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>Z< The Geometry for Detection Theory Anand G. Dabak and Don H. Johnson

Using the tools of category theory and differential geometry, we extend the geometric notions consequent of Gaussian detection problems to non-Gaussian ones. The nominal probability measures associated with the hypotheses form points on an infinite dimensional manifold. These measures may correspond to non-additive as well as additive noise situations and can express a variety of dependence structures. By imposing detection theoretic constraints on the manifold's structure, we show that the natural geometry for detection theory is non-Riemannian and that the nominal measures are connected by geodesic curves formed from the exponential mixtures of these measures. While no distance measure exists between nominals on this manifold, the Kullback-Leibler information is shown to be related to squared intermeasure distance. The geometry is used to pose and solve classic robust detection problems and to find biasing densities that guarantee significant importance sampling gain in detector performance simulation.

>Z< Mean-Squared Error Analysis of Kernel Regression Estimator for Time Series Y. Kang Lee and Don H. Johnson

Because of a lack of a priori information, the minimum mean-squared error predictor, the conditional expectation, is often not known for a non-Gaussian time series. We show that the nonparametric kernel regression estimator of the conditional expectation is mean-squared consistent for a time series: When used as a predictor, the estimator asymptotically matches the mean-squared error produced by the true conditional expectation. We also describe a more computationally efficient predictor based on the recursive kernel regression estimator, and show it can asymptotically achieve mean-squared errors arbitrarily close to the conditional expectation. Numerical examples are provided to demonstrate the effectiveness of nonparametric prediction.

>Z< Wiener-Optimum Linear Detectors for Non-Gaussian Channels Don H. Johnson

Optimal detectors for non-Gaussian channels are nonlinear, and are therefore sensitive to problem parameters. We describe a technique for designing linear detectors for arbitrary additive noise channels that takes into account dependence structure, amplitude distribution, and transmitted signals. This technique is based on Wiener's theory of nonlinear systems, and amounts to cross-correlating the optimal detector's output with its input when it is driven by white Gaussian noise. We have not proven this linear detector's optimality; simulations demonstrate that it has some of the properties required of optimal detectors in several cases.

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>Z< Optimal Linear Detectors for Additive Noise Channels Don H. Johnson

We describe procedures for designing optimal and suboptimal linear detectors (discrete-time case) for additive white noise channels. We first describe a technique for designing linear detectors based on the Wiener theory of nonlinear systems. Next, exact expressions for optimal linear detector are found when the noise amplitude's probability distribution is stable. We then describe how to computationally design the optimal linear detector when the noise is Laplacian. Considering all the linear detectors thus derived, the ad hoc Wiener approach is shown to be suboptimal, and no general form for the optimal linear detector's unit-sample response is apparent. Performance analyses and simulations indicate substantial performance losses occur when linear detectors are used instead of optimal (likelihood ratio) ones.