We present a new formula for computing arbitrary moments of unitarily invariant matrix distributions. The Schur-Weyl duality is used to decompose the expected value of tensor powers of the random matrices as a linear combination of projection operators onto unitary irreducible representations. The coefficients in this combination, which are labeled by Young diagrams, are expectations of products of determinants of the random matrices. It is demonstrated in a number of important cases, including matrix gamma and matrix beta distributions, that these coefficients can be simply computed from a knowledge of the normalization factors of the distributions.

Radars use coherence to increase the signal-processing gain. In electronic support, that same coherence can be exploited to accurately estimate carrier frequency when measured modulo the pulse repetition frequency. Such estimates can be used for passive emitter geolocation, for example. In the pioneering work of Howard & Lavoie on this topic [1], [2], results are demonstrated only where the pulse repetition interval (PRI) is an integer and where pulses are sinusoidal. In this paper, we show how the results can be generalised to a broad class of pulse shapes, without needing to know the pulse shape in advance. We also show how the signals may be efficiently processed when the PRI is non-integer.

Eigenvalues of the Gram matrix formed from received data frequently appear in sufficient detection statistics for multi-channel detection with Generalized Likelihood Ratio (GLRT) and Bayesian tests. In a frequently presented model for passive radar, in which the null hypothesis is that the channels contain only complex white Gaussian noise and the alternative hypothesis is that the channels contain a common rank-one signal in the mean, the GLRT statistic is the largest eigenvalue λ1of the Gram matrix formed from data, which has a Wishart distribution. Although exact expressions for the distribution of λ1are known under both hypotheses, numerically calculating values of these distribution functions presents difficulties in cases where the dimension of the data vectors is large. Following on recent work addressing this issue under the null hypothesis, this paper presents a method to calculate values of this distribution under the alternative hypothesis, allowing tractable computation of receiver operating characteristic curves.

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.

In multi-channel detection, sufficient statistics for Generalized Likelihood Ratio and Bayesian tests are often functions of the eigenvalues of the Gram matrix formed from data vectors collected at the sensors. When the null hypothesis is that the channels contain only independent complex white Gaussian noise, the distributions of these statistics arise from the joint distribution of the eigenvalues of a complex Wishart matrix G. This paper considers the particular case of the largest eigenvalue λ1of G, which arises in passive radar detection of a rank-one signal. Although the distribution of λ1is known analytically, calculating its values numerically has been observed to present formidable difficulties. This is particularly true when the dimension of the data vectors is large, as is common in passive radar applications, making computation of accurate detection thresholds intractable. This paper presents results that significantly advance the state of the art for this problem.

This paper considers the problem of testing for mutual independence of multiple sets of complex Gaussian vectors. This problem has classical roots in statistics and has been of recent interest in the signal processing literature in connection with multi-channel signal detection. The maximal invariant statistic for this problem is described both as a collection of subspaces of the data space (i.e., points on a complex Grassmannian manifold) and as a corresponding set of complex matrices. The distribution of the maximal invariant is also derived under both hypotheses in the testing problem.

We consider the problem of fusing measurements in a sensor network, where the sensing regions overlap and data are nonnegative real numbers, possibly resulting from a count of indistinguishable discrete entities. Because of overlaps, it is generally impossible to fuse this information to arrive at an accurate value of the overall amount or count of material present in the union of the sensing regions. Here we study the computation of the range of overall values consistent with the data and provide several results. Posed as a linear programming problem, this leads to questions associated with the geometry of the sensor regions, specifically the arrangement of their nonempty intersections. We define a computational tool called the fusion polytope , based on the geometry of the sensing regions. Its properties are explored, and in particular, a topological necessary and sufficient condition for this to be in the positive orthant, a property that considerably simplifies calculations, is provided. We show that in two dimensions, inflated tiling schemes based on rectangular regions fail to satisfy this condition, whereas inflated tiling schemes based on hexagons do.

A geometrical perspective is introduced that enables unification and generalization of several results regarding the distributions of quantities that arise in connection with an important class of multiple-channel detectors. Standard models on sets of normalized vectors following from joint Gaussian assumptions in this context are relaxed to the geometrically appealing model of uniform distributions on the Stiefel manifold of K-frames in N-dimensional space. In addition to bolstering geometric insight, several prior results are subsumed and strengthened by results obtained under this formulation. Additionally, a generalization of a classical theorem of W. G. Cochran is enabled by this framework.

A measure of the smallest angle between two p-dimensional subspaces of a higher-dimensional vector space is introduced. This p-coherence is seen to generalize standard measures of coherence, and its relationship to canonical correlation is noted. The use of p-coherence as a test statistic for the presence of a common p-dimensional signal subspace in data collected by two arrays of K > p sensors is proposed.

Estimating the number of linearly independent signals impinging on a collection of receivers is of increasing recent interest, particularly in connection with MIMO systems for communications and sensing. This paper derives maximum a posteriori estimators for signal rank from noisy data collected at multiple receivers. Situations in which noise variance is known and unknown are both treated. These estimators are shown to significantly outperform maximum-likelihood/BIC rank estimators.

Bayesian and generalized likelihood ratio tests are derived for detection of a common unknown signal of known rank K in M > K independent channels of white gaussian noise. The cases of known and unknown noise variance are both treated. These derivations encompass the development of explicit expressions for an invariant measure on the grassmannian manifold of K-dimensional subspaces of complex N-dimensional space and parameterization of this manifold to enable the calculation of the necessary marginalization integrals. Performance of the detectors is compared by simulation.

Tracking of the orientation of a rigid body based on directional measurements is a key issue in many applications. Configurations in this sense are precisely representable as elements of the rotation group SO(3), and the issue devolves to one of tracking on this group, for which and algorithm is described here. Its novelty derives from the use of maximum entropy distributions on these groups as models for the priors, and from the approximation algorithms that permit numerical implementation of such a model. These solutions can be written in a recursive form. While the general ideas apply in all dimensions, the focus of this paper is on the important 3-dimensional case. It is impossible to compute the exact solution; instead, obtained here is a highly effective approximation. It is shown that, in contrast with other approaches, the algorithm described here produces outputs which are both very accurate and statistically meaningful.

An information-geometric approach to sensor management is introduced that is based on following geodesic curves in a manifold of possible sensor configurations. This perspective arises by observing that, given a parameter estimation problem to be addressed through management of sensor assets, any particular sensor configuration corresponds to a Riemannian metric on the parameter manifold. With this perspective, managing sensors involves navigation on the space of all Riemannian metrics on the parameter manifold, which is itself a Riemannian manifold. Existing work assumes the metric on the parameter manifold is one that, in statistical terms, corresponds to a Jeffreys prior on the parameter to be estimated. It is observed that informative priors, as arise in sensor management, can also be accommodated. Given an initial sensor configuration, the trajectory along which to move in sensor configuration space to gather most information is seen to be locally defined by the geodesic structure of this manifold. Further, divergences based on Fisher and Shannon information lead to the same Riemannian metric and geodesics.

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 Ramírez et al.

We study the detection error probability associated with a balanced binary relay tree, where the leaves of the tree correspond toNidentical and independent sensors. The root of the tree represents a fusion center that makes the overall detection decision. Each of the other nodes in the tree is a relay node that combines two binary messages to form a single output binary message. Only the leaves are sensors. In this way, the information from the sensors is aggregated into the fusion center via the relay nodes. In this context, we describe the evolution of the Type I and Type II error probabilities of the binary data as it propagates from the leaves toward the root. Tight upper and lower bounds for the total error probability at the fusion center as functions ofNare derived. These characterize how fast the total error probability converges to 0 with respect toN, even if the individual sensors have error probabilities that converge to 1/2.

Tracking in a multipath environment poses many challenges. It is worth while to quantify the achievable performance bounds in such an environment. However finding the performance bounds could be challenging. This requires calculating derivatives of parameters that are functions of reflection points on walls, with respect to target related quantities. We propose a method to calculate the lower bound for minimum mean squared error (MMSE) in target state estimation in a multipath environment. A novel measurement model is introduced taking into account a general setup with locations of obstacle walls known. Furthermore we model the wall reflectivities as random variables. Simulation results are provided in which we have obtained the bound for various setups.

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.

This paper introduces the Reed Muller Sieve, a deterministic measurement matrix for compressed sensing. The columns of this matrix are obtained by exponentiating codewords in the quaternary second order Reed Muller code of length N. For k = O(N), the Reed Muller Sieve improves upon prior methods for identifying the support of a k-sparse vector by removing the requirement that the signal entries be independent. The Sieve also enables local detection; an algorithm is presented with complexity N2log N that detects the presence or absence of a signal at any given position in the data domain without explicitly reconstructing the entire signal. Reconstruction is shown to be resilient to noise in both the measurement and data domains; the ℓ2/ℓ2error bounds derived in this paper are tighter than the ℓ2/ℓ1bounds arising from random ensembles and the ℓ1/ℓ1bounds arising from expander-based ensembles.

This paper focuses on conditional optimization as a decoding primitive for high rate space-time codes that are obtained by multiplexing in the spatial and code domains. The approach is a crystallization of the work of Hottinen which applies to space-time codes that are assisted by quasi-orthogonality. It is independent of implementation and is more general in that it can be applied to space-time codes such as the Golden Code and perfect space-time block codes, that are not assisted by quasi-orthogonality, to derive fast decoders with essentially maximum likelihood (ML) performance. The conditions under which conditional optimization leads to reduced complexity ML decoding are captured in terms of the induced channel at the receiver. These conditions are then translated back to the transmission domain leading to codes that are constructed by multiplexing orthogonal designs. The methods are applied to several block space-time codes obtained by multiplexing Alamouti blocks where it leads to ML decoding with complexityO(N2) whereNis the size of the underlying QAM signal constellation. A new code is presented that tests commonly accepted design principles and for which decoding by conditional optimization is both fast and ML. The two design principles for perfect space-time codes are nonvanishing determinant of pairwise differences and cubic shaping, and it is cubic shaping that restricts the possible multiplexing structures. The new code shows that it is possible to give up on cubic shaping without compromising code performance or decoding complexity.

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 times 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.

Multiple-input multiple-output (MIMO) functionality has been shown to dramatically increase the capacity of wireless communication systems when the environment provides rich multipath scattering. In a predominantly line-of-sight (LOS) environment, the loss of diversity reduces the potential gain considerably. This can be remedied in part by the use of dual-polarized antennas, which increases the rank of the wireless channel and introduces diversity, while minimizing the antenna's form factor. However the performance of a dualpolarized antenna is still degraded by antenna rotations that are typical of mobile terminal operation. This paper presents a solution which uses a triad antenna at the transmitter and a triad at the receiver, to provide a 8-10 dB gain over the baseline dual-polarized system. A triad is composed of three orthogonal dipoles oriented in perpendicular directions. A triad antenna can generate an arbitrary oscillating dipole moment at the transmitter and consequently an arbitrary polarized electric field at the receiver, subject only to the constraints imposed by the physics of the electromagnetic (EM) field. We show that, in LOS environments, the capacity of the channel is invariant under arbitrary rotations of the transmit and/or receive antennas about their centres. Simulation results show that the performance is stable as the propagation environment varies from rich scattering to pure LOS. A full rate 3 × 3 space-time block code (STBC) is proposed for the triad system that is designed for low complexity decoding.

The use of complementary codes as a means of reducing radar range sidelobes is well-known, but lack of resilience to Doppler is often cited as a reason not to deploy them. This work describes techniques for providing Doppler resilience with an emphasis on tailoring Doppler performance to the specific aim of target tracking. The Doppler performance can be varied by suitably changing the order of transmission of multiple sets of complementary waveforms. We have developed a method that improves Doppler performance significantly by arranging the transmission of multiple copies of complementary waveforms according to the first order Reed-Muller codes. Here we demonstrate significant tracking gains in the context of accelerating targets by the use of adaptively chosen waveform sequences of this kind, compared to both a fixed sequence of similar waveforms, and an LFM waveform.

We describe a method of constructing a sequence (pulse train) of phase-coded waveforms, for which the ambiguity function is free of range sidelobes along modest Doppler shifts. The constituent waveforms are Golay complementary waveforms which have ideal ambiguity along the zero Doppler axis but are sensitive to nonzero Doppler shifts. We extend this construction to multiple dimensions, in particular to radar polarimetry, where the two dimensions are realized by orthogonal polarizations. Here we determine a sequence of two-by-two Alamouti matrices where the entries involve Golay pairs and for which the range sidelobes associated with a matrix-valued ambiguity function vanish at modest Doppler shifts. The Prouhet–Thue–Morse sequence plays a key role in the construction of Doppler resilient sequences of Golay complementary waveforms.

Space–time codes built out of Alamouti components have been adopted in wireless standards such as UMTS, IEEE 802.11n, 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 correspondence provides new theoretical insight into different algorithms for interference cancellation through a Bayesian analysis that expresses performance as a function of signal-to-noise ratio (SNR) in terms of the “angles” between different space–time coded data streams.

In this paper, we propose a method to exploit waveform agility in modern radars to improve performance in the challenging task of detecting small targets on the ocean surface in heavy clutter. The approach exploits the compound-Gaussian model for sea clutter returns to achieve clutter suppression by forming an orthogonal projection of the received signal into the clutter subspace. Waveform scheduling is then performed by incorporating the information about the clutter into the design of the next transmitted waveform. A simulation study demonstrates the effectiveness of our approach.

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.

This paper describes a new radar primitive that enables instantaneous radar polarimetry at essentially no increase in signal processing complexity. This primitive coordinates transmission of distinct waveforms on orthogonal polarizations and applies a unitary matched filter bank on receive. This avoids the information loss inherent in single-channel matched filters. A further advantage of this scheme is the elimination of range sidelobes

We investigate the theory of the finite discrete Heisenberg-Weyl group in relation to the development of adaptive radar. We contend that this group can form the basis for the representation of the radar environment in terms of operators on the space of waveforms. We also demonstrate, following recent developments in the theory of error correcting codes, that the finite discrete Heisenberg-Weyl group provides a unified basis for the construction of useful waveforms/sequences for radar, communications and the theory of error correcting codes.

Modern radar systems have considerable flexibility in their modes of operation. In particular, it is possible to modify the waveform on a pulse to pulse basis, and electronically steered phased arrays can quickly point the radar beam in any feasible direction. Such flexibility calls for new methods of scheduling, both of the waveform and the beam direction so as to optimize the radar performance. We consider a radar system capable of rapid beam steering and of waveform switching. The transmit waveform is chosen from a small library. The operational requirement of the radar is to track a number of manoeuvring targets while performing surveillance for new potential targets. Tracking is accomplished by means of an LMIPDA (linear multitarget integrated probabilistic data association) tracker. An interacting multiple models (IMM) method is used to model manoeuvering targets in the tracker. LMIPDA provides a probability of track existence, permitting adoption of a track-before-detect technique. False alarm tracks are maintained until the probability of track existence falls below a threshold. Our aim is to maintain the tracks of the existing targets to within a specified accuracy as determined by the absolute value of the track error covariance matrix. However, this has to be done within the time available, given that a full scan has to be performed within a prescribed interval. We give an algorithm for scheduling revisits to measure the targets while maintaining surveillance.

There is a signal-to-noise ratio (SNR) below which the frequency of a coherent radar pulse train can be estimated no more accurately than the incoherent combination of the frequency estimates from the individual pulses. This SNR threshold applies to all frequency estimators, and defines a fundamental limit for passive location of radar emitters using differential Doppler. In this paper, an approximation to the estimation variance of any maximum likelihood (ML) type estimator is derived. The approximation is shown to be a very good predictor of the mean square error performance of ML estimators and the SNR at which the threshold occurs. A rigorous Barankin bound on any unbiased estimator of frequency is derived and compared to the above approximate estimation variance.

A wavelet transform is introduced as a means of detecting the characteristic scale or period T of a radar pulse sequence in an incoming stream of pulses. A particular choice of mother wavelet contains a fixed number, M, of cycles of a complex exponential, which provides a resolution of 1/M times T. The transform operates on interleaved guise sequences and, by thresholding, determines T for each component sequence. The detector is robust against missing pulses and timing jitter and is sensitive to simple, staggered and complex pulse sequences. The method is an improvement on other established approaches, such as the time-difference-of-arrival (TDOA) histogram and the periodogram.