IT needs courage to produce a text-book on the geometry of N dimensions. The subject is not unduly difficult to those who take an interest in it, but most people have deeply seated prejudices which prevent them from taking it seriously into consideration. One of the pioneers, Schlafli, in spite of his reputation in other branches of mathematics, failed to secure publication for his valuable memoir on hyperspace, and in fact it did not appear in full until after the author's death and fifty years after it was written. An Introduction to the Geometry of N Dimensions. By Prof. D. M. Y. Sommerville. Pp. xvii + 196. (London: Methuen and Co., Ltd., 1929.) 10s. net.

An Introduction to the Geometry of N Dimensions

Crami??r-Rao bounds on estimation accuracy are established for estimation problems on arbitrary manifolds in which no set of intrinsic coordinates exists. The frequently encountered examples of estimating either an unknown subspace or a covariance matrix are examined in detail. The set of subspaces, called the Grassmann manifold, and the set of covariance (positive-definite Hermitian) matrices have no fixed coordinate system associated with them and do not possess a vector space structure, both of which are required for deriving classical Crami??r-Rao bounds. Intrinsic versions of the Crami??r-Rao bound on manifolds utilizing an arbitrary affine connection with arbitrary geodesics are derived for both biased and unbiased estimators. In the example of covariance matrix estimation, closed-form expressions for both the intrinsic and flat bounds are derived and compared with the root-mean-square error (RMSE) of the sample covariance matrix (SCM) estimator for varying sample support K. The accuracy bound on unbiased covariance matrix estimators is shown to be about (10/log 10)n/K 1/2 dB, where n is the matrix order. Remarkably, it is shown that from an intrinsic perspective, the SCM is a biased and inefficient estimator and that the bias term reveals the dependency of estimation accuracy on sample support observed in theory and practice. The RMSE of the standard method of estimating subspaces using the singular value decomposition (SVD)is compared with the intrinsic subspace Crami??r-Rao bound derived in closed form by varying both the signal-to-noise ratio (SNR) of the unknown p-dimensional subspace and the sample support. In the simplest case, the Crami??r-Rao bound on subspace estimation accuracy is shown to be about (p(n - p)1/2 K-1/2SN-1/2 rad for p-dimensional subspaces. It is seen that the SVD-based method yields accuracies very close to the Crami??r-Rao bound, esta blishing that the principal invariant subspace of a random sample provides an excellent estimator of an unknown subspace. The analysis approach developed is directly applicable to many other estimation problems on manifolds encountered in signal processing and elsewhere, such as estimating rotation matrices in computer vision and estimating subspace basis vectors in blind source separation.

Covariance, Subspace, and Intrinsic CramrRao Bounds

This paper considers the problem of passive detection with a multistatic radar system involving a noncooperative illuminator of opportunity (IO) and multiple receive platforms. An unknown source signal is transmitted by the IO, which illuminates a target of interest. These receive platforms are geographically dispersed, and collect independent target echoes due to the illumination by the same IO. We consider a generalized canonical correlation (GCC) detector for passive detection which requires the knowledge of the noise power. We derive closed-form expressions for the probabilities of false alarm and detection of this detector. For the case where the noise power is unknown, we propose a generalized likelihood ratio test (GLRT) detector to deal with the passive detection problem. Moreover, a closed-form expression for the probability of false alarm of this GLRT detector is given, which shows that the proposed GLRT detector exhibits a constant false alarm rate property with respect to the noise power. Numerical simulations demonstrate that the proposed GLRT detector generally outperforms the generalized coherence detector, a previous popular passive detector that neither requires the knowledge of the noise power. In addition, the GLRT also outperforms the GCC detector when the latter has an uncertainty in its knowledge of the noise power.

Two Target Detection Algorithms for Passive Multistatic Radar

The existing multiobjective evolutionary algorithms (EAs) based on nondominated sorting may encounter serious difficulties in tackling many-objective optimization problems (MaOPs), because the number of nondominated solutions increases exponentially with the number of objectives, leading to a severe loss of selection pressure. To address this problem, some existing many-objective EAs (MaOEAs) adopt Euclidean or Manhattan distance to estimate the convergence of each solution during the environmental selection process. Nevertheless, either Euclidean or Manhattan distance is a special case of Minkowski distance with the order ${P=2}$ or ${P=1}$ , respectively. Thus, it is natural to adopt Minkowski distance for convergence estimation, in order to cover various types of Pareto fronts (PFs) with different concavity–convexity degrees. In this paper, a Minkowski distance-based EA is proposed to solve MaOPs. In the proposed algorithm, first, the concavity–convexity degree of the approximate PF, denoted by the value of ${P}$ , is dynamically estimated. Subsequently, the Minkowski distance of order ${P}$ is used to estimate the convergence of each solution. Finally, the optimal solutions are selected by a comprehensive method, based on both convergence and diversity. In the experiments, the proposed algorithm is compared with five state-of-the-art MaOEAs on some widely used benchmark problems. Moreover, the modified versions for two compared algorithms, integrated with the proposed ${P}$ -estimation method and the Minkowski distance, are also designed and analyzed. Empirical results show that the proposed algorithm is very competitive against other MaOEAs for solving MaOPs, and two modified compared algorithms are generally more effective than their predecessors.

An Evolutionary Algorithm Based on Minkowski Distance for Many-Objective Optimization

The Golden code is a full-rate full-diversity space-time code which has been incorporated in the IEEE 802.16 (WiMAX) standard. The worst case complexity of a tree-based sphere decoder for a square QAM constellation is O(N3), where N is the size of the underlying QAM constellation; the worst case will dominate average decoding complexity on any channel with a significant line of sight component. In this paper, we present a simple algorithm with quadratic complexity for decoding the Golden code that can be employed by mobile terminals with either one or two receive antennas, that is resilient to near singularity of the channel matrix, and that gives essentially maximum likelihood (ML) performance. Dual use is an advantage, since there will likely be some IEEE 802.16 mobile terminals with one receive antenna and some with two antennas. The key to the quadratic algorithm is a maximization of the likelihood function with respect to one of the pair of signal points conditioned on the other. This choice is made by comparing the determinants of two covariance matrices, and the underlying geometry of the Golden code guarantees that one of these choices is good with high probability.

Fast Essentially Maximum Likelihood Decoding of the Golden Code

Space-time codes built out of Alamouti components have been adopted in wireless standards such as UMTS, IEEE 802.1 In and IEEE 802.16 where they facilitate higher data rates through multiplexing of parallel data streams and the addition of two or more antennas at the receiver that perform interference cancellation. This paper provides new theoretical insight into an algorithm for interference cancellation through a Bayesian analysis that expresses performance as a function of SNR in terms of the ''angles" between different space-time coded data streams. Our approach provides insights into the coupling of channel coding to spatial and polarization degrees of freedom.

https://www.ese.wustl.edu/~nehorai/DARPA/publications/ITW07_Srianunpiboon.pdf

Diversity Gains Across Line of Sight and Rich Scattering Environments from Space-Polarization-Time Codes

Perfect space-time block codes (STBCs) were first introduced by Oggier et al. to have full rate, full diversity and non-vanishing determinant. A maximum likelihood decoder based on the sphere decoder has been used for efficient decoding of perfect STBCs. However the worst-case complexity for the sphere decoder is an exhaustive search. In this paper we present a reduced complexity algorithm for 3 × 3 perfect STBC which gives essentially maximum likelihood (ML) performance and which can be extended to other perfect STBC. The algorithm is based on the conditional maximization of the likelihood function with respect to one of the set of signal points given another. There are a number of choices for which signal points to condition on and the underlying structure of the code guarantees that one of the choices is good with high probability. Furthermore, the approach can be integrated with the sphere decoding algorithm with worst case complexity corresponding exactly to that of our algorithm.

Low complexity essentially maximum likelihood decoding of perfect space-time block codes

The generalized coherence (GC) estimate is a well studied statistic for detection of a common but unknown signal on several noisy channels. In this paper, it is shown that the GC detector arises naturally from a Bayesian perspective. Specifically, it is derived as a test of the hypothesis that the signals in the channels are independent Gaussian processes against the hypothesis that the processes have some arbitrary correlation. This is achieved by introducing suitable non-informative priors for the covariance matrices across the channels under the two hypotheses. Subsequently, reduced likelihoods are obtained by marginalizing the joint distribution of the data and the covariance matrix in each case. The likelihood ratio is then shown to be a monotonic function of the GC detection statistic. This derivation extends to the case of time-correlated signals, allowing comparison with the generalized likelihood ratio test (GLRT) recently proposed by Ramirez et al.

A Bayesian derivation of generalized coherence detectors

Target detection for a multi-static passive radar, in which the radar uses multiple illuminators of opportunity, as well as multiple target surveillance receivers, is considered. It is shown that this detection problem can be formulated as a statistical test of whether a signal of known rank is present in both the reference and target surveillance channels, or only in the reference channels, in the presence of independent gaussian white noise across all channels. Bayesian and generalized likelihood ratio tests for target detection are derived for both known and unknown receiver noise variances and are compared through simulation.

Passive radar detection using multiple transmitters

A class of detectors for cyclostationarity is introduced These detectors are based on the use of generalized coherence to measure correlation among two or more collections of random vectors The generalized coherence framework allows any finite collection of pertinent samples of the cyclic auto-correlation function estimates formed from the measured signal data to be combined into the detection statistic The performance of this approach is demonstrated and compared against other established cyclostationarity detectors in both a cognitive radio scenario and a multi-channel passive surveillance scenario

Songsri Sirianunpiboon

Papers

Diversity Gains Across Line of Sight and Rich Scattering Environments from Space-Polarization-Time Codes

Low complexity essentially maximum likelihood decoding of perfect space-time block codes

A Bayesian derivation of generalized coherence detectors

Passive radar detection using multiple transmitters

Detection of Cyclostationarity Using Generalized Coherence