Browsing by Author "Gillis, J.T."
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Item Computation of the Circular Error Probability Integral(1991) Gillis, J.T.; ISRThis note describes a simplified derivation of the representation of the circular error probability (CEP) integral, which is the integral over a disk centered at the origin of a zero mean two dimensional Gaussian random variable, as a one-dimensional integral. In addition, a rapidly converging series expression is derived for the CEP.The integral occurs in the evaluation of communication and radar signals, and other statistical applications.
Item Conditions for the Equivalence of ARMAX and ARX Systems(1991) McGraw, G.A.; Gustafson, C.L.; Gillis, J.T.; ISRIt is shown that an autoregressive moving average with exogenous input (ARMAX) system can be represented as an autoregressive with exogenous input (ARX) model if and only if the transfer function from the noise port to the output port has no transmission zeros. A construction using the matrix fractional description of the system is used to prove this result. This construction shows that, by proper addition of sensor measurements and extending the order of the ARX model, accurate parameter estimates of systems driven by unmeasured disturbances can be obtained.Item Random Sampling of Random Fields: Least Squares Estimation(1991) Gillis, J.T.; ISRThe paper begins with a discussion of deterministic sampling, where it is observed that when one can reconstruct the covariance one can also reconstruct the sample path (in quadratic mean). Then the theorem of Shapiro and Silverman, which states that Poisson based sampling allows reconstruction of the covariance at any sampling rate and a construction of an estimator of the covariance (due to Papoulis) are presented. A class of estimators for random fields using Poisson (and Poisson like) sampling is developed. The optimal estimator (minimum mean square error) is shown to exist and the error is shown to go to zero only as the sampling rate goes to infinity; Poisson sampling behaves differently from regular sampling in this respect. Poisson sampling is shown to be the best (lowest error) for a wide class of multidimensional point processes (sampling meassures). One feature of the development is that it applies directly in IR (Euclidian N-Space). It is shown that the optimal estimator has many desirable properties (continuity, etc.); however, recursion in terms of the density of the sampling processes is not easily developed. A sub-optimal estimator with this desirable property is also discussed. In the case that the random field is Gaussian, the proposed estimator is seen to be the conditional mean.Item Reconstruction of Stochastic Processes Using Frames(1991) Gillis, J.T.; ISRThis note discusses sampling in a general context and shows that the (dual) frame reconstruction formula holds for stochastic processes, in quadratic mean. Specifically we show that if the covariance can be reconstructed using frames then the sample path can also be reconstructed. The application of the result for the generation of approximate sample paths for simulation is discussed. Ergodic properties of the approximate estimators are also investigated.