Array algorithms for H∞ estimation

We develop array algorithms for H^∞ filtering. These algorithms can be regarded as the Krein space generalizations of H ² array algorithms, which are currently the preferred method fur implementing H² filters. The array algorithms considered include two main families: square-root array algorithms, which are typically numerically more stable than conventional ones, and fast array algorithms which, when the system is time-invariant, typically offer an order of magnitude reduction in the computational effort. Both have the interesting feature that one does not need to explicitly check for the positivity conditions required for the existence of H^∞ filters, as these conditions are built into the algorithms themselves. However, since H^∞ square-root algorithms predominantly use J-unitary transformations, rather than the unitary transformations required in the H² case, further investigation is needed to determine the numerical behavior of such algorithms     

Discrete-time complementary models and smoothing

The fact that the smoothing error is a (wide sense) Markov process is somehow surprising since smoothed estimates depend upon both past and future data. In this paper we first give a simple and general proof of this fact. Then we use the so-called complementary models introduced by Weinert and Desai to derive forwards and backwards markovian models for the smoothing error in state-space models. By exploring the structure of the complementary models we show that, under certain restrictions, only two simple structured models exist, one that runs forwards in time and the other that runs backwards in time. The forwards complementary model leads to the forward Rauch-Tung-Striebel (RTS) smoothing formula and to a backwards markovian model for the error, whereas the backwards model leads to the backward RTS formula and to a forwards error model. The two models for the smoothing error can be derived one from the other by a forward-backward transformation that preserves the sample paths. Finally, by using a combination of the two complementary models we give yet another proof for the two-filter smoothing formula.     

A performance analysis of subspace-based methods in the presence ofmodel error. II. Multidimensional algorithms

For pt.I, see ibid., vol.40, no.7, p.1758-74 (1992). In pt.I the performance of the MUSIC algorithms for narrowband direction-of-arrival (DOA) estimation when the array manifold and noise covariance are not correctly modeled was investigated. This analysis is extended to multidimensional subspace-based algorithms including deterministic (or conditional) maximum likelihood, MD-MUSIC, weighted subspace fitting (WSF), MODE, and ESPRIT. A general expression for the variance of the DOA estimates that can be applied to any of the above algorithms and to any of a wide variety of scenarios is presented. Optimally weighted subspace fitting algorithms are presented for special cases involving random unstructured errors of the array manifold and noise covariance. It is shown that one-dimensional MUSIC outperforms all of the above multidimensional algorithms for random angle-independent array perturbations     

Linear estimation in Krein spaces. II. Applications

We have shown that several interesting problems in H^∞ -filtering, quadratic game theory, and risk sensitive control and estimation follow as special cases of the Krein-space linear estimation theory developed in Part I. We show that all these problems can be cast into the problem of calculating the stationary point of certain second-order forms, and that by considering the appropriate state space models and error Gramians, we can use the Krein-space estimation theory to calculate the stationary points and study their properties. The approach discussed here allows for interesting generalizations, such as finite memory adaptive filtering with varying sliding patterns     

Estimation of skew angle in text-image analysis bySLIDE: Subspace-based line detection

A new signal processing method is developed for estimating the skew angle in text document images. Detection of the skew angle is an important step in text processing tasks such as optical character recognition (OCR) and computerized filing. Based on a recently introduced multiline-fitting algorithm, the proposed method reformulates the skew detection problem into a special parameter-estimation framework such that a signal structure similar to the one in the field of sensor array processing is obtained. In this framework, straight lines in an image are modeled as wavefronts of propagating planar waves. Certain measurements are defined in this virtual propagation environment such that the large amount of coherency that exists between the locations of the pixels on parallel lines is exploited to enhance a subspace in the space spanned by the measurements. The well-studied techniques of sensor array processing (e.g., the ESPRIT algorithm) are then exploited to produce a closed form and high-resolution estimate for the skew angle.     

On the sensitivity of the ESPRIT algorithm to non-identical subarrays

ESPRIT (estimation of signal parameters via rotational invariance techniques) is a recently introduced algorithm for narrowband direction-of-arrival (DOA) estimation. Its principal advantage is that the DOA parameter estimates are obtained directly, without knowledge (and hence storage) of the array manifold and without computation or search of some spectral measure. This advantage is achieved by constraining the sensor array to be composed of two identical, translationally invariant subarrays. In this paper, we analyse the sensitivity of ESPRIT to the assumption that the subarrays are identical. The analysis is applicable to a wide variety of array errors, including non-identical angle-dependent and angle-independent gain and phase perturbations, errors in the locations of the subarray elements, and mutual coupling effects. A representative simulation example will be presented to validate the analysis and compare the performance degradation of ESPRIT with that of the MUSIC algorithm.     

The asymptotic stability of nonlinear (Lur'e) systems with multipleslope restrictions

The authors present the analysis of the asymptotic stability of multiple slope-restricted nonlinear (Lur'e) systems. By providing a Lyapunov function, they obtain a matrix-language criterion in terms of algebraic Riccati equations and linear matrix inequalities, which are discussed at the point of computational issues. Additionally, they consider the frequency-domain interpretation of the result     

Detection and estimation in sensor arrays using weighted subspacefitting

The problem of signal parameter estimation of narrowband emitter signals impinging on an array of sensors is addressed. A multidimensional estimation procedure that applies to arbitrary array structures and signal correlation is proposed. The method is based on the recently introduced weighted subspace fitting (WSF) criterion and includes schemes for both detecting the number of sources and estimating the signal parameters. A Gauss-Newton-type method is presented for solving the multidimensional WSF and maximum-likelihood optimization problems. The global and local properties of the search procedure are investigated through computer simulations. Most methods require knowledge of the number of coherent/noncoherent signals present. A scheme for consistently estimating this is proposed based on an asymptotic analysis of the WSF cost function. The performance of the detection scheme is also investigated through simulations     

Immittance-domain Levinson algorithms

Several computationally efficient versions of the Levinson algorithm for solving linear equations with Toeplitz and quasi-Toeplitz matrices are presented, motivated by a new stability test. The new versions require half the number of multiplications and the same number of additions as the conventional form of the Levinson algorithm. The saving is achieved by using three-term (rather than two-term) recursions and propagating them in an impedance/admittance (or immittance) domain rather than the conventional scattering domain. One of the recursions coincides with recent results of P. Delsarte and Y. Genin (IEEE Trans., Acoust. Speech, Signal Proc., vol.ASSP-34, p.470-8, June 1986) on split Levinson algorithms for symmetric Toeplitz matrices, where the efficiency is gained by using the symmetric and skew-symmetric versions of the usual polynomials. This special structure is lost in the quasi-Toeplitz case, but one still can obtain similar computational reductions by suitably using three-term recursions in the immittance domain     

SLIDE: subspace-based line detection

An analogy is made between each straight line in an image and a planar propagating wavefront impinging on an array of sensors so as to obtain a mathematical model exploited in recent high resolution methods for direction-of-arrival estimation in sensor array processing. The new so-called SLIDE (subspace-based line detection) algorithm then exploits the spatial coherence between the contributions of each line in different rows of the image to enhance and distinguish a signal subspace that is defined by the desired line parameters. SLIDE yields closed-form and high resolution estimates for line parameters, and its computational complexity and storage requirements are far less than those of the standard method of the Hough transform. If unknown a priori, the number of lines is also estimated in the proposed technique. The signal representation employed in this formulation is also generalized to handle grey-scale images as well. The technique has also been generalized to fitting planes in 3-D images. Some practical issues of the proposed technique are given