Showing papers on "Gaussian process published in 1971"

The information capacity of amplitude- and variance-constrained sclar gaussian channels

Abstract: The amplitude-constrained capacity of a scalar Gaussian channel is shown to be achieved by a unique discrete random variable taking on a finite number of values. Necessary and sufficient conditions for the distribution of this random variable are obtained. These conditions permit determination of the random variable and capacity as a function of the constraint value. The capacity of the same Gaussian channel subject, additionally, to a nontrivial variance constraint is also shown to be achieved by a unique discrete random variable taking on a finite number of values. Likewise, capacity is determined as a function of both amplitude- and variance-constraint values.

Optimal adaptive estimation: Structure and parameter adaption

Abstract: Optimal structure and parameter adaptive estimators have been obtained for continuous as well as discrete data Gaussian process models with linear dynamics. Specifically, the essentially nonlinear adaptive estimators are shown to be decomposable (partition theorem) into two parts, a linear nonadaptive part consisting of a bank of Kalman-Bucy filters and a nonlinear part that incorporates the adaptive nature of the estimator. The conditional-error-covariance matrix of the estimator is also obtained in a form suitable for on-line performance evaluation. The adaptive estimators are applied to the problem of state estimation with non-Gaussian initial state, to estimation under measurement uncertainty (joint detection-estimation) as well as to system identification. Examples are given of the application of the adaptive estimators to structure and parameter adaptation indicating their applicability to engineering problems.

RKHS approach to detection and estimation problems--I: Deterministic signals in Gaussian noise

Abstract: First it is shown how the Karhunen-Loeve approach to the detection of a deterministic signal can be given a coordinate-free and geometric interpretation in a particular Hilbert space of functions that is uniquely determined by the covariance function of the additive Gaussian noise. This Hilbert space, which is called a reproducing-kernel Hilbert space (RKHS), has many special properties that appear to make it a natural space of functions to associate with a second-order random process. A mapping between the RKHS and the linear Hilbert space of random variables generated by the random process is studied in some detail. This mapping enables one to give a geometric treatment of the detection problem. The relations to the usual integral-equation approach to this problem are also discussed. Some of the special properties of the RKHS are developed and then used to study the singularity and stability of the detection problem and also to suggest simple means of approximating the detectability of the signal. The RKHS for several multidimensional and multivariable processes is presented; by going to the RKHS of functionals rather than functions it is also shown how generalized random processes, including white noise and stationary processes whose spectra grow at infinity, are treated.

Signal detection in Gaussian noise of unknown level: An invariance application

Abstract: The concept of invariance in hypothesis testing is brought to bear on the problem of detecting signals of known form and unknown energy in Gaussian noise of unknown level. The noise covariance function is assumed to be K(t,u) = \sigma^2 \pho(t,u) where \rho(t,u) is the known form of the covariance function and \sigma^2 is the unknown level. Classical approaches to signal detection depend on the assumption that K(t,u) is known completely. Then, a correlation-type receiver that is the uniformly most powerful (UMP) test of H_o (signal absent) versus H_1 (signal present) can be derived. When \sigma^2 is unknown, there exists no UMP test. However, it is shown in this paper that there exists a test of H_o versus H_1 that is UMP-invariant for a very natural group of transformations on the space of observations. The derived test is found to be independent of knowledge about the noise level \sigma^2 , since the derived test (receiver) contains an error-free estimate of \sigma^2 . This utopian conclusion is reconciled by noting that the derived receiver can never be physically realized. It is shown that any physically realizable version of the receiver has a t -distributed test statistic. This permits choice of operating receiver thresholds and evaluation of performance characteristics.

On the First-Excursion Probability in Stationary Narrow-Band Random Vibration, II

Abstract: The first-excursion probability of a stationary narrow-band Gaussian process with mean zero has been studied. Within the framework of point process approach, series approximations derived from the theory of random points and approximations based on the maximum entropy principle have been developed. With the aid of numerical examples, merits of the approximations proposed previously as well as of those developed in this paper have been compared. The results indicate that the maximum entropy principle has not produced satisfactory approximations but the approximation based on nonapproaching random points is found to be the best among all the approximations proposed herein. A conclusion drawn from the present and the previous studies is that the point process approach produces a number of useful approximations for the first-excursion probability, particularly those based on the concepts of the Markov process, the clump-size, and the nonapproaching random points.

Abstract Wiener processes and their reproducing Kernel Hilbert spaces

Abstract: : The paper explores the relationship between Gaussian processes and their associated RKH Spaces. A simple proof of Gross's theorem on abstract Wiener spaces is given. For a Gaussian measure mu with continuous covariance R defined on the Banach space C(T) of real continuous functions on T (T being a separable complete metric space) it is shown that the closure of H(R) in C(T) is the support of mu. This result is extended to Gaussian measures on arbitrary separable Banach spaces. A necessary and sufficient criterion for a separable Gaussian process x(t) (0 = or < t = or < 1) with continuous covariance R to have continuous sample paths is furnished by the following result to the effect that the canonical normal distribution on H(R) extends to a Gaussian measure on C(0,1) if and only if the sup-norm on H(R) is measurable in the sense of Gross. (Author)

Complex Gaussian noise moments

Abstract: The problem of the computation of moments of nonzero mean circularly complex Gaussian noise is treated. The noise need not be symmetric about the carrier frequency. In particular, the second-order moments are computed, and expansions are given. The necessary univariate and bivariate complex Hermite polynomials are discussed. The means of some rational functions useful in FM theory are given. This paper extends work of Rice, Middleton, and Zadeh to complex Gaussian noise with nonzero mean and nonsymmetrical power spectrum.

An Asymptotic 0-1 Behavior of Gaussian Processes

Abstract: : The paper presents a result which generalizes the paper of Watanabe (Trans. AMS 148, 233-248). The result is extended also to the nonstationary process treated by Watanabe.

Some geometric properties of the likelihood ratio (Corresp.)

Abstract: A fundamental spatial property is noted that relates the likelihood ratio for random signals in Gaussian noise to conditional-mean parameter estimation. This allows a geometric interpretation of the estimation-correlation operation. Some additional properties exhibited by the likelihood ratio are also presented.

Random Walk Model for First-Passage Probability

Abstract: A numerical method is developed for the calculation of first-passage time probability of single degree-of-freedom, linear and nonlinear, dynamic oscillators excited by Gaussian white noise. The random walk model is a difference equation which governs the diffusion of the oscillators response probability in the phase plane and is a discrete analog to the continuous theory Fokker-Planck equation. First-passage is examined by considering the diffusion process when absorbing barriers are superimposed upon the phase plane. Two linear systems are studied for first-passage and compared to results from another numerical approach with good correlation. First-passage is also examined for several nonlinear systems to demonstrate its applicability. It is the study of the latter for which the technique has particular value.

The Generation of Correlated Log-Normal Clutter for Radar Simulations

Abstract: From a set of N statistically independent Gaussian random variables, N correlated log-normal random variables may be generated to represent the amplitude of correlated clutter signals. Means are developed to realize specified mean value and covariance matrices of the desired clutter correlated log-normal variables. The realization is in the form of a weighting matrix to produce a set of correlate Gaussian random variables which become log-normal after a suitable nonlinear transformation.

Plane-Wave-Gaussian Energy-Band Study of Nb

Abstract: The numerical accuracy of the plane-wave-Gaussian (PWG) mixed-basis method of calculating crystalline energy bands is displayed for Nb. Atomic studies involving the Gaussian basis and crystalline studies involving the PWG mixed basis are systematically developed to establish the dependence of the energy eigenvalues upon the number of Gaussians, the number of plane waves, and the Gaussian overlap parameter. Use of the Nb crystalline potential of Deegan and Twose allows comparison with their results.

Optimum waveforms for sonar velocity discrimination

Abstract: Velocity resolvent wide-band sonar waveforms are derived by using the variational calculus together with some properties of the wide-band ambiguity function.

Group codes for phase- and amplitude-modulated signals on a Gaussian channel

Abstract: A group code is a set of vectors, generated by a group of orthogonal matrices transforming a starting vector. Ingemarsson [5] has shown that, if the group is commutative, the code is equivalent to a set of phase-modulated signals. Codes generated by noncommutative groups have been described by Slepian [9]. In this paper another class of codes generated by noncommutative groups is described. This class is called group codes for PHAMP signals and can be described as a combination of phase- and amplitude-modulated signals. The performance of these codes, when communicating over the Gaussian channel, is analyzed, and it is shown that some of them have a performance close to the bounds given by Slepian [8].

Signal design for fast-fading Gaussian channels

Abstract: The design of signals for digital communication over fast-fading Gaussian channels is considered; the emphasis is on nonorthogonal signaling schemes. A discrete channel model is used. Necessary and sufficient conditions on the transmitted signals are found that make the Bayes receiver independent of the channel parameters. By using a geometric interpretation of the resultant receiver a heuristic design criterion is developed. Then union bounds for particular nonorthogonal signaling schemes are evaluated. Nonorthogonal schemes based on modulation similar to differential-phase-shift-keyed (DPSK) modulation are found to use substantially less bandwidth than equivalent schemes based on generalized frequency-shift-keyed modulation.

An upper bound for probability of error related to a given decision region in N-dimensional signal set (Corresp.)

Abstract: An upper bound for the probability of error related to a given decision region, in an arbitrary N -dimensional signal set transmitted over a coherent Gaussian white noise channel, is described here. This bound is valid for any number of dimensions and for any closed decision region that satisfies a very weak condition.

Supervised learning sequential structure and parameter adaptive pattern recognition: Discrete data case (Corresp.)

Abstract: In a previous paper [1], Bayes-optimal recursive supervised learning structure and parameter adaptive pattern recognition systems were derived for continuous "lumped" Gaussian processes. In this paper, the discrete data case is considered, and the discrete data version of the partition theorem is derived. Several examples are also presented of the application of the adaptive detectors, and computational results are given indicating their learning capacity and convergence rate.

Convergence to the rate-distortion function for Gaussian sources

Abstract: In this paper we derive an expression for the minimum-mean-square error achievable in encoding t samples of a stationary correlated Gaussian source. It is assumed that the source output is not known exactly but is corrupted by correlated Gaussian noise. The expression is obtained in terms of the covariance matrices of the source and noise sequences. It is shown that as t \rightarrow \infty , the result agrees with a known asymptotic result, which is expressed in terms of the power spectra of the source and noise. The rate of convergence to the asymptotic results as a function of coding delay is investigated for the case where the source is first-order Markov and the noise is uncorrelated. With D the asymptotic minimum-mean-square error and D_t the minimum-mean-square error achievable in transmitting t samples, we find \mid D_t - D \mid \leq O((t^{-1} \log t) ^ {1/2}) when we transmit the noisy source vectors over a noiseless channel and \mid D_t - D \mid \leq O((t^{-1} \log t)^ {1/3}) when the channel is noisy.

Variances of spectral parameters with a Gaussian shape (Corresp.)

Abstract: In the paper [1] by the late M. J. Levin, maximum likelihood estimates and limits on their variances are derived for the parameters of the power spectrum of a zero-mean stationary Gaussian process. In this note we rederive the variances of these estimates in a somewhat different manner and give numerical results for power spectra with a Gaussian shape.

Realization of the Gauss-in-Gauss detector using minimum-mean-squared-error filters (Corresp.)

Abstract: Alternative structures for the optimum detection of Gaussian signals in Gaussian noise are derived that can be interpreted in terms of minimum-mean-squared-error (MMSE) estimators of signal and noise. The realization is useful when the statistics of the signal or noise or both are unknown since the detector can be implemented in an adaptive mode by using tapped delay lines whose weights are adjusted recursively to yield the minimum-mean-squared-error estimate of certain components of the incoming waveforms.

Estimation of randomly occurring stochastic signals in Gaussian noise (Corresp.)

Abstract: This note applies the general likelihood ratio for the minimum-mean-square-error estimation of the signal when the presence of the signal at the receiver is uncertain during the entire observation interval.

On the correlation function of signal plus Gaussian noise passed through nonlinear devices (Corresp.)

Abstract: This correspondence discusses the transformations between the output autocorrelation functions of functionally related zero-memory nonlinear devices excited by signal plus Gaussian noise. The results are applied to show how output autocorrelation functions can easily be obtained for exponential and half-wave power law devices.

Rate distortion over band-limited feedback channels (Corresp.)

Abstract: Although linear feedback is by itself sufficient to achieve capacity of an additive Gaussian white noise (AGWN) channel, it can not, in general, achieve the theoretical minimum mean-squared error for analog Gaussian data. This correspondence gives the necessary and sufficient conditions under which this optimum performance can be achieved.