Showing papers by "Louis L. Scharf published in 2005"

The adaptive coherence estimator: a uniformly most-powerful-invariant adaptive detection statistic

Abstract: We show that the Adaptive Coherence Estimator (ACE) is a uniformly most powerful (UMP) invariant detection statistic. This statistic is relevant to a scenario appearing in adaptive array processing, in which there are auxiliary, signal-free, training-data vectors that can be used to form a sample covariance estimate for clutter and interference suppression. The result is based on earlier work by Bose and Steinhardt, who found a two-dimensional (2-D) maximal invariant when test and training data share the same noise covariance. Their 2-D maximal invariant is given by Kelly's Generalized Likelihood Ratio Test (GLRT) statistic and the Adaptive Matched Filter (AMF). We extend the maximal-invariant framework to the problem for which the ACE is a GLRT: The test data shares the same covariance structure as the training data, but the relative power level is not constrained. In this case, the maximal invariant statistic collapses to a one-dimensional (1-D) scalar, which is also the ACE statistic. Furthermore, we show that the probability density function for the ACE possesses the property of "total positivity," which establishes that it has monotone likelihood ratio. Thus, a threshold test on the ACE is UMP-invariant. This means that it has a claim to optimality, having the largest detection probability out of the class of detectors that are also invariant to affine transformations on the data matrix that leave the hypotheses unchanged. This requires an additional invariance not imposed by Bose and Steinhardt: invariance to relative scaling of test and training data. The ACE is invariant and has a Constant False Alarm Rate (CFAR) with respect to such scaling, whereas Kelly's GLRT and the AMF are invariant, and CFAR, only with respect to common scaling.

Interference cancellation in adjoint operators for communication receivers

Abstract: A receiver in a wireless communication system comprises a reverse transform configured to produce a vector of baseband signal values, and a projection canceller configured to project the vector of baseband signal values onto at least one subspace that is substantially orthogonal to an interference subspace. The reverse transform may be adjoint to a forward transform employed by at least one transmitter in the wireless communication system. The combination of interference cancellation with one or more receiver operations may be a substantially adjoint operation relative to one or more transmitter operators and channel-propagation effects. The reverse transform may include a Fourier transform, a wavelet transform, or any other well known invertible transforms. Reverse transforms may include spread-spectrum multiple-access coding and may be implemented in systems configured to perform single-input, multiple output or multiple-input, multiple-output operations. Interference components may be selected in a projection canceller relative to predetermined ratios of interference in the received signal.

Detection and estimation of improper complex random signals

Abstract: Nonstationary complex random signals are in general improper (not circularly symmetric), which means that their complementary covariance is nonzero. Since the Karhunen-Loeve (K-L) expansion in its known form is only valid for proper processes, we derive the improper version of this expansion. It produces two sets of eigenvalues and improper observable coordinates. We then use the K-L expansion to solve the problems of detection and estimation of improper complex random signals in additive white Gaussian noise. We derive a general result comparing the performance of conventional processing, which ignores complementary covariances, with processing that takes these into account. In particular, for the detection and estimation problems considered, we find that the performance gain, as measured by deflection and mean-squared error (MSE), respectively, can be as large as a factor of 2. In a communications example, we show how this finding generalizes the result that coherent processing enjoys a 3-dB gain over noncoherent processing.

Matched direction detectors and estimators for array processing with subspace steering vector uncertainties

Abstract: In this paper, we consider the problem of estimating and detecting a signal whose associated spatial signature is known to lie in a given linear subspace but whose coordinates in this subspace are otherwise unknown, in the presence of subspace interference and broad-band noise. This situation arises when, on one hand, there exist uncertainties about the steering vector but, on the other hand, some knowledge about the steering vector errors is available. First, we derive the maximum-likelihood estimator (MLE) for the problem and compute the corresponding Crame/spl acute/r-Rao bound. Next, the maximum-likelihood estimates are used to derive a generalized likelihood ratio test (GLRT). The GLRT is compared and contrasted with the standard matched subspace detectors. The performances of the estimators and detectors are illustrated by means of numerical simulations.

Two-channel constrained least squares problems: solutions using power methods and connections with canonical coordinates

Abstract: The problem of two-channel constrained least squares (CLS) filtering under various sets of constraints is considered, and a general set of solutions is derived. For each set of constraints, the solution is determined by a coupled (asymmetric) generalized eigenvalue problem. This eigenvalue problem establishes a connection between two-channel CLS filtering and transform methods for resolving channel measurements into canonical or half-canonical coordinates. Based on this connection, a unified framework for reduced-rank Wiener filtering is presented. Then, various representations of reduced-rank Wiener filters in canonical and half-canonical coordinates are introduced. An alternating power method is proposed to recursively compute the canonical coordinate and half-canonical coordinate mappings. A deflation process is introduced to extract the mappings associated with the dominant coordinates. The correctness of the alternating power method is demonstrated on a synthesized data set, and conclusions are drawn.

The Hilbert space geometry of the Rihaczek distribution for stochastic analytic signals

Abstract: The Rihaczek distribution for stochastic signals is a time- and frequency-shift covariant bilinear time-frequency distribution (TFD) based on the Crame/spl acute/r-Loe/spl grave/ve spectral representation for a harmonizable process. It is a complex Hilbert space inner product (or cross correlation) between the time series and its infinitesimal stochastic Fourier generator. To this inner product, we may attach an illuminating geometry, wherein the cosine squared of the angle between the time series and its infinitesimal stochastic Fourier generator is given by the Rihaczek distribution. The Rihaczek distribution also determines a time-varying Wiener filter for estimating a time series from its infinitesimal stochastic Fourier generator and measures the resulting error covariance. We propose a factored kernel to construct estimators of the Rihaczek distribution that are contained in Cohen's class of bilinear TFDs.

Matched direction detectors

Abstract: In this paper, we address the problem of detecting a signal whose associated spatial signature is subject to uncertainties, in the presence of subspace interference and broadband noise, and using multiple snapshots from an array of sensors. To account for steering vector uncertainties, we assume that the spatial signature of interest lies in a given linear subspace while its coordinates in this subspace are unknown. The generalized likelihood ratio test (GLRT) for the problem at hand is formulated. We show that the GLRT amounts to searching for the best direction in the subspace after projecting out the interferences. The distribution of the GRLT under both hypotheses is derived and numerical simulations illustrate its performance.

Multi-Rank Adaptive Beamforming

Abstract: This paper presents a multi-rank generalization of the Capon beamformer to accommodate model mismatch in situations where the unknown signal of interest lies in a multidimensional subspace. By expanding the beamforming subspace robustness (or diversity) is achieved at the expense of resolution. The generalization involves solving a quadratically-constrained quadratic minimization problem, and designing a constraint matrix. Three strategies for designing this constraint matrix are discussed. Simulation examples are presented to demonstrate the performance of the multi-rank Capon beamformer

Multi-Rank Adaptive Beamforming with Linear and Quadratic Constraints

Abstract: Extensions of MVDR and Capon estimation techniques are presented for the situation in which the signal is either rank-one of unknown orientation in a subspace or multi-rank. Only signal-plus-noise snapshots are available. The relationships among linearly and quadratically-constrained approaches are clarified and a unified treatment is given that includes both direct and sidelobe canceller architectures. The unifying component is the multi-rank MVDR beamformer followed by post processing. Detection statistics are presented for the situation in which there is no signal-free training data. Simulations are used to compare rank-one and multi-rank performance.

Time-frequency and dual-frequency representation of fractional brownian motion

Abstract: Fractional Brownian motion (fBm) is a useful non-stationary model for certain fractal and long-range dependent processes of interest in telecommunications, physics, biology, and finance. Conventionally, the power spectrum of fBm is claimed to be a fractional power-law. However, fBm is not a wide-sense stationary process, so the precise meaning of this spectrum is unclear. In this paper, we model and analyze fBm in the context of harmonizable random processes. We derive and interpret exact expressions for novel useful complex valued second-order moment functions for fBm. These moment functions are time-frequency and dual-frequency correlation functions, connecting the random process to its infinitesimal random Fourier generator. In particular, we derive and discuss the time-frequency Rihaczek spectrum, and the dual-frequency Loeve spectrum. Our main finding is that the dual-frequency spectrum of fBm has its spectral support confined to three discrete lines. This leads to the surprising conclusion that for fBm, the DC component of the infinitesimal Fourier generator is correlated with ail other frequencies of the Fourier generator. We propose and apply multitaper based estimators for the moment functions, and numerical estimates based on synthetic fBm data and real world earthquake data confirm our theoretical results

Constrained Quadratic Minimizations for Signal Processing and Communications

Abstract: Constrained minimization problems considered here arise in the design of multi-dimensional subspace beamformers for radar, sonar, seismology, and wireless communications, and in the design of precoders and equalizers for digital communications. The problem is to minimize a quadratic form, under a set of linear or quadratic constraints. We derive the solutions to these problems and establish connections between them. We show that the quadratically-constrained problem can be solved by solving a set of linearly-constrained problems and then using a majorization argument and Poincare's separation theorem to determine which linearly-constrained problem solves the quadratically-constrained one. In addition, we present illuminating circuit diagrams for our solutions, called generalized sidelobe canceller (GSC) diagrams, which allow us to tie our constrained minimizations to linear minimum mean-squared error (LMMSE) estimations.

A geometric interpretation of the rihaczek time-frequency distribution for stochastic signals

Abstract: Based on the Cramer-Loeve spectral representation for a harmonizable random process, the Rihaczek distribution is a time- and frequency-shift covariant, bilinear time-frequency distribution. It can be expressed as a complex Hilbert space inner product between the time series and its infinitesimal stochastic Fourier generator. We show that we may attach an illuminating geometry to this inner product, wherein the cosine-squared of the angle between the time series and its infinitesimal stochastic Fourier generator is given by the Rihaczek distribution. We propose to construct estimators of the Rihaczek distribution using a factored kernel in Cohen's class of bilinear time-frequency distributions

Unattended sparse acoustic array configurations and beamforming algorithms

Abstract: Various sparse array configurations have been studied to improve spatial resolution for separating several closely spaced targets in tight formations using unattended acoustic arrays. To extend the array aperture, it is customary to employ sparse array configurations with uniform inter-array spacing wider than the half-wavelength intra-subarray spacing, hence achieving more accurate direction of arrival (DOA) estimates without using extra hardware. However, this larger inter-array positioning results in ambiguous DOA estimates. To resolve this ambiguity, sparse arrays with multiple invariance properties could be deployed. Alternatively, one can design regular or random sparse array configurations that provide frequency diversity, in which case every subarray is designed for a particular band of frequencies. These different configurations are investigated in this paper. Additionally, we present a Capon DOA algorithm that exploits the specific geometry of each array configuration. Simulation results are presented to study the pros and cons of different sparse configurations.© (2005) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Balanced classifiers and their fusion

Abstract: The problem of constructing a single classifier when multiple phenomenologies are measured by different sensor types is made more difficult because features take diversified forms, and classifiers built from them have variable performance. For example, features can be continuous or binary valued (as in discrete labels), or be composed of incompatible structural primitives. Therefore, it is difficult to lump all of these features together into a single classifier for decision making. This realization leads to the combined use of multiple classifiers. The solution presented in this paper describes the formulation and development of: A computational procedure for computing approximate hyperplane decision boundaries to achieve a balanced classifier. Achieving a minimum Bayes-risk balanced classifier as a convex combination of balanced classifiers. This is done for both independent and correlated cases. Convex combinations of balanced classifiers are balanced. However, our research has further generalized this concept by computing optimal convex combinations of classifiers so as to also attain the property of being minimum Bayes-risk for the combined classifier. The principle exploited was to incorporate either the decisions or the decision statistics of the individual classifiers within a combined confusion matrix considering both the correlated and independent cases. This was posed as an optimization problem to be approached via Markov-Chain Monte Carlo methods. Some preliminary results are shown.