Showing papers on "White noise published in 1971"

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.

Stochastic Control for Small Noise Intensities

Abstract: This paper is concerned with the approximate solution of stochastic optimal control problems which arise by perturbing the system equations in the deterministic Pontryagin control model, through an additive white noise term with small coefficient. The system states are assumed completely observable. Mathematically the problem becomes one of singular perturbation of the Hamilton–Jacobi equation by a small second order term. Our main results concern expansions of solutions of the perturbed equation in powers $\varepsilon ,\varepsilon ^2 ,\varepsilon ^3 , \cdots $ of the noise variance coefficients. The results obtained hold in regions where the corresponding solution of the Hamilton-Jacobi equation is sufficiently well-behaved.

An innovations approach to least-squares estimation--Part III: Nonlinear estimation in white Gaussian noise

Abstract: In Parts I and II of this paper, we presented the innovations approach to linear least-squares estimation in additive white noise. In the present paper, we show how to extend this technique to the nonlinear estimation (filtering and smoothing) of non-Gaussian signals in additive white Gaussian noise. The use of the innovations allows us to obtain formulas and simple derivations that are remarkably similar to those used for the linear case thereby distinguishing clearly the essential points at which the nonlinear problem differs from the linear one.

Optimal Stationary Control with State Control Dependent Noise

Abstract: The steady state optimal linear regulator with state and control dependent noise is analyzed in a manner similar to that developed by Wonham [1]. By state dependent noise we mean Gaussian white noise with coefficient linear in the state variable, and similarly for control dependent noise. Using a Lyapunov criterion for the existence of stationary probability distributions due to Zakai, it is possible to treat equations leading to diffusion processes with degenerate differential generators. It is found that if the noise is sufficiently small, then an optimal control exists. Further analysis, again using Lyapunov methods, yields conditions under which an optimal control exists no matter how large the noise is.

Effects of intermittent noise on the performance of a complex psychomotor task

Abstract: Manual image-motion compensation, a complex psychomotor task involved in certain photographic activities from orbit, was investigated as a function of the temporal pattern (aperiodic, periodic, or continuous) and intensity level (50, 70, or 90 db.) of white noise. Performance was measured in terms of the total amount of time image motion was held at or below a 40-microradians/second criterion for specific blocks of trials. The results of the investigation showed that white noise had a detrimental effect on image motion compensation performance, and that the magnitude of the decrement varied as a function of both the temporal pattern and intensity level of this noise.

A masking noise with speech-envelope characteristics for studying intelligibility.

Abstract: A noise whose amplitude envelope followed closely that of a concomitant speech signal was generated by multiplying white noise and the amplitude envelope of the speech, permitting the signal‐to‐noise (S/N) ratio to be specified on a short‐time nonvarying basis. The spectrum of the amplitude envelope for continuous speech was studied, and the distributions of the vowel and consonant levels in articulation test materials were determined. Articulation functions in such noise and in continuous white noise were generated. Within the range of S/N ratios studied, the gains of the functions for vowels and consonants were 4% and 2.5% per decibel, respectively, in both types of noises. The results clearly depict the operational differences between conventional and envelope‐noise S/N‐ratio specification and suggest that use of the envelope‐noise masker may eliminate some of the problems associated with current methods.

A High-Speed Sequential Decoder: Prototype Design and Test

Abstract: We describe the design of a rate-1/2 hard-decision sequential decoder capable of operation at data rates up to 5 M bit/s. Test results are given for digitally generated errors, white noise, and real channels. The results are substantially in agreement with predictions of a coding gain of the order of 5 dB at a 10-5error rate.

Detection of a Distributed Target

Abstract: The influence of increasing range resolution on the detectability of targets with dimensions greater than the resolution cell is studied. An N-cell target model is assumed, which contains k reflecting cells, each reflecting independently according to the same Rayleigh amplitude distribution. It will be referred to as the (N,k) target. Detection based on one transmitted pulse is performed against a background of white normal noise. Detection in stationary clutter is also considered. The optimum detector is obtained but, in view of its complexity, the performance of a simpler detector, the square-law envelope detector with linear integrator (SLEDLI), is analyzed, and a formula for the probability of detection is obtained. Graphs are presented which show the probability of detection as a function of signal-to-noise ratio (SNR) for various values of N k, and false alarm probability. For N/k not too large it is shown that the SLEDLI is near optimum.

Dynamic state estimation in power systems

Abstract: Estimation of the dynamic state of a power system is the first prerequisite for control and stability prediction under transient conditions. Since the magnetic flux linkages which characterize the instantaneous state of the machines and the system are not directly measurable, a variable-dimension stage-invariant Kalman filter was developed and used to estimate the state of the system based on measurements at each individual machine of instantaneous field and armature current, field and armature voltage, and angular deviation of machine rotor shaft from a synchronous phasor reference. Gaussian white noise in the measurements was assumed, and the filter provided a recursive near-optimal minimum variance estimate of the state of the nonlinear system.

Random Processes for Earthquake Simulation

Abstract: Criteria presented for the evaluation of earthquake simulation processes relate to the autocovariance function, maximum acceleration response spectra, and nonstationarity of the process. Two previously used and two new processes are examined with respect to these criteria by means of theoretical and simulated results. The previous processes develop a nonstationary white noise for input to a linear filter and take the filter output as the simulated ground acceleration. They are identical except for filter properties. Neither of these processes complies with all the criteria. One fails to provide reasonable response spectra; the other produces an unrealistic ground-velocity variance function that does not disappear with time. The new processes incorporate weighting functions that produce correlated, nonwhite filter inputs. One of these produces response spectra with undesirable irregularities. The other, which employs an exponential decay-type weighting function, complies with all criteria and is recommended as a suitable model for earthquake simulation. Additional relationships are examined for rms and maximum oscillator response spectra.

Fitting Time Series Models for Prediction

Abstract: The mathematical theory of the best linear prediction of stationary time series presumes that the model generating the series, which can be specified by either the autocovariance function or the spectral density, is known. The true model is, of course, not known in practice, and the procedure is to fit a model and predict as if this fitted model were the truth. The question then is one of deciding whether the resulting predictions are about as good as could be gotten if the truth were known. This paper describes a method for assessing the predictions of the fitting model by an analysis of residuals. In particular, it is argued that the traditional tests of hypothesis for white noise are inappropriate if prediction is the goal, and a method is described for determining whether or not the mean square errors of the predictions arising from the fitted model can be measurably reduced.

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.

A model for the acoustic impedance of a perforated plate liner with multiple frequency excitation

Abstract: A nonlinear resistance model is used in the one-dimensional equations of motion with an arbitrary exciting pressure function. The effects of high amplitude fluid motion, grazing flow, and spectral excitation can be studied together. Sample calculations of acoustic resistances are presented using a high amplitude discrete tone superimposed upon a simulated white noise spectrum. The tone amplitude is varied and its effect is shown both with and without a grazing flow velocity.

Optimal state-vector estimation for non-Gaussian initial state-vector

Abstract: The optimal estimate, in the mean-square-error sense, of state-vector of a linear system excited by zero-mean white Gaussian noise with non-Gaussian initial state-vector is obtained. Both the optimal estimate and the corresponding error covariance matrix are given. It is shown that the optimal estimator consists of two parts: a linear estimator that is obtained from a Kalman filter and a nonlinear estimator. In addition, the a posteriori probability p(x_{k}/\lambda_{k}) is also given.

Smoothing as an improvement on filtering: a universal bound

Abstract: Comparison is made between the mean-square errors Pf and Ps associated with the linear least-squares filtered and smoothed estimates of a stationary process of spectral density S(ω) in white noise of spectral density No. A universal curve is obtained which relates the minimum possible value of Ps/Pf to ωmax{S(ω) No}. The curve sets a bound on the maximum improvement over filtering which smoothing will offer, in terms of the maximum signal/noise ratio.

Shaping filter representation of nonstationary colored noise

Abstract: The problem of determining a shaping filter for nonstationary colored noise is considered. The shaping filter transforms white noise into a possibly nonstationary random process (having no white noise component) with a specified covariance function. A set of conditions to be satisfied by the covariance function leads to the determination of a shaping filter. The shaping filter coefficients are simply related to the solution of a matrix Riccati equation. In order to formulate the Riccati equation, basic results concerning the mean-square differentiability of a random process are developed. If the Riccati equation can not be defined, an autonomous (zero-input) shaping filter may be easily determined.

Frequency-Counted Measurements and Phase Locking to Noisy Oscillators

Abstract: The phase-error variance of a phase-locked loop is dependent on the stability of its voltage-control oscillator and that of the source oscillator being tracked. Statistics relative to oscillator stability are commonly gathered by counted-frequency techniques and are so specified in manufacturers' data. In order thus to predict the performance of a loop using such an oscillator, it is necessary to know how to relate counter data to loop error. This paper presents such a method based on a simple but realistic model of oscillator noises and shows that the mean sample variance of the counted-frequency method converges rather curiously and slowly to the actual variance. The model for the flicker component of the noise is physically realistic (finite power) and allows one to find the range of validity for the usual formal calculations for the sample variance of the counted frequency. In addition, insight is gained into the relation between the sample variance and the actual finite variance of the realistic model. The effect of oscillator instability on a first-order loop is longterm steady-state phase-error drift and short-term zero-mean fluctuation about this steady state. For the second-order loop, the steady-state drift disappears.

Some Properties of Transmission Systems with Minimum Mean-Square Error

Abstract: Three properties of a linear transmission system optimized with respect to the mean-square error of the signal received are discussed. These are the attenuation of the main peak of a transmitted pulses the ratio of the components of the mean-square error (signal distortion and noise), and the increase of the error caused by imperfect implementation of the phases of the optimal transfer functions.

Random Vibration of Thin Elastic Plates and Shallow Shells

Abstract: : The large amplitude vibrations of thin elastic plates and shallow shells having boundary conditions and subjected to random excitation are investigated by using various approximate techniques. The random vibrations of rectangular plates and circular plates subjected to white random excitation are simulated numerically by two different methods. The first method is that the governing equations are reduced to a single-degree-of-freedom dynamical system and the reduced equation is then integrated numerically by the Runge-Kutta method employing the simulated approximate white noise as an input. The second method consists in integrating the equation of motion and the compatibility equation numerically by a finite-difference method employing the simulated approximate white noise as an input.

On the Performance of Some Sequential Multiple-Decision Procedures

Abstract: The performance of several sequential procedures for the following multiple-decision problem is investigated. Samples from k random processes (or populations) are available, k at a time (one from each process), to a receiver or data processor. One process contains a signal; the other k - 1 are statistically identical noise. The receiver is to select the odd process (locate the signal), with prescribed probability of error. The optimal receiver makes the selection in minimum average time. Analytical and Monte Carlo computations were performed under the hypothesis that the processes sampled are Rayleigh; however, a method for extrapolating results to other cases is given. The parameter k is allowed to vary from 2 to 1000.

Rate distortion over band-limited feedback channels

Abstract: Rate distortion over band limited feedback channels, considering capacity of additive Gaussian white noise channel

Central Limit Theorems for Conditionally Linear Random Processes

Abstract: A random process is called a linear process if it is an infinite sum of statistically independent component random processes. A particular example of a linear process is the output of a filter driven by a sequence of impulses whose times of occurrence are the times of a Poisson process. If the filter responses are also random and the responses to impulses applied at different times are statistically independent, the filter output is still a linear process. If, however, either the impulses do not occur according to a Poisson process or the filter responses are not independent, the process is called conditionally linear. The latter situation can be used to describe the radar echoes from randomly dispersed scatterers which, however, exhibit some phase coupling.It has been shown in many special cases that when a linear process is passed through a low-pass filter, the output is approximately Gaussian. These are the well-known central limit theorems for linear processes. This paper presents a very general and w...

Radio frequency measurements

Abstract: There is disclosed a method for determining the absolute response of a signal strength meter, or the absolute response of a device under test with the use of a previously calibrated signal strength meter. In the former case, a continuous wave signal of predetermined amplitude and frequency is applied to the input of the meter while it is tuned to the frequency of the continuous wave signal. A white noise signal is then applied to the input of the meter while it is tuned to the same frequency, and the level of the white noise signal is adjusted so that the meter reading is the same as the reading taken for the continuous wave signal. Thereafter, as the signal strength meter is tuned throughout the frequency range of interest, the absolute response of the signal strength meter can be determined by observing the meter reading even though the bandwidth of the filter of the meter may not be known. In a similar manner, a previously calibrated meter can be used to measure the absolute response of a device under test to whose input there are applied the white noise and continuous wave signals, and whose output is connected to the meter.