Showing papers on "White noise published in 1984"

The Output Distribution of Median Type Filters

Abstract: The distribution of the output of the one-dimensional median filter is derived for several cases including the k th-order output distribution with any input distribution. This is then used in several illustrative examples of median filtering a signal plus white noise.

Robust Wiener- Kolmogorov theory

Abstract: A minimax formulation is considered for the problem of designing robust linear causal estimators of linear functions of discrete-time wide-sense stationary signals when knowledge of the signal and/or noise spectra is inexact. The solution is given (under mild regularity conditions) in terms of a least favorable pair of spectra, thus reducing the minimax problem to a direct maximization problem which in many cases can be solved easily. It is noted that this design method leads, in particular, to robust n -step predictors, robust causal filters, and robust n -lag smoothers. The method of design is illustrated by a thorough development of the special case of one-step noiseless prediction. Further, solutions are given explicitly for the problem of robust causal filtering of an uncertain signal in white noise, and numerical examples are given for this case which illustrate the effectiveness of this design.

Restoration of Digital Multiplane Tomosynthesis by a Constrained Iteration Method

Abstract: Tomosynthetic reconstructions suffer from the disadvantage that blurred images of object detail lying outside the plane of interest are superimposed over the desired image of structures in the tomosynthetic plane. It is proposed to selectively reduce these undesired superimpositions by a constrained iterative restoration method, suitably generalized to permit simultaneous deconvolution of multiple planes. Sufficient conditions are derived ensuring the convergence of the iterations to the exact solution in the absence of noise and constraints. Although in practice the restoration process must be left incomplete because of inescapable noise and quantization artifacts, the experimental results demonstrate that for reasons of stability the convergence conditions derived for the noise-free, unconstrained case should be satisfied. In order to establish a basis for a formal stopping criterion of the iteration procedure, the buildup of noise in the sequence of iterative restorations arising from white noise in the original radiographs is investigated theoretically and experimentally. This results in the derivation of an approximation to the limiting noise variance in the reconstructions which is verified experimentally.

Method and apparatus for measuring flow

Abstract: Flow, and in particular nonuniform flow, as in a vascular system, is measured by applying a stochastic excitation signal to a system inlet which results in a measurable output signal at a downstream system outlet. Flow rate may be extracted by cross-correlating the excitation signal and the output signal. Calibration may be effected by conservation of mass principles whereby quantity type parameters are related to concentration type parameters. The stochastic signal has the characteristics of white noise, such that simplified spread spectral detection and signal extraction techniques may be employed to recover the desired intelligence.

Accurate frequency estimation at low signal-to-noise ratio

Abstract: A frequency estimator for sinusoids in white noise is described. Convergence results are obtained for the single sinusoid case and a simulation described for the multiple sinusoid case. The estimator is shown to be capable of providing accurate frequency estimates at lower SNR's than currently existing techniques. Furthermore, the simplicity of the algorithm lends itself to a simple and efficient implementation.

Asymptotic variance expressions for identified black-box transfer function models

Abstract: Identification of black-box transfer function models is considered. It is assumed that the transfer function models possess a certain shift-property, which is satisfied for example by all polynomial-type models. Expressions for the variances of the transfer function estimates are derived, that are asymptotic both in the number of observed data and in the model orders. The result is that the joint covariance matrix of the transfer functions from input to output and from driving white noise source to the additive output disturbance, respectively, is proportional to the inverse of the joint spectrum matrix for the input and driving noise multiplied by the spectrum of the additive output noise. The factor of proportionality is the ratio of model order to number of data. This result is independent of the particular model structure used. The result is applied to evaluate the performance degradation due to variance for a number of typical model uses. Some consequences for input design are also drawn.

Predictive deconvolution and the zero‐phase source

Abstract: Predictive deconvolution is commonly applied to seismic data generated with a Vibroseisr® source. Unfortunately, when this process invokes a minimum‐phase assumption, the phase of the resulting trace will not be correct. Nonetheless, spiking deconvolution is an attractive process because it restores attenuated higher frequencies, thus increasing resolution. For detailed stratigraphic analyses, however, it is desirable that the phase of the data be treated properly as well. The most common solution is to apply a phase‐shifting filter that corrects for errors attributable to a zero‐phase source. The phase correction is given by the minimum‐phase spectrum of the correlated Vibroseis wavelet. Because no minimum‐phase spectrum truly exists for this bandlimited wavelet, white noise is added to its amplitude spectrum in order to design the phase‐correction filter. Different levels of white noise, however, produce markedly different results when field data sections are filtered. A simple argument suggests that th...

Statistical analysis of Pisarenko's method for sinusoidal frequency estimation

Abstract: Statistical analysis is performed for Pisarenko's method of sinusoidal frequency estimation by using the periodogram technique. Under the assumption that the data consist of several sinusoids plus white noise, we give an asymptotic expression of the error variance of the sinusoidal frequency estimator. Also, a modified method is proposed. This is computationally efficient but the statistical property is not improved. We also present simulation results and compare them to the theoretical ones. Lastly, we point out that the Pisarenko's method can be implemented as an (on-line) bias compensated least-squares algorithm, thus avoiding the eigenvalue and eigenvector calculations.

On the Probability Density of Signal-to-Noise Ratio in an Improved Adaptive Detector. Part 1

Abstract: : Document describes an approximate probability distribution for the SNR (signal-to-noise ratio) of an improved adaptive detector in near rank-1 gaussian noise where the filter weights are computed using the principal eigenvectors of the estimated noise covariance matrix. The noise consists of two components, a strong rank-1-covariance interference component plus white noise. Computer simulation is used to verify the approximating SNR distribution and show that it is accurate even for small sample size and low interference-to- noise ratios.

Identification of nonminimum phase linear stochastic systems

Abstract: We consider the identification of time-invariant, nonminimum phase, stochastic systems driven by non-Gaussian white noise, given only the noisy observations of the system output A two-step procedure is proposed In the first step a spectrally equivalent minimum phase (SEMP) system is estimated using a standard technique that exploits only the second order statistics of the measurements In the second step partial, 4th order cumulants of the measurements are exploited to resolve the location of the system zeros Knowledge of the probability distribution of the driving noise is not required Strong consistency of the proposed estimator is proved under certain mild sufficient conditions Simulation results are also presented in support of the theory

Information tradeoffs in using the sample autocorrelation function in ARMA parameter estimation

Abstract: This paper considers bounds on the statistical efficiency of estimators of the poles and zeros of an ARMA process based on estimates of the process autocorrelation function (ACF). Special attention is paid to autoregressive (AR) and AR plus white noise processes. It is seen that reducing the ARMA process data to a given set of consecutive lags of the popular lagged-product ACF estimates prior to parameter estimation increases Cramer-Rao bounds on the generalized error covariance. A parametric study of the bound deterioration for some illustrative signal and noise situations reveals some empirical strategies for choosing ACF estimate lags to preserve statistical information. Analysis is based on the relative information index (RII) [2], and derivations of the large sample Fisher's information matrix for the raw data and for the lagged-product ACF estimate of an ARMA process are included.

Robustness and Approximation of Escape Times and Large Deviations Estimates for Systems with Small Noise Effects

Abstract: For the purposes of estimating escape time from a given set, or other statistical properties of systems with small noise effects, it is generally assumed in applications that the system noise is white Gaussian.The Gaussian assumption greatly simplifies the computation, but is not adequate for many important classes of applications in control and communication theory. For example, when the noise is small, the mean escape time from a set can be quite sensitive to the underlying statistics even though, in the study of the effects of the noise over any fixed finite time interval, the Gaussian approximation might be a good one. This paper is concerned with the sensitivity of these statistical quantities to the underlying statistical structure, when the noise effects are small, and also with the question of when the Gaussian assumption makes sense. Consider a sequence of systems with small noise effects whose statistics converge in some sense to those of a “limit” system. The techniques developed involve approx...

Application of high‐resolution processing to range and depth estimation using ambiguity function methods

Abstract: For many acoustic environments a target's acoustic field incident on a hydrophone array segment is not representable by a plane wave, but is a function, generally, of three coordinates: range, depth, and bearing. In these cases a conventional beamformer, which is designed to detect plane waves, cannot localize the target accurately. Techniques have been developed recently to exploit the complexity of the field to estimate the source location coordinates by correlating the received field on the array with accurate replicas of the acoustic field, derived from knowledge of the environment. The potential utility of such techniques has been demonstrated in determining range and depth for simulated high‐SNR signals. In this paper, however, they are shown to exhibit excessive sidelobes for low SNR. To alleviate this problem, two high‐resolution techniques, the Maximum Likelihood Method (MLM I and “Alternate Orthogonal Projection” (AOP), or linear predictor, are applied to the simulated case of one target in white noise in a Pekeris environment. MLM is seen to produce stable main peaks which localize targets precisely with low sidelobes, while AOP is shown to be unstable in the presence of random noise and to produce false peaks even when the noise fields are stable.

Approximation of nonlinear filtering problems and order of convergence

Abstract: In this paper, we consider a filtering problem where the observation is a function of a diffusion corrupted by an independent white noise. We estimate the error caused by a discretization of the time interval ; we obtain some approximations of the optimal filter which can be computed with Monte-Carlo methods and we study the order of convergence.

An approach to Ito linear equations in Hilbert spaces by approximation of white noise with coloured noise

Abstract: We consider the stochastic problem , in a Hilbert space H, where f,X are prescribed data, Wt is a real Brownian motion, and A(t), B generate an analytic semi-group and a strongly continuous group respectively. The domains D(A (t)) may vary with t and we only require D(A(t))CD(B) for each t. A unique generalized solution is constructed as the pathwise uniform limit of solutions of suitable approximating deterministic problems, which are obtained by approaching the white noise dWt with a sequence of regular coloured noises

Biofeedback method and apparatus

Abstract: The new biofeedback technique permits simultaneous, preferably redundant visual and auditory presentation of any intrinsically motivating stimuli together with continous information pertaining to the physiological parameter to be controlled. Essentially, it varies the signal to noise ratio (S/N) of an audio or video signal as a function of any physiological parameter or combination of several parameters. That is, intrinsically motivating stimuli, visual and auditory, are presented through a color TV set; image and sound are initially masked by white noise, set to a level just above perception (minimum signal and maximum noise). As the experimental subject changes a certain physiological parameter, image and sound become clearer if the change occurs in the desired direction. The video signal remains synchronized at any noise level. The final S/N ratio has been utilized as an index of motivation in an experiment to evaluate the efficiency of the new technique.

A spectral matching technique for ARMA parameter estimation

Abstract: An iterative frequency domain technique is presented for estimating AR-plus-noise and autoregressive moving-average (ARMA) parameters. The technique is based on minimizing the error between the sample power spectrum and a spectral model. The variance of the estimation error is shown to be close to the Cramer-Rao bound for some examples.

Robust Smoothing Applied to White Noise and Single Outlier Contaminated Raman Spectra

Abstract: There are several smoothing procedures for spectral data which are affected by occasionally occurring outliers. Most of the known methods are based on local averages (or fits) of the spectral data. We introduce here an outlier-insensitive, robust smoothing method which rejects the influence of huge noise spikes. The proposed smoothing algorithm can be tuned by two parameters. The first corresponds to the signal-to-noise ratio, the second to the halfwidths of the spectral bands. We apply this new technique to several spectra and prove the advantages of our method of identifying peaks and baselines in Raman spectroscopy.

Clock Characterization Tutorial

Abstract: Managers ar9 often required to make key program decisions based on the performance of some elements of a large system. This paper is intended to assist the manager in this important task in so far as it relates to the proper use of precise and accurate clocks. An intuitive approach will be used to show how a clock’s stability is measured, why it is measured the way i t i s , and why it is described the way it is. 4n intuitive explanation of the meaning of time domain and frequency domain measures as well as why they are used will also be given. Explanations of when an “Allan variance” plot should be used and when it should not be used will also be given. The relationship of the rms time error of a clock to a CI~(T) diagram will also be given. The environmental sensitivities of a clock are often the most important effects determining its performance. Typical environmental parameters of concern and nomj.nsl sensitivity values for commonly used clocks will be reviewed. SYSTEMATIC AND RANDOM DEVIATIONS IN CLOCKS This paper is tutorial in nature with a minimum of mathematics -the goal being to characterize clock behavior. First, time deviations or frequency deviations in clocks may be categorized into two types: systematic deviations and random deviations. The systematic deviations come in a variety of forms. Typical examples are frequency sidebands, diurnal or annual variations in a clock’s behavior, time offset, frequency offset and frequency drift. Figure 1 illustrates some of these. If a clock has a frequency offset, the time deviations will appear as a ramp function. On the other hand, if a clock has a frequency drift, then the resulting time deviations will appear as a quadratic time function --the time deviations will be proportional to the square of the running time. There are many other systematic effects that are very important to consider in understanding a clock’s characteristics and Figure 1 is a very simplistic picture or nominal model of most. precision osci l lators. The random fluctuations or deviations in precision oscillators can often be characterized by power law spectra. In other words, if the time residuals are examined, after removing the systematic effects, one or more of the power law spectra shown in Figure 2 are typically observed. The meaning of power law spectra is that if a Fourier analysis or spectral density analysis is proportional to fB; B designates the particular power law process (8 = 0, -l,-2,-3,-4 and u = 2nf). The first process shown in Figure 2 is called white noise phase modulation (PM)., The noise is typically observed in the short term fluctuations, for example, of an active hydrogen maser for sample times of from one second to about 100 seconds. This noise is also observed In quartz crystal oscillators for sample times in the vicinity of a millisecond and shorter. The f l icker noise P M , f-l, is the second line in Figure 2. This kind of noise is often found for sample times of one millisecond to one second in quartz crystal oscillators. The f-* or random walk PM indicated by the third line is what is observed for the time deviations of rubidium, cesium, or hydrogen frequency standards. If the first difference is taken of a series of discrete time readings from the third line, then the result is proportional to the frequency deviations, which will be an f” process or a white noise frequency modulation (FM) process. In other words the time and the frequency are related through a derivative or an integral depending upon which way one goes. The derivative of the time deviations yields the frequency deviations, and the integral of the frequency deviations yields the time deviations. So, random walk time deviations result from white noise Ft4. In general, the spectral density of the frequency fluctuations is w* times the spectral density of the time flu tuations. The fourth line in Figure 2 is an f-s process. I f this were representative of the time fluctdations then the frequency would be an f-’ or a flicker noise FM process. This process is typical of the output of a quartz crystal oscillator for sample times longer than one second or the output of rubidium or cesium standards in the long term (on the order of a few hours, few days, or few weeks

Estimating the Variances of Autocorrelations Calculated from Financial Time Series

Abstract: SUMMARY Autocorrelation coefficients calculated from n observations are known to have variances approxi- mately equal to 1/n, for a series of independent and identically distributed variables. The variances can be higher for a general uncorrelated process. Estimates of the variances are derived, assuming only that the process is uncorrelated with symmetric distributions. Results are presented for 17 financial time series. Most estimates exceed 2.5/n for daily returns from commodities, 1.6/n for currencies and 1.3/n for a share index. Standard tests for zero autocorrelation are therefore un- reliable. Suitably rescaled data have autocorrelation variances close to 1/n. Daily changes in the prices of a stock, currency or commodity are either uncorrelated or very weakly autocorrelated. Tests for zero autocorrelation are needed to help discover how quickly and how accurately prices respond to relevant information. This is an important issue for all users of financial markets. Tests are usually derived from an asymptotic theorem about the distributions of sample autocorrelations, proved by Anderson and Walker (1964). The theorem implies that the variance of a sample autocorrelation is approximately 1 /n for n observations from a finite variance, strict white noise process. A process is strict white noise if its variables are independent and identically distributed (i.i.d.). Zero autocorrelation does not imply a strict white noise process. Indeed the variances of price-changes appear to fluctuate. Consequently, l/n may not be an appropriate sampling variance for autocorrelation coefficients. This paper presents a method for estimating the sampling variance. Results are given for 17 financial time series. The median estimate of the sampling variance is about 2.5/n. Therefore standard tests, based on an assumed variance 1 /n, are most unreliable. It is also shown that certain rescaled data have autocorrelation variances close to 1/n. Thus reliable tests can be performed by using the rescaled data.

On adaptive implementations of Pisarenko's harmonic retrieval method

Abstract: The problem of locating spectral lines (sinusoids) in white noise has been studied intensively in the past few years, especially since the work of Pisarenko who showed that unbiased estimates of the sinusoidal frequencies could be obtained from the λ 0 eigenvector (i.e. the eigenvector associated with the smallest eigenvalue) of the covariance matrix of the process. Recently, adaptive implementations of Pisarenko's method have been developed in which the estimates can be updated as new data is observed, and so these algorithms have the ability to track slowly time-varying processes. In this paper, a new adaptive algorithm based on the inverse power method is developed and compared by simulation to other methods. Conclusions are drawn which affect any adaptive implementation of Pisarenko's method.

The equivalent discrete-time optimal control problem for continuous-time systems with stochastic parameters

Abstract: This paper considers the transformation of the digital optimal control problem to an equivalent discrete-time optimal control problem for linear continuous-time systems with continuous-time white stochastic parameters and additive continuous-time white noise. The observations available at the sampling instants are in general non-linear and corrupted by discrete-time noise. The equivalent discrete-time system has white stochastic parameters. Expressions are derived for the first and second moments of these parameters and for the cost criterion parameters which are explicit in the parameters and the statistics of the continuous-time system.

The interspike interval of a cable model neuron with white noise input

Abstract: The firing time of a cable model neuron in response to white noise current injection is investigated with various methods. The Fourier decomposition of the depolarization leads to partial differential equations for the moments of the firing time. These are solved by perturbation and numerical methods, and the results obtained are in excellent agreement with those obtained by Monte Carlo simulation. The convergence of the random Fourier series is found to be very slow for small times so that when the firing time is small it is more efficient to simulate the solution of the stochastic cable equation directly using the two different representations of the Green's function, one which converges rapidly for small times and the other which converges rapidly for large times. The shape of the interspike interval density is found to depend strongly on input position. The various shapes obtained for different input positions resemble those for real neurons. The coefficient of variation of the interspike interval decreases monotonically as the distance between the input and trigger zone increases. A diffusion approximation for a nerve cell receiving Poisson input is considered and input/output frequency relations obtained for different input sites. The cases of multiple trigger zones and multiple input sites are briefly discussed.

Artifacts in Wiener Kernels Estimated Using Gaussian White Noise

Abstract: Wiener's nonlinear system identification theory characterizes a system function with a set of kernels of integrals. One method of determining these Wiener kernels is the cross-correlation technique proposed by Lee and Schetzen, which uses Gaussian white noise as the input to the unknown system. Because a test stimulus is only an approximation of infinitely long Gaussian white noise, it is possible that artifacts are generated during the estimation of the kernels. To help identify and characterize these artifacts, Wiener kernel estimates for two simple nonlinear model systems were made using a pseudorandom Gaussian white noise sequence. The results showed that because of the approximation of a Gaussian distribution, artifacts appear in the estimated kernels due to a form of aliasing. These artifacts can be reduced by increasing the sequence length of the input noise.

Detection of multiple sinusoids using a parallel ALE

Abstract: This paper introduces an Adaptive Line Enhancer (ALE) whose parallel structure enables the detection and enhancement of multiple sinusoids. A function describing the performance surface is derived for the case where several line signals are buried in white noise. A steepest descent adaptive algorithm is derived, and simulations are used to demonstrate its performance.