Showing papers on "White noise published in 1970"

Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models

Abstract: Many statistical models, and in particular autoregressive-moving average time series models, can be regarded as means of transforming the data to white noise, that is, to an uncorrelated sequence of errors. If the parameters are known exactly, this random sequence can be computed directly from the observations; when this calculation is made with estimates substituted for the true parameter values, the resulting sequence is referred to as the "residuals," which can be regarded as estimates of the errors. If the appropriate model has been chosen, there will be zero autocorrelation in the errors. In checking adequacy of fit it is therefore logical to study the sample autocorrelation function of the residuals. For large samples the residuals from a correctly fitted model resemble very closely the true errors of the process; however, care is needed in interpreting the serial correlations of the residuals. It is shown here that the residual autocorrelations are to a close approximation representable as a singular linear transformation of the autocorrelations of the errors so that they possess a singular normal distribution. Failing to allow for this results in a tendency to overlook evidence of lack of fit. Tests of fit and diagnostic checks are devised which take these facts into account.

Speech analysis and synthesis by linear prediction of the speech wave.

Abstract: A method of representing the speech signal by time‐varying parameters relating to the shape of the vocal tract and the glottal‐excitation function is described. The speech signal is first analyzed and then synthesized by representing it as the output of a discrete linear time‐varying filter, which is excited by a suitable combination of a quasiperiodic pulse train and white noise. The output of the linear filter at any sampling instant is a linear combination of the past output samples and the input. The optimum linear combination is obtained by minimizing the mean‐squared error between the actual values of the speech samples and their predicted values based on a fixed number of preceding samples. A 10th‐order linear predictor was found to represent the speech signal band‐limited to 5kHz with sufficient accuracy. The 10 coefficients of the predictor are shown to determine both the frequencies and bandwidths of the formants. Two parameters relating to the glottal‐excitation function and the pitch period are determined from the prediction error signal. Speech samples synthesized by this method will be demonstrated.

On small random perturbations of dynamical systems

Abstract: In this paper we study the effect on a dynamical system of small random perturbations of the type of white noise: where is the -dimensional Wiener process and as . We are mainly concerned with the effect of these perturbations on long time-intervals that increase with the decreasing . We discuss two problems: the first is the behaviour of the invariant measure of the process as , and the second is the distribution of the position of a trajectory at the first time of its exit from a compact domain. An important role is played in these problems by an estimate of the probability for a trajectory of not to deviate from a smooth function by more than during the time . It turns out that the main term of this probability for small and has the form , where is a certain non-negative functional of . A function , the minimum of over the set of all functions connecting and , is involved in the answers to both the problems. By means of we introduce an independent of perturbations relation of equivalence in the phase-space. We show, under certain assumption, at what point of the phase-space the invariant measure concentrates in the limit. In both the problems we approximate the process in question by a certain Markov chain; the answers depend on the behaviour of on graphs that are associated with this chain. Let us remark that the second problem is closely related to the behaviour of the solution of a Dirichlet problem with a small parameter at the highest derivatives.

The innovations approach to detection and estimation theory

Abstract: Given a stochastic process, its innovations process will be defined as a white Gaussian noise process obtained from the original process by a causal and causally invertible transformation. The significance of such a representation, when it exists, is that statistical inference problems based on observation of the original process can be replaced by simpler problems based on white noise observations. Seven applications to linear and nonlinear least-squares estimation. Gaussian and non-Gaussian detection problems, solution of Fredholm integral equations, and the calculation of mutual information, will be described. The major new results are summarized in seven theorems. Some powerful mathematical tools will be introduced, but emphasis will be placed on the considerable physical significance of the results.

Spectral analysis of EEG's by autoregressive decomposition of time series

Abstract: A technique for scalar and multidimensional spectral analysis based on the autoregressive representation of the observed data records is presented and illustrated. In an autoregressive representation the observed data set is regressed on its own past history. This results in a formula that expresses the observed data as the output of a linear filter excited by an uncorrelated sequence (“discrete white noise”). Energy spectral densities, transfer functions, and coherences are computed from the autoregressive formula. The results of this technique are compared with the older windowed periodogram methods of spectral analysis. Two potential advantages over the latter methods are observed. For spectral estimates of comparable statistical performance, the autoregressive method analysis appear smoother and easier to interpret than the older windowed periodogram analysis. Also, in contrast with the expertise required to apply the windowed periodogram analysis, use of the autoregressive representation spectral analysis method appears to require little or no subjective judgment.

Comment on "An innovations approach to least-squares estimation, part I: Linear filtering in additive white noise"

Abstract: The innovations approach to linear least-squares aIF proximation problems is ûrst to \"whiten' the observed data by a causal and invertible operation, ând 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 ûltering problem for stationary processes over & semi-inûnite interval. Ilere we shall extend the technique to handle nonstationary conlinuous-time processes over ûnite intervals. In Part I we shall apply this method to obtain a simple derivation of the Kalman-Bucy recursive ûltering formulas (for both continuous-time and discrete-time processes) and also some minor generalizations thereof.

A Flexible Neural Analog Using Integrated Circuits

Abstract: An electronic neural analog is described which contains variable absolute and relative refractory periods, two time constants, and separate control of the ``accommodation'' to sub-threshold voltage changes and the ``adaptation'' produced by the occurrence of output pulses (spikes). The extensive use of integrated-circuit operational amplifiers permits an accurate description of input-output relationships over a wide range of values for all parameters. This facilitates comparison of the results obtained both with neural data and mathematical or digitally simulated models of neural activity. The effect of white noise on interval histograms and their parameters is described and its effect when added to other inputs. Noise disrupts the phase-locked patterns produced by sinusoidal stimuli and the averaged response may become a smooth sinusoidal function in the presence of added noise. Adaptation may produce a phase lead to sinusoidal stimuli, while accommodation may produce a phase lag. Corresponding overshoots or undershoots are seen with square-wave inputs.

Photon counting: A problem in classical noise theory

Abstract: In this paper we formulate the general problem of determining the photoelectron "counting" distribution resulting from an electromagnetic field impinging on a quantum detector. Although the detector model used was derived quantum mechanically, our treatment is wholly classical and includes all results known to date. This combination is commonly referred to as the semiclassical approach. The emphasis, however, lies in directing the problem towards optical communication. The electromagnetic field is assumed to be the sum of a deterministic signal and a zero-mean narrow-band Gaussian random process, and is expanded in a Karhunen-Loeve series of orthogonal functions. Several examples are given. It is shown that all the results obtainable can be written explicitly in terms of the noise covariance function. Particular attention is given to the case of a signal plus white Gaussian noise, both of which are band-limited to \pm B Hz. Since the result is a fundamental one, to add some physical insight, we show four methods by which it can be obtained. Various limiting forms of this distribution are derived, including the necessary conditions for those commonly accepted. The likelihood functional is established and is shown to be the product of Laguerre polynomials. For the problem of continuous estimation, the Fisher information kernel is derived and an important limiting form is obtained. The maximum a posteriori (MAP) and maximum-likelihood (ML) estimation equations are also derived. In the latter case the results are also functions of Laguerre polynomials.

The vibrating string forced by white noise

Abstract: The equation of the vibrating string forced by white noise is formally solved, using stochastic integrals with respect to a plane Brownian motion, and it is proved that a certain process associated to the energy is a martingale. Then Doob's martingale inequality is used to furnish some probability bounds for the energy.

Likelihood ratios for Gaussian processes

Abstract: We give a comprehensive discussion of the structure of the likelihood ratio (LR) for discrimination between two Gaussian processes, one of which is white. Several more general problems can be reduced, usually by differentiation, to this form. We shall show that nonsingular detection problems of this form can always be interpreted as problems of the apparently more special "signal-in-noise" type, where the cross-covariance function of the signal and noise must be of a special "one-sided" form. Moreover, the LR for this equivalent problem can be written in the same form as that for known signals in white Gaussian noise, with the causal estimate of the signal process replacing the known signal. This single formula will be shown to be equivalent to a variety of other formulas, including all those previously known. The proofs are based on a resolvent identity and on a representation theorem for second-order processes, both of which have other applications. This paper also contains a discussion of the various stochastic integrals and infinite determinants that arise in Gaussian detection problems

First-excursion failure of randomly excited structures

Abstract: The probability for a randomly excited structure to survive a service time interval without suffering a first-excursion failure is determined analytically. The first-excursion failure occurs when, for the first time, the structural response passes out of a prescribed safety domain. The problem is formulated from the viewpoint of the Stratonovich-Kuznetsov theory of random points. The exact solution is expressed in two equivalent series forms, one reducible to Rice's "in and exclusion" series. The first order truncation of the second series corresponds to Poisson random points and the second order truncation to random points with "pseudo" Gaussian arrival rate. Numerical results are presented for a single-degree-offreedom linear oscillator under Gaussian white noise excitation based on these truncations and the model of nonapproaching random points suggested by Stratonovich.

Filtering for Linear Distributed Parameter Systems

Abstract: Nonlinear filtering for linear parabolic distributed parameter systems with white noise, considering stochastic boundary value problem

A further note on a general likelihood formula for random signals in Gaussian noise

Abstract: A general likelihood ratio formula of the author's for the detection of signals in independent white Gaussian noise is extended to allow a "one-sided" dependence in which only the future white noise is required to be independent of past signal and noise. The assumption of Gaussian additive noise is also somewhat relaxed. The proof is based on some recent martingale theorems and on the concept of the innovations process.

Effect of noise on the modulation transfer function of the visual channel.

Abstract: The effect of noise on the modulation transfer function was studied by means of threshold measurements. White noise and 1/f noise of various levels and different cutoff frequencies were displayed on a television screen together with a sinusoidally modulated bar pattern. The signal-to-noise threshold necessary for perception was measured as a function of the spatial frequency of the bar pattern. This signal-to-noise threshold, in addition to being strongly dependent on the bar-pattern frequency is also dependent on the rms value and the frequency distribution of the noise as well as the difference between the bar-pattern frequency and medium frequency of the noise. An attempt was made to explain the results by visual observation of the bar pattern in the presence of narrow-bandwidth noise.

State estimation by orthogonal expansion of probability distributions

Abstract: A recursive estimation scheme suitable for real-time implementation is derived for a class of nolinear systems and observations expressed as nonlinear functions in discrete time, corrupted by a non-Gaussian mutually correlated random white noise sequence. The probability densities are expanded as a Gram-Charlier series and a Gauss-Hermite quadrature formula is used for computing the expectations. In the multidimensional case an expansion about a density of mutually independent Gaussian variables is used instead of a general multidimensional Gaussian density, which may result in a poorer performance in linear systems with Gaussian noise. However, in the case of nonlinear systems and non-Gaussian noise, the computational simplifications which result, outweigh the impairment in performance if any. A computational example is included.

Ear Asymmetry in the Threshold of Fusion of Two Clicks: A Signal Detection Analysis:

Abstract: In the three experiments described, a comparison is made of two methods of measuring the threshold of fusion of two clicks: a signal detection method, and a modified method of constants. The former method reveals a significant ear asymmetry effect, which is accentuated when a burst of white noise is presented contralaterally with the clicks. Results are discussed with reference to differentiation of function of the cerebral hemispheres.

Likelihood Functions for Stochastic Signals in White Noise

Abstract: For a general stochastic signal in white noise absolute continuity is proved and the Radon-Nikodym derivative is given. These results were stated in a previous paper (Duncan 1968). Independent of the absolute continuity result, a modification is proved for the hypothesis with signal present. This paper is a sequel to an earlier paper by the author (Duncan 1968) where likelihood functions were obtained for diffusion process signals. While the general result was noted there, in this paper we explicitly prove the more general result and show that the proof easily follows from the techniques used in the previous work. We shall also indicate in a rigorous mathematical way how the hypotheses may be changed using some results for the decomposition of supermartingales.

Determination of signal and noise statistics using correlation theory

Abstract: A seismic trace may be represented as the sum of a signal and noise series. Each of the series may further be represented by convolution of a finite wavelet and a random series. With this representation, and provided that the signal and noise are uncorrelated, it is possible, in theory, to extract signal and noise statistics from two adjacent traces of a reflection seismogram. Some experiments are shown on model seismic traces, and it is shown that within the time‐duration of the seismic wavelet, the estimates of signal and noise statistics are reasonable for low signal‐to‐noise ratio. There remains, however, the problem of determining the optimum time lengths of the estimates.

Directional audiometry. I. Directional white-noise audiometry.

Abstract: The construction and calibration of a device for the angular localization of white noise is described. The apparatus has been used to measure directional hearing abilities in the horizontal plane in 30 normally-hearing subjects using white noise. A standard graph of the results of the tests has been prepared, showing the limits within which 95 per cent of the localizations were made. This diagram will serve as a basis for comparison when examining the angular localization of noise in patients with hearing loss.

Stationary and nonstationary characteristics of gyro drift rate

Abstract: The full text of the paper describes in detail: 1) the identification of the time series model from autocorrelation and partial correlation of the data; 2) the estimation of the , 6, ramp and random walk parameters using maximum likelihood and nonlinear least squares; and 3) the means by which one would deduce model adequacy through autocorrelation of the white noise residuals and confidence limit theory. As an example of the theory, consider Fig. 1, which shows a sample of normalized long term gyro drift rate. Since the process is nonstationary, the data is differenced as is shown in Fig. 2. The analysis indicated that the math model for this gyro drift rate sample is

Spectral factorization of time-varying covariance functions: The singular case

Abstract: If a known linear system is excited by Gaussian white noise, the calculation of the output covariance of the system is relatively straightforward. This paper considers the harder converse problem, that of passing from a known covariance to a system which will generate it. The problem is solved for covariancesR y (t, τ) with |R y (t, t)| < ∞ for allt and such that they-process is Gauss-Markov, i.e., it may be obtained as the output of a linear finite-dimensional system excited by white noise.

Nonsingular detection and likelihood ratio for random signals in white Gaussian noise

Abstract: This paper is concerned with the mathematical aspect of a detection problem (a random signal in white Gaussian noise). Specifically, we obtain a sufficient condition for nonsingular detection and derive a likelihood-ratio expression in terms of least-mean-square estimates. The problem itself is old, and the likelihood-ratio expression is also well known. The contribution of this paper is a relatively elementary and self-contained derivation of the likelihood-ratio expression as well as the nonsingularity condition.

Absolute sensitivity to white noise under auxiliary visual stimulation1

Abstract: Absolute sensitivity to white noise was measured with the O in the dark, in constant illumination, and in the dark but with sound-synchronized flashes of light at three different intensities. A confidence rating procedure was used, and the results were analyzed in terms of the theory of signal detectability. There appeared to be no consistent effect of auxiliary visual stimulation on absolute auditory sensitivity for the four Os examined.

On system identification from noise-obscured input and output measurements†

Abstract: This paper deals with maximum-likelihood system identification when both the input and the output signals are corrupted by Gaussian observation noise. A derivation of exact maximum-likelihood estimation for this problem is included, but the difficulty of implementing it numerically precludes its practical evaluation at this time. A new approximate method is introduced, called the ‘output reference’ method, in which the input noise is referred to the output, and an iterative gradient search method used. This technique requires no a priori knowledge of the noise covariance matrix. The method of Koopmans—Levin, which does require knowledge of the noise covariance matrix, is then reviewed in detail, and experimental results are presented for the white noise case which indicate that the output reference method is more accurate.

Response of periodically varying systems to shot noise — application to switched rc circuits

Abstract: This paper is concerned with the statistical properties of the output y(t) of a periodically varying linear system when the input is random shot noise. Usually y(t) can be divided into a noise part, y N (t), and a periodic part, y per (t). Expressions are obtained for the Fourier components of y per (t) and the power spectrum of y N (t). Various averages associated with y(t) are studied. Some of the results for shot noise input can be converted into corresponding results for white noise input. Some of the theoretical results are illustrated by applying them to two examples. In both examples the system consists of an arrangement of a resistance, a condenser, and a switch which opens and closes periodically. The output is the voltage across the condenser.

Apparatus and systems for testing and monitoring receiver equipment, including system capability for continuous in-service performance monitoring

Abstract: Apparatus and systems for testing and monitoring receiver equipment providing automatic noise figure indication, and capable of continuous in-service performance monitoring. The automatic noise figure indicator has a logarithmic voltagecontrolled attenuator with a fast-acting automatic gain control (AGC) feedback loop for keeping the attenuator output constant, with read out means connected to the AGC loop and responsive to its alternating voltage level for providing information on the noise figure. A white noise test source is gated on-and-off at predetermined frequency F1; for continuous in-service performance monitoring this noise source is also modulated at higher frequency F2 preferably at least ten times frequency F1. In continuous monitoring this test noise is introduced at an energy level substantially below the receiver noise energy level, i.e., it is buried to avoid interference with normal receiver operation, and this buried test signal is later retrieved by a signal enhancing correlator. Receiver parameter monitoring occurs continuously while receivers are in-service without interfering with normal operation. Among the monitored parameters disclosed are gain tracking characteristics and phase/gain characteristics of companion receivers, antenna VSWR, amplitude non-linearity in a receiver, and R.F. gain and transmission flatness of equipment having the same input and output frequency. Prompt, accurate and direct information about receiver operational status is provided to the operator while equipment is actually in use; a higher average quality of performance can be maintained because degradation is immediately detectable and thus correctable. Also, potential failures can thereby be anticipated and ''''down time'''' reduced. Maintenance costs are reduced because service is done in pre-scheduled normal time.

Transient Behavior of a Phase-Locked Loop in the Presence of Noise

Abstract: Transient behavior of the first-order phase-locked loop in the presence of noise is investigated. Numerical integration is used to obtain solutions for the Fokker-Planck equation which represents loop dynamics, Results illustrate quantitative behavior of the time-varying probability density function of phase error when the input is a constant frequency sinusoid in the presence of additive white Gaussian noise. Several selected values of signal-to-noise ratio (SNR) are considered.

White noise, instructions, and two-flash tusion with two signal-detection procedures

Abstract: Two signal-detection procedures were used to parcel out the effects of white noise and instructions with a two-flash discrimination task. Neither condition influenced the sensitivity scores of either model Variations in instructions changed the criterion and threshold measures. High correlation coefficients were obtained for the corresponding scores of the two signal-detection analyses.

Operator-Valued wide-sense Markov processes and solutions of infinite-dimensional linear differential systems driven by white noise

Abstract: Introduction. The study of finite linear stochastic differential systems with white-noise input has been the subject of several papers starting with Doob [4] (see also [2], [5], [7], [10], [14]). Some recent results of Falb [8] on infinitedimensional filtering depend on the structure of the solution of an infinitedimensional stochastic differential system. One also encounters an infinite stochastic differential system in the study of heat equations with random source (see [1] and [19]). All such processes arising from linear stochastic differential systems in both the finite-dimensional (see [2]), and infinite-dimensional (see [8]) cases, constitute appropriate generalizations of wide-sense Markov processes introduced by Doob [4]. We obtain in this paper some results on the structure of general wide-sense Markov processes which constitute generalizations of those in [14]. These generalizations constitute, in addition to the extension to the infinite-dimensional case, a sharpening of the finite-dimensional results of [14]. As applications of our results in the infinite-dimensional case, we present generalizations of the results of Beutler [2] and some results of Falb [8]. The approach of this paper is similar to that in [4] (stationary case) and [14] (for the non-stationary case), in that it depends on the structure of L2,M for an operator-valued measure M and the appropriate generalization of stochastic integrals. In view of some recent work of the authors [15], this approach seems extendible to the infinite-dimensional case. However, such an extension, although apparent, is not direct in view of the usual difficulties of working with infinite-dimensional processes; one of the difficulties is that the inverse of a covariance operator is unbounded, since, throughout this paper, Hilbert-Schmidt operator-valued stochastic processes as in [18] are considered. The paper is divided into six sections. After the preliminary Section 1, Section 2 contains the basic structure theorem of wide-sense Markov processes in terms of wide-sense martingales. In Section 4, we study the solutions of a