Showing papers on "Gaussian noise published in 1978"

All-pole modeling of degraded speech

Abstract: This paper considers the estimation of speech parameters in an all-pole model when the speech has been degraded by additive background noise. The procedure, based on maximum a posteriori (MAP) estimation techniques is first developed in the absence of noise and related to linear prediction analysis of speech. The modification in the presence of background noise is shown to be nonlinear. Two suboptimal procedures are suggested which have linear iterative implementations. A preliminary illustration and discussion based both on a synthetic example and real speech data are given.

Stability and Control of Stochastic Systems with Wide-band Noise Disturbances. I

Abstract: For dynamical systems with external influences which are approximately white noise (wide-band noise), we show that stability and other properties of the white noise problem that depend on the infinite time interval, continue to hold away from white noise but not far from it.

Decision and estimation theory

Abstract: This textbook provides a useful and compact introduction to the fundamentals of decision and estimation theory by developing the subject in a most logical and lucid manner. Moreover, the approach is scholarly and presents new views of control problems. The crux of the text is contained in the chapter of sequential decision theory which is exceptionally clearly presented and very comprehensive. The book is divided into two parts. Part one covers decision theory in Chapters 3-7, and part two covers estimation theory in Chapters 8-11. A summary of the essential points is presented at the end of each chapter. Chapter 1 is an introduction to the decision and estimation problem and includes an outline of the book. Chapter 2 reviews briefly the basic concepts of probability theory. A good knowledge of this chapter is essential in order to understand the subsequent chapters. However, readers familiar with the contents of this chapter, may proceed directly to Chapter 3. The topics considered in this chapter are the following: discrete probability theory, random variables, and random processes. Chapter 3 treats the decision problem in its simplest form, that is, binary decision problems with a single scalar observation. Here the reader will find an excellent treatment of five different criteria: maximumlikelihood decision criterion, the Neyman-Pearson criterion, probabilityof-error criterion, the Bayes risk criterion, and the min-max criterion. These criteria are useful for designing the decision rule for a simple binary decision problem. In Chapter 4 the authors extend the results of Chapter 3 by treating multiple-variable observations. The observation here is expressed in vector form and as a continuous function. Vector observations, the general Gaussian problem, and waveform observations and additive Gaussian noise are examined in depth. Chapter 5 is also an extension of Chapter 3, but the emphasis here is on multiple decisions. This chapter begins with the general Bayesian approach by examining problems with multiple messages, then proceeds to discuss the general and Gaussian cases of probability error and erasure decision problems. The discussion on erasure decision problems is particularly important in that it is a prerequisite for understanding the sequential decision problem. Chapter 6 deals with the binary erasure criteria, sequential Bayes tests, the Wald sequential test, and the average sample number. The discussion on sequential decision tests is particularly well done and adds much to the usefulness of this text. Chapter 7, entitled "Composite and Nonparametric Decision Theory," concludes part one. The nonparametric statistics approach attempts to develop tests that do not depend on the density function. This chapter discusses composite decisions, the sign test, and the Wilcoxon test. In the composite decision problem one is concerned with problems in which the conditional density is known but certain other parameters such as the mean and variance may be unspecified. The sign test, which depends on the sign of the observations, requires a minimum amount of information about the probability distribution of the observations. The Wilcoxon test

The Cramér-Rao estimation error lower bound computation for deterministic nonlinear systems

Abstract: For continuous-time nonlinear deterministic system models with discrete nonlinear measurements in additive Ganssian white noise, the extended Kalman filter (EKF) convariance propagation equations linearized about the true unknown trajectory provide the Cramer-Rao lower bound to the estimation error covariance matrix. A useful application is establishing the optimum filter performance for a given nonlinear estimation problem by developing a simulation of the nonlinear system and an EKF linearized about the true trajectory.

The noise power spectrum in computed X-ray tomography.

Abstract: An expression is derived showing that the two-dimensional noise power spectrum of computed X-ray tomography is proportional to mod G(k) mod 2/k where k is the radial spatial frequency and G(k) is the one-dimensional corrective filter used in the filtered back-projection reconstruction technique. It is shown that predicted noise power spectra compare well with those estimated from CT reconstructions of simulated noise for both the ramp filter and the Hanning-weighted ramp filter. A consequence of the non-uniform shape of the noise power spectrum is that statistical noise in CT reconstructions is correlated from point to point. Because of this correlation when the reconstructed CT values are averaged over some region, the uncertainty of the average depends on the shape of the region as well as its area. This dependence is confirmed by computer simulations.

On the capacity of channels with additive non-gaussian noise

Abstract: We give upper and lower bounds of the capacity C ( X ) of a channel with additive noise X under a constraint on input signals in terms of the second order moments It is proved that C ( X 0 ) ⩽ C ( X ) ⩽ C ( X 0 )+ H X 0 ( X ), where C ( X 0 ) is the capacity of the channel with additive Gaussian noise X 0 with same covariance as that of X and H X 0 ( X ) is the entropy of the measure induced by X in the functional space with respect to that induced by X 0

Method and means for adaptively filtering near-stationary noise from an information bearing signal

Abstract: An input signal containing information such as speech or music as well as near-stationary noise is applied in parallel to a noise-analysis circuit and a noise-reduction circuit, each of which comprises a plurality of bandpass filters covering the range of frequencies associated with the information. The absolute value, or a function thereof, of the output of each bandpass filter in the noise-analysis circuit is produced and smoothed. The presence of near-stationary noise in the input signal is determined by examining the nature of the smoothed signal in each band assuming noise has a frequency spectrum which does not vary with time or varies only within a narrow range over a predetermined period of time with respect to the spectral parameters of the information signal. If noise is detected, the noise-analysis circuit identifies spectral parameters of the information and/or noise in each band using the smoothed signal therein. In the preferred embodiment of the invention, the bandpass filters of the noise-reduction circuit have gain elements that are adjusted in accordance with the identified parameters to minimize, under some continuous minimization criterion, the effect of the noise in the input signal thus enhancing intelligibility of the information therein. Minimization can be such that the gain-to-parameter relationships are similar to those in Weiner or Kalman filtering theory with a-priori knowledge of the noise, or of the noise and information, except that in this case, a-priori knowledge of the noise is acquired via identification and is not preassumed.

Study of the Random Vibration of Nonlinear Systems by the Gaussian Closure Technique

Abstract: A technique is developed to study random vibration of nonlinear systems. The method is based on the assumption that the joint probability density function of the response variables and input variables is Gaussian. It is shown that this method is more general than the statistical linearization technique in that it can handle non-Gaussian excitations and amplitude-limited responses. As an example a bilinear hysteretic system under white noise excitation is analyzed. The prediction of various response statistics by this technique is in good agreement with other available results.

Detection in Laplace Noise

Abstract: The discrete time detection of a known constant signal in white stationary Laplace noise is considered. Exact expressions describing the performance of both the Neyman-Pearson optimal detector and the suboptimal linear detector are presented. Also, graphs of the receiver operating characteristics are given. The actual performance of the Neyman-Pearson optimal detector is compared to that predicted by a Gaussian approximation to the distribution of the test statistic.

Digital Removal of Random Media Image Degradations by Solving the Diffusion Equation Backwards in Time

Abstract: We consider the image restoration problem with Gaussian-like point spread functions and reformulate it as an initial value problem for the backwards diffusion equation. This approach leads to rigorous bounds on the reliability of the restoration, as a function of the noise variance, without any assumptions on the spectral characteristics of either signal or noise.In the latter half of the paper, we then describe a powerful algorithm, based on the above reformulation, and successfully use it to restore a turbulence degraded image. Typically, a complete restoration and display requires 10 seconds of CDC 7600 computing time, for a $128 \times 128$ image. We also describe a restoration experiment where Gaussian blur was simulated, and multiplicative noise added according to Huang’s model. The algorithm performs competently even at low signal to noise ratios.

Estimation of images degraded by film-grain noise

Abstract: Film-grain noise describes the intrinsic noise produced by a photographic emulsion during the process of image recording and reproduction. In this paper we consider the restoration of images degraded by film-grain noise. First a detailed model for the over-all photographic imaging system is presented. The model includes linear blurring effects and the signal-dependent effect of film-grain noise. The accuracy of this model is tested by simulating images according to it and comparing the results to images of similar targets that were actually recorded on film. The restoration of images degraded by film-grain noise is then considered in the context of estimation theory. A discrete Wiener filer is developed which explicitly allows for the signal dependence of the noise. The filter adaptively alters its characteristics based on the nonstationary first order statistics of an image and is shown to have advantages over the conventional Wiener filter. Experimental results for modeling and the adaptive estimation filter are presented.

Restoring with maximum entropy. III. Poisson sources and backgrounds

Abstract: The maximum entropy (ME) restoring formalism has previously been derived under the assumptions of (i) zero background and (ii) additive noise in the image. However, the noise in the signals from many modern image detectors is actually Poisson, i.e., dominated by single-photon statistics. Hence, the noise is no longer additive. Particularly in astronomy, it is often accurate to model the image as being composed of two fundamental Poisson features: (i) a component due to a smoothly varying background image, such as caused by interstellar dust, plus (ii) a superimposed component due to an unknown array of point and line sources (stars, galactic arms, etc.). The latter is termed the “foreground image” since it contains the principal object information sought by the viewer. We include in the background all physical backgrounds, such as the night sky, as well as the mathematical background formed by lower-frequency components of the principal image structure. The role played by the background, which may be separately and easily estimated since it is smooth, is to pointwise modify the known noise statistics in the foreground image according to how strong the background is. Given the estimated background, a maximum-likelihood restoring formula was derived for the foreground image. We applied this approach to some one-dimensional simulations and to some real astronomical imagery. Results are consistent with the maximum-likelihood and Poisson hypotheses: i.e., where the background is high and consequently contributes much noise to the observed image, a restored star is broader and smoother than where the background is low. This nonisoplanatic behavior is desirable since it permits extra resolution only where the noise is sufficiently low to reliably permit it.

Noise and coherence in optical image processing. II. Noise fluctuations

Abstract: Fluctuations in image illuminance resulting from various sources of optical noise are studied as a function of the spatial coherence of the illumination. It is shown that noise fluctuations caused by the pupil plane can be reduced considerably by using incoherent, or even partially coherent, rather than coherent illumination. Conversely, noise caused by defects in the object plane is not affected by the degree of coherence, except for phase noise which is suppressed in incoherent light. Expressions for noise fluctuations are developed on the basis of a simplifying Gaussian assumption for the noise sources; the validity of this assumption is justified.

Fast time-invariant implementations of Gaussian signal detectors

Abstract: A new implementation is presented for the optimum likelihood ratio detector for stationary Gaussian signals in white Gaussian noise that uses only two causal time-invariant filters. This solution also has the advantage that fast algorithms based on the Levinson and Chandrasekhar equations can he used for the determination of these time-invariant filters. By using a notion of "closeness to stationarity,' there is a natural extension of the above results for nonstationary signal processes.

On the choice of noise for the analysis of the peripheral auditory system

Abstract: The cross-correlation between output and input of a system containing nonlinearities, when that system is stimulated with Gaussian white noise, is a good estimate of the linear properties of the system. In practice, however, when sequences of pseudonoise are used, great errors may be introduced in the estimate of the linear part depending on the properties of the noise. This consideration assumes special importance in the analysis of the linear properties of the peripheral auditory system, where the rectifying properties of the haircells constitute a second order nonlinearity. To explore this problem, a simple model has been designed, consisting of a second order nonlinearity without memory and sandwiched between two bandpass filters. Different types of pseudonoise are used as input whereupon it is shown that noise based on binary m-sequences, which is commonly used in noise generators, will yield totally incorrect information about this system. Somewhat better results are achieved with other types of noise. By using inverse-repeat sequences the results are greatly improved. Furthermore, certain anomalies obtained in the analysis of responses from single fibers in the auditory nerve are viewed in the light of the present results. The theoretical analysis of these anomalies reveals some information about the organization of the peripheral auditory system. For example, the possibility of the existence of a second bandpass filter in the auditory periphery seems to be excluded.

Locally optimum detection of discrete‐time stochastic signals in non‐Gaussian noise

Abstract: Two basic models are considered for the problem of detecting small, zero‐mean stochastic signals in non‐Gaussian noise processes. Appying the theory of local tests, a general detector structure is derived for each of these models. A third (suboptimal) scheme based on null‐zone detection is also proposed and a technique for optimizing its parameters is discussed. Using the criterion of asymptotic relative efficiency, the performance of these three detectors and of a power detector are compared under each of the proposed models.

Recursive causal linear filtering for two-dimensional random fields

Abstract: The causal estimation of a two-parameter Gaussian random field in the presence of an additive, independent, white Gaussian noise is studied. The dynamics of this random field are modeled by partial differential equations from which the recursive filtering equations and the generalized Riccati equation are derived. A specific example is solved in detail.

Characterization of Frequency Stability: Uncertainty Due to the Autocorrelation of the Frequency Fluctuations

Abstract: We considered the general sampling form for the estimate of the Allan variance which is the proposed measure of frequency stability in the time domain, and we defined a variable proportional to the difference between the average fractional frequency fluctuations over the time interval ? to derive the autocorrelation coefficient of the process to which the variable belongs. Calculations of the variance of the estimated Allan variance proved that it may be convergent to its true value with infinite sample number for considered spectral densities of frequency noise. We also applied the results to estimations of frequency measurements to know the influence of the autocorrelation of the process considered. In order to obtain some direct estimates of the confidence of the estimate, distributions of the estimate were plotted by means of computer simulations, and were compared with the chi-square distribution. Those results suggested that for white-and flicker-phase noises (and white-frequency noise) we have to take into account the autocorrelation of the process, while for flicker-or random-walk-frequency noise we may regard the process as a nearly independent (and Gaussian) one.

On-Line Pseudo-Error Monitors for Digital Transmission Systems

Abstract: Pseudo-error detectors are devices which show great potential for the measurement of the bit error rate of an on-line digital communications link. They are implemented in the form of a second detector (in addition and in parallel to the traffic data bit detector) which is very perturbation-sensitive. They do not compromise the traffic handling capacity of the system. Four methods of generating the pseudo-error characteristic are described: i) shifted detection threshold; ii) intersymbol interference enhancement; iii) noise addition; iv) sampling phase offset. Practical considerations generally govern the choice of method. Experimental results of pseudo-error detector behavior in the presence of Gaussian noise show that stable characteristics can be achieved to estimate a wide range of bit error rate (BER's) in very modest time intervals. In addition, experimental results in a real complex environment consisting of a 1.544 Mbits/s T1 capacity digital link using QPSK modulation techniques, including both Gaussian and non-Gaussian perturbations, show pseudo-error detection to be reliable for the measurement of BER and for controlling channel switching.

Estimation-detection of object boundaries in noisy images

Abstract: Estimation of boundaries of objects in noisy images is considered when the objects and the background are statistically characterized. The noise is assumed white, additive, and Gaussian. Optimal recursive estimators in a joint estimation-detection context are derived. Applications to binary pictures are illustrated.

Preservation of the rescaled adjusted range: 3. Fractional Gaussian noise algorithms

Abstract: Improved modeling procedures are available for use with a discrete time fractional Gaussian noise (FGN) model. These advancements in FGN modeling consist of an exact simulation procedure, a maximum likelihood approach to obtain efficient estimates of the model parameters, and a technique for calculating the model residuals so that they can be subjected to rigorous diagnostic checks. The computer algorithms for the aforementioned modeling improvements are available on microfiche. The intention of this article is to summarize the essential ingredients of the complete paper, which appears on microfiche. Summary and entire article are available on microfiche. Order from American Geophysical Union, 1909 K Street, N.W., Washington, D.C. 20006. Document W78-004; $1.00. Payment must accompany order.

Optimum detection of signals in non‐Gaussian noise

Abstract: The purpose of this paper is to derive optimum processing structures for use in the detection of signals in additive non‐Gaussian noise. The cases chosen for analysis are of particular interest in sonar detection problems since it has been reported that ambient oceannoise may, under sone conditions, deviate from the Gaussian model. The processing structures are considered to be models of the likelihood ratio which is an optimum test no matter what the signal and noise statistics, and optimum single channel and array processors are derived for the small signal‐to‐noise ratio cases where (1) the signal is completely known, and (2) the signal is a noiselike, not necessarily Gaussian, zero‐mean process. Expressions are derived which compare the performance of processors optimized for non‐Gaussian noise with those optimized for Gaussian noise for each of the two cases, with transfer functions for the required optimum nonlinear filters and the expected improvements in performance determined using some ’’typical’’ non‐Gaussian probability density functions. Justification of these particular density functions is beyond the scope of this paper except to note that very good agreement is obtained with some published experimental data. New results include the derivation of optimum array processors for the detection of plane wave signals when the array is steered at the signal arrival angle, in a non‐Gaussian noise field and the development of expressions to predict their performance for the case where the signal is a zero‐mean, noiselike process.

Error probability in digital fiber optic communication systems

Abstract: In a digital fiber optic transmission system, during photodetection process, a shot noise is produced that is neither stationary nor independent of the digital message. The evaluation of the average error probability in the presence of such a message-dependent shot noise, of additive Gaussian noise, and of intersymbol interference is considered. Two methods of calculation are outlined: an exhaustive method and a Gram-Charlier series expansion method. The latter is preferred when the number of interferers is moderately large. Some numerical examples for binary independent-symbol transmission are presented.

Asymptotically robust quantization for detection

Abstract: The problem of designing quantizer-detectors whose performance is insensitive to small deviations in noise statistics is considered. The problem is approached on a small-signal asymptotic basis using the Huber-Tukey [1] contaminated density class to model the noise. By applying a formulation originally noted by Martin and Schwartz, it is shown that the proposed robust quantizer has properties exhibited by established robust procedures. As an example, results are presented for the case of contaminated Gaussian noise for various degrees of contamination.

Single-threshold detection of a random signal in noise with multiple independent observations. 1: Discrete case with application to optical communications.

Abstract: A single-threshold processor is derived for a wide class of classical binary decision problems involving the likelihood-ratio detection of a signal embedded in noise. The class of problems we consider encompasses the case of multiple independent (but not necessarily identically distributed) observations of a nonnegative (nonpositive) signal, embedded in additive, independent, and noninterfering noise, where the range of the signal and noise is discrete. We show that a comparison of the sum of the observations with a unique threshold comprises optimum processing, if a weak condition on the noise is satisfied, independent of the signal. Examples of noise densities that satisfy and violate our condition are presented. The results are applied to a generalized photocounting optical communication system, and it is shown that most components of the system can be incorporated into our model. The continuous case is treated elsewhere [ IEEE Trans. Inf. TheoryIT-25, (March, 1979)].