terms of parameters such as the signal-to-noise ratio, dimensionality of the MIMO clutter subspace, and the mismatch angle between the virtual array vector for an assumed target scatter model and the signal component of the actual virtual array vector. Illustrative results are presented using a fast-time orthogonal waveforms proposed in the literature as an example. Pulses in the example signal set comprise a sequence of frequency hopping subpulses and the frequency hopping sequences of these waveforms are derived from a subset of the Reed-Solomon code and the waveforms have several interesting properties including insensitivity to frequency errors.
