Showing papers on "White noise published in 1968"

An innovations approach to least-squares estimation--Part II: Linear smoothing in additive white noise

Abstract: The innovations approach to linear least-squares approximation problems is first to "whiten" the observed data by a causal and invertible operation, and then to treat the resulting simpler white-noise observations problem This technique was successfully used by Bode and Shannon to obtain a simple derivation of the classical Wiener filtering problem for stationary processes over a semi-infinite interval Here we shall extend the technique to handle nonstationary continuous-time processes over finite intervals In Part I we shall apply this method to obtain a simple derivation of the Kalman-Bucy recursive filtering formulas (for both continuous-time and discrete-time processes) and also some minor generalizations thereof

Optimum Signal-Processing for Pulse-Amplitude Spectrometry in the Presence of High-Rate Effects and Noise

Abstract: In this paper the idea of a general signal processing system which should satisfy various pulse rate and noise requirements is explored. Optimum processing functions (weighting functions) are considered for an ideal system, and for real conditions where effects like imperfect pole-zero cancellation are present. Time-variant filters of the gain-varying class are used to realize the required optimum weighting functions of finite width. It is shown how nonfinite-width weighting functions of some time-invariant filters can be modified into finite-width functions by switching. These switched-gain time-variant filters are somewhat limited in choice of weighting functions. A general processing system can be realized employing filters with continuously time-variant elements. In particular, a gain-varying element (i.e., an analog multiplier) can be used in conjunction with an integrator to realize arbitrary weighting functions, and therefore the theoretically maximum signal-to-noise ratio. The system is time-variant only for the noise and not for the signal, so that it does not require high precision of the time-variant element. The system output is independent of the gating interval, and does not require precise timing. A method for evaluation of such systems in terms of noise, ballistic deficit and sensitivity to parameter variations is given.

Evaluation of likelihood functions

Abstract: An expression is obtained for the likelihood function for the detection of a stochastic signal (diffusion process) in white noise. A stochastic differential equation is then obtained for the evolution of the likelihood function and the coefficients of this differential equation are related to a corresponding nonlinear filtering problem. Some extensions are noted to diffusion process signals in correlated noise and to more general stochastic signals.

Estimation of the Innovation Variance of a Stationary Time Series

Abstract: The log of the periodogram is used to estimate the innovation variance of a stationary time series. The difference between the log of the estimated variance of the process and the log of the estimated innovation variance is used as a test for white noise. It is shown that this test is equivalent to Bartlett's test for homogeneity of variances applied to the periodogram without grouping and the asymptotic properties are derived.

A new algorithm for linear system identification

Abstract: This correspondence considers the on-line parameter identification of a forced linear discrete-time dynamic system from a sequence of white-noise-corrupted output measurements. In contrast to other approaches, the proposed stochastic approximation algorithm does not require knowledge of the noise statistics and converges to the true value of the parameters in the mean-square sense. If the input measurements are also corrupted with white noise, an additional term depending on the variance of the noise is required.

Center-of-gravity information feedback

Abstract: We consider M signals in a D -dimensional signal space. These M signals are used to communicate over an additive Gaussian white noise channel. It is shown that if a noiseless feedback channel is available one could use this feedback channel to inform the transmitter of the location of the center of gravity of the signal structure and thus obtain a very efficient signaling scheme. Each signal point is assigned a mass proportional to its posterior probability at the particular instant. The center of gravity is used by the transmitter as a new origin for the signal space. It is shown that some previously considered coding schemes for channels with feedback are particular cases of center-of-gravity feedback. The probability of error decreases as a double exponential function of the coding delay as opposed to an exponential decrease for one-way systems. The effect of noise in the feedback path is briefly considered.

Lower order optimal linear filtering of nonstationary random sequences

Abstract: The following deals with the discrete-time linear minimum-variance filtering of nonstationary random processes. The dynamics of the signal and colored noise processes are represented by a combined random process model.[1] Some of the measurement elements contain additional white noise, others do not. Similar to the continuous-time case of Bryson and Johansen,[3] the white-noise-free measurements will be used to reduce the order of the Kalman filter,[1],[2].

On the Optimal Control of a System Governed by a Linear Parabolic Equation with White Noise Inputs

Abstract: Optimal control of system governed by linear parabolic equation with white noise inputs, using mathematical model to generate distributed system analog

Random Vibrations in Discrete Nonlinear Dynamic Systems

Abstract: A one-degree-of-freedom system and a two-degree-of-freedom system containing Dis-placement and velocity dependent nonlinearities subjected to stationary gaussian white noise excitation have been studied by the method of the Fokker-Planck equation. Non-linearities have been represented by suitable polynomials. The Fokker-Planck equations governing the stationary probability density function for these systems have been solved by representing the density function by a multiple series of Hermite polynomials. The constants in the series expansion were determined by Galerkin's method. Analysis is developed for the system containing nonlinearities described by suitable polynomials in velocity and displacement dependent forces. Comparisons were made between series and exact solutions for those special cases for which exact solutions are known.

Linear smoothing using measurements containing correlated noise with an application to inertial navigation

Abstract: Kalman and Bucy [1] derived the maximum likelihood filter for continuous linear dynamic systems where all measurements contain white noise, i.e., noise with short correlation times compared to response times of the dynamic system. The corresponding maximum likelihood smoother was described in [2]. The maximum likelihood filter was presented in [3] for the case in which some measurements contain either no noise or colored noise, i.e., noise with correlation times comparable to or larger than the response times of the dynamic system. In this paper the maximum likelihood smoother for this latter case is derived by formulating the estimation problem as a problem in the calculus of variations having state variable equality constraints. An application of the results is made to estimating gyro drift rates of an inertial navigation system.

Spectral density of piecewise linear first order systems excited by white noise

Abstract: The Fokker-Planck equation is used to develop a general method for finding the spectral density and other properties of first order systems governed by stochastic differential equations of the form dx dt + f(x) = n(t), where f(x) is piecewise linear and n(t) is stationary Gaussian white noise. For such systems, it is shown how the Laplace transformed Fokker-Planck equation can be solved to obtain the transformed transition probability density. By manipulation of this equation and its adjoint, an expression is derived for the spectral density and moments in terms of the transformed transition density and its derivatives. The results in several special cases are presented.

Rapid acquisition sequences

Abstract: A recurring communications systems problem involves the determination, with relatively high precision, of the phase of a periodic, noise-corrupted signal. Perhaps the most commonly proposed signals for such purposes are pseudorandom sequences. Indeed, such signals are known to be optimum, at least when the noise is white and Gaussian, in the sense of minimizing the time needed to find the correct phase with a specified reliability. Equipment limitations, however, often preclude the efficient use of these sequences. In particular, suppose the received signal is known to be of the form y(t) = x(t - u \Delta T) + n(t) , with x(t) a signal periodic with period T , and n(t) additive white Gaussian noise. The problem is to determine as reliably and rapidly as possible the "phase" u of the received signal. The optimum detector involves the formation of the N = \Delta T/T correlations I_{ u} = \int y(t)x(t - u \Delta T) dt and the selection of the largest of these. Frequently, however, equipment constraints force these correlations to be made serially. In this event, the required search time can be as much as N times as great as that needed in the absence of such constraints. Even the most sophisticated sequential search algorithms require search times directly proportional to N , the number of contending phases. In this paper a class of binary sequences is demonstrated which require a search time on the order of only (log2 N)2, a substantial improvement when N is large. These sequences are shown to be essentially optimum under the conditions outlined.

Trajectory shaping for the minimization of state-variable estimation errors

Abstract: A technique is devised for synthesizing nonlinear controls that optimally shape nominal trajectories with respect to state-variable estimation errors propagating along the nominal trajectory. Both the nonlinear process and the observations upon which the estimation is based are corrupted by Gaussian white noise. A numerical example is presented that demonstrates the pronounced coupling effect of the nominal trajectory upon the propagation of state-variable estimation errors. Numerous practical implications of this work are cited.

Realizable whitening filters and state-variable realizations

Abstract: This letter discusses the whitening-filter approach to the detection of known signals in colored Gaussian noise when there is a residual white noise component present. The existence of a realizable (i.e., nonanticipatory) whitening filter is demonstrated and the form of the processor is explicitly given in terms of the Kalman-Bucy filter.

Optimal signal design for sequential signaling over a channel with feedback

Abstract: Turin [1] has studied the problem of optimal sequential detection over a channel with feedback when the signals used are subject to peak and average power constraints. With special additional constraints on the signals he was able to obtain a series solution for the first passage probability density and hence find optimal signals when the ratio of peak power to average power was infinity or one. Horstein,[2] concurrently with this study, was able to find optimal signals for arbitrary ratios of peak to average power when the signals were subject to the same special constraints. This paper presents a simpler and more direct approach for obtaining optimal signals which minimize the expected time to decision for arbitrary peak power constraints and various average power constraints. Explicit closed form solutions are derived for the expected energies and times to decision without having to first obtain the probability distribution of the first passage time. No special constraints are required on the signals. For communication with white noise in the feedback link, optimal stationary signals are found for peak and average power constraints. It is shown that communication rates up to but strictly less than channel capacity are possible with noisy feedback. However, it is also noted that for the schemes considered here, the stationary signals and white noise in the feedback link imply infinite feedback power.

First order piecewise linear systems with random parametric excitation

Abstract: The Fokker-Planck equation is used to develop a general method for finding the spectral density and other properties of first order systems governed by stochastic differential equations of the form dx dt + f(x)[1 + m(t)] = n(t) , where f(x) is piecewise linear and m(t) and n(t) represent stationary Gaussian white noise. The method is similar to one used by the authors to deal with the case m(t) = 0, but is complicated by the possible existence of irregular (singular) points of the Fokker-Planck equation. Graphical results for some special cases are presented.

Lagrangian Wiener‐Hermite Expansion for Turbulence

Abstract: The use of an advected white noise base for a truncated expansion is shown to provide an approximation for turbulence which has the inviscid equipartition solution and is invariant to random Galilean transformations.

Crowd Noise and Conformity

Abstract: 90 college men and women worked problems in the presence of crowd noise (CN), white noise (WN), or no noise (NN). CN produced more tendency to conform to answers chosen by others than NN among men (p < .05) but not among women.

An analysis of the effect of integral and sigma pulse frequency modulation on white noise

Abstract: Integral and sigma pulse frequency modulation effects on white noise, analyzing autocorrelation and spectral density functions of PFM system output

Optimal control and filtering of linear stochastic systems

Abstract: This paper is in two parts The first part treats the optimal control of a linear stochastic system with a quadratic performance criterion An explicit solution for the cost functional is given The second port deals with the optimal filtering of the linear system considered in the first part Modifications in the filtering equation have been made for the case in which the observation noise is of the truncated white noise type An explicit solution for the a posteriori density function, defined by the filtering equation, is given The paper concludes with a numerical solution to both the optimal control and the filtering problem associated with a six-plate chemical absorption tower

Examples of optimum detection of Gaussian signals and interpretation of white noise

Abstract: Simultaneously orthogonal expansion of two processes is one of the major mathematical tools for solving the problem of optimum detection of Gaussian signals in Gaussian noise. This expansion yields two integral equations: a homogeneous equation for the threshold and an inhomogeneous one for the test statistic of an optimum decision rule. After reviewing the optimum detection theory leading to the integral equations, four examples are presented to illustrate techniques of solving these equations and determination of the thresholds and test statistics. These techniques involve only elementary calculus and simple linear algebra. Finally, by way of example, an asymptotic interpretation of "white noise" in the context of optimum detection theory is given.

Effects of low-intensity acoustical stimulation on visual thresholds.

Abstract: The effects of low-intensity acoustical stimulation on the visual threshold of 2 Ss were investigated. 4 levels of white noise intensity were used, paired in random sequence with 10 different intensities of light, and randomly alternated with periods of quiet. It was found that white noise affects visual threshold, although the exact nature of this effect was not determined.

A low-frequency gaussian white-noise generator†

Abstract: The various methods of generating low-frequency noise are discussed. The merits of using a binary random signal, particularly the binary random square wave, as a noise source are considered, The advantages of applying the binary random signal to a low-pass filter to produce white gaussian noise are pointed out. A general method of designing filters which compensate for the fall in spectral response of the binary signal over the pass-band is described. Also, a theoretical method of determining the amplitude probability distribution of the filter output is described. The equations derived are solved for the case of third-order filters. Experimental and theoretical characteristics are given for the case of a noise generator using a third-order filter.

An Analog Technique for the Equalization of Multiple Electromagnetic Shakers for Vibration Testing

Abstract: A method that uses an analog computer to perform an open-loop control (equalization) for multishaker environmental vibration testing is presented. The computer is programmed to simulate the equations of motion of the electromechanical system. The input of the computer is an electrical signal representing the acceleration time histories to be reproduced at the base of the structure. Examples of multishaker equalization are shown for transient testing and random testing on a beam excited at two points. The method is valid for random, transient, and harmonic vibration testing.

Optimum differentiation using Kalman filter theory

Abstract: Consideration is given to the construction of an optimum differentiator to give the minimum-variance unbiased estimate of the first derivatives of random signals corrupted by white noise It is assumed that the signals are differentiable and are the outputs of a known linear finite-dimensional (possibly time-varying) system excited by white noise Extension of the results to consider higher-order differentiation is straightforward

An approximation method for nearly linear, first–order stochastic differential equations†

Abstract: An approximation method is developed for nearly linear, first-order stochastic differential equations with additive white noise. The method is based on an asymptotic expansion of eigenfunctions and...