Showing papers on "White noise published in 1988"

A decomposition approach to CPM

Abstract: It is shown that any continuous-phase-modulation (CPM) system can be decomposed into a continuous-phase encoder and a memoryless modulator in such a way that the former is a linear (modulo some integer P) time-invariant sequential circuit and the latter is also time invariant. This decomposition is exploited to obtain alternative realizations of the continuous-phase encoder (and hence of CPM) and also to obtain alternative forms of the optimum decoding algorithm. When P is a prime p so that the encoder is linear over the finite field GF(p), it is shown that cascading it with an outside convolutional encoder is equivalent to a single convolutional encoder. It is pointed out that the cascade of the modulator, the waveform channel (which it is assumed is characterized by additive white Gaussian noise), and the demodulator that operates over one symbol interval yield a discrete memoryless channel that can be studied without the distractions introduced by continuous-phase encoding. >

Stochastic Modeling of Seafloor Morphology: Inversion of Sea Beam Data for Second-Order Statistics

Abstract: At scale lengths less than 100 km or so, statistical descriptions of seafloor morphology can be usefully employed to characterize ridge crest processes, off-ridge tectonics and vulcanism, sedimentation, and postdepositional transport. We seek to develop methods for the estimation of seafloor statistics that take into account the finite precision, resolution, and sampling obtained by actual echo sounding systems. In this initial paper we restrict our attention to the problem of recovering second-order statistics from data sets collected by multibeam devices such as Sea Beam. The seafloor is modeled as a stationary, zero-mean, Gaussian random field completely specified by its two-point covariance function. We introduce an anisotropic two-point covariance function that has five free parameters describing the amplitude, orientation, characteristic wave numbers, and Hausdorff (fractal) dimension of seafloor topography. We formulate the general forward problem relating this model to the statistics of an ideal multibeam echo sounder, in particular the along-track autocovariance functions of individual beams and the cross-covariance functions between beams of arbitrary separation. Using these second moments as data functionals, we then pose the inverse problem of estimating the seafloor parameters from realistic, noisy data sets with finite sampling and beamwidth, and we solve this inverse problem by an iterative, linearized, least squares method. The inversion method is applied to Sea Beam transit data from both the Pacific and Atlantic oceans. Sea Beam system noise stands out as a sharp spike on the along-track autocovariance function and can be modeled as a white noise process whose amplitude generally increases with beam angle. The five parameters in our second-order model can be estimated from the inversion of data sets comprising ∼100–200 km of track length. In general, the cross-track wave number is the most poorly determined, although uncertainties in the assumed Sea Beam response may bias the values of the fractal dimension. Using the assumed beamwidth, the measured noise values, and the seafloor parameters recovered from the inversion, we generate Sea Beam “synthetics” whose statistical character can be directly compared with raw Sea Beam data. For most of the track segments we have processed thus far the synthetics are similar to the data. In the case of one Atlantic profile, however, the comparison clearly indicates the necessity of incorporating higher-order statistics. The space domain procedures described in this paper can be extended for this purpose.

Fast, accurate algorithm for numerical simulation of exponentially correlated colored noise.

Abstract: Traditionally, stochastic differential equations used in the physical sciences have involved Gaussian white noise. ' In recent times, however, white noise has been replaced by colored noise in a variety of contexts. Laser noise problems and first passage time problems have been shown to necessitate the use of colored noise instead of white noise. Even the mathematical foundations for the theory of stochastic differential equations call for colored noise if the Stratonovich perspective is adopted, as it is when physical arguments are invoked. ' In each of these contexts, many speci6c problems require numerical simulation as a component of a complete analysis. This is usually a consequence of nonlinearity and the resulting intractability in purely analytic terms. Consequently, numerical-simulation algorithms have been developed, originally for white noise, and recently for colored noise as well. The simplest type of colored noise to generate is exponentially correlated colored noise. Such noise introduces only one more parameter, the correlation time for the exponential correlation, and it is easily generated by a linear damping equation driven by white noise. Our new algorithm is for this kind of colored noise. In Sec. II we review the white-noise algorithm and the differential version of the exponentially correlated, colored-noise algorithm. In Sec. III we present the integral version of the colored-noise algorithm and demonstrate its superior properties.

A time-frequency formulation of optimum detection

Abstract: A time-frequency formulation is proposed for the optimum detection of Gaussian signals in white Gaussian noise. By choosing the Wigner-Ville distribution as the basic time-frequency tool, it is shown that the corresponding receivers generally take the form of a correlation between time-frequency structures, matching mathematical optimality with a physically meaningful interpretation. The case of low SNR is examined and various examples are considered: deterministic signal, Rayleigh fading signal, random jitter, and random time-varying channel. A general class of time-frequency receivers is proposed which admits as limiting cases different known structures, and its suboptimum performance is evaluated. Possible extensions to more elaborate situations (including parameter estimation) are mentioned. >

Signal detection in fractional Gaussian noise

Abstract: Several results related to the reproducing kernel Hilbert space of fractional Brownian motion are presented to facilitate the study of signal detection in additive fractional Gaussian noise. This Hilbert space is completely characterized, and an alternative characterization for the restriction of this class of functions to a compact interval (0.T) is given. Infinite- and finite-interval whitening filters for fractional Brownian motion are also developed. The application of these results to the signal detection problem yields necessary and sufficient conditions for a deterministic or stochastic signal to produce a nonsingular shift when embedded in additive fractional Gaussian noise. A formula for the likelihood ratio corresponding to any deterministic nonsingular shift is developed. >

Single-user detectors for multiuser channels

Abstract: Optimum decentralized demodulation for asynchronous Gaussian multiaccess channels is considered. It is assumed that the receiver is the destination of the information transmitted by only one active user, and single-user detectors that take into account the existence of the other active users in the channel are obtained. The problem considered is one of signal detection in additive colored nonGaussian noise, and attention is focused on one-shot structures where detection of each symbol is based only on the received process during its corresponding interval. Particular emphasis is placed on asymptotically optimum detectors for weak interferers, for CDMA (code-division multiple-access) signature waveforms with long spreading codes, and for low background Gaussian noise level. >

Detection of coded modulation signals on linear, severely distorted channels using decision-feedback noise prediction with interleaving

Abstract: On linear bandlimited Gaussian noise channels with sufficiently high SNR, channel capacity can be approached by combining powerful coded modulation schemes designed for Nyquist channels with the equalization power of decision-feedback equalization (DFE). However, this combination may not be realized in a straightforward manner, since, in general, DFE requires delay-free decisions for feedback, and in a coded system such decisions are not sufficiently reliable. A technique is proposed that combines periodic interleaving with noise-predictive DFE, so that delayed reliable decisions can be used for feedback. When sufficient delay in the interleavers can be tolerated, this technique can attain the DFE performance. On severely distorted channels, modest delays can be sufficient to obtain respectable gains over linear equalization. >

A comparative study and assessment of Doppler ultrasound spectral estimation techniques. Part II: Methods and results.

Abstract: Various alternative spectral estimation methods are examined and compared in order to assess their possible application for real-time analysis of Doppler ultrasound arterial signals. Specifically, five general frequency domain models are examined, including the periodogram, the general autoregressive moving average (ARMA) model which has the autoregressive (AR) and moving average (MA) models as special cases, and Capon's maximum likelihood spectral model. A simulated stationary Doppler signal with a known theoretical spectrum was used as the reference test sequence, and white noise was added to enable various signal/noise conditions to be created. The performance of each method representative of each spectral model was assessed using both qualitative and quantitative schemes that convey information related to the bias and variance of the spectral estimates. Three integrated performance indices were implemented for quantitative analysis. The relative computational complexity for each algorithm was also investigated. Our results indicate that both the AR(Yule-Walker) and ARMA(singular value decomposition) models of orders (8) and (4, 4), respectively, show good agreement with the theoretical spectrum, and yield estimates with variances considerably less than the Fast Fourier Transform (FFT). Preliminary results obtained with these methods using a clinical, non-stationary Doppler signal supports these observations.

Auto‐Regressive Model for Nonstationary Stochastic Processes

Abstract: An autoregressive model for univariate, one‐dimensional, nonstationary, Gaussian random processes with evolutionary power spectra is introduced. At the same time, an efficient technique for numerically generating sample functions of such nonstationary processes is developed. The technique uses a recursive equation which: (1) Reflects the nature of the nonstationarity of the process whose sample functions are to be generated; and (2) involves a normalized univariate, one‐dimensional white noise sequence. The coefficients of the recursive equation are determined using the autocorrelation function of the process, which in turn is calculated from the evolutionary power spectrum at every time instant. Using the recursive equation with those coefficients, sample functions over a specified domain can be generated with substantial computational ease. Univariate, one‐dimensional, nonstationary processes with three different forms of the evolutionary power spectrum are modeled, and their sample functions are genera...

Finding motion parameters from spherical motion fields (or the advantages of having eyes in the back of your head)

Abstract: A theory is developed for determining the motion of an observer given the motion field over a full 360 degree image sphere. The method is based on the fact that for an observer translating without rotation, the projected circular motion field about any equator can be divided into disjoint semicircles of clockwise and counterclockwise flow, and on the observation that the effects of rotation decouple around the three equators defining the three principal axes of rotation. Since the effect of rotation is geometrical, the three rotational parameters can be determined independently by searching, in each case, for a rotational value for which the derotated equatorial motion field can be partitioned into 180 degree arcs of clockwise and counterclockwise flow. The direction of translation is also obtained from this analysis. This search is two dimensional in the motion parameters, and can be performed relatively efficiently. Because information is correlated over large distances, the method can be considered a pattern recognition rather than a numerical algorithm. The algorithm is shown to be robust and relatively insensitive to noise and to missing data. Both theoretical and empirical studies of the error sensitivity are presented. The theoretical analysis shows that for white noise of bounded magnitude M, the expected errors is at worst linearly proportional to M. Empirical tests demonstrate negligible error for perturbations of up to 20% in the input, and errors of less than 20% for perturbations of up to 200%.

Optimal linear-quadratic systems for detection and estimation

Abstract: The problem of linear-quadratic systems for detection has long been solved by assuming the deflection criterion and Gaussian noise. It is shown here that the Gaussian assumption can be removed, and a complete solution is presented for an arbitrary probability distribution with finite fourth-order moments. The optimal solution can always be obtained by solving a linear system of equations. Some properties of the optimal systems are developed for particular examples of nonGaussian noise. It is shown that there is a strong relationship between linear-quadratic optimal detection and optimal estimation, which extends results known for the purely linear case. >

Estimating the number of sinusoids in additive white noise

Abstract: The test proposed uses the eigenvector decomposition of the estimated autocorrelation matrix and is based on matrix perturbation analysis. The estimator is shown to be able to resolve closely spaced sinusoids at lower signal-to-noise ratios than heuristic tests. Simulation results for two closely spaced sinusoids are detailed. Several unanswered questions are discussed. >

Calculus on Gaussian and Poisson white noises

Abstract: Recently one of the authors has introduced the concept of generalized Poisson functionals and discussed the differentiation, renormalization, stochastic integrals etc ([8], [9]), analogously to the works of T Hida ([3], [4], [5]) Here we introduce a transformation for Poisson fnnctionals with the idea as in the case of Gaussian white noise (cf [10], [11], [12], [13]) Then we can discuss the differentiation, renormalization, multiple Wiener integrals etc in a way completely parallel with the Gaussian case The only one exceptional point, which is most significant, is that the multiplications are described by for the Gaussian case, for the Poisson case, as will be stated in Section 5 Conversely, those formulae characterize the types of white noises

Robust image modeling techniques with an image restoration application

Abstract: A robust parameter-estimation algorithm for a nonsymmetric half-plane (NSHP) autoregressive model, where the driving noise is a mixture of a Gaussian and an outlier process, is presented. The convergence of the estimation algorithm is proved. An algorithm to estimate parameters and original image intensity simultaneously from the impulse-noise-corrupted image, where the model governing the image is not available, is also presented. The robustness of the parameter estimates is demonstrated by simulation. Finally, an algorithm to restore realistic images is presented. The entire image generally does not obey a simple image model, but a small portion (e.g. 8*8) of the image is assumed to obey an NSHP model. The original image is divided into windows and the robust estimation algorithm is applied for each window. The restoration algorithm is tested by comparing it to traditional methods on several different images. >

Bayesian clustering for unsupervised estimation of surface and texture models

Abstract: A method of calculating the maximum-likelihood clustering for the unsupervised estimation of polynomial models for the data in images of smooth surfaces or for range data for such surfaces is presented. An image or a depth map of a region of smooth 3-D surface is modeled as a polynomial plus white noise. A region of physically meaningful textured-image such as the image of foliage, grass, or road in outdoor scenes or conductor or lintburn on a thick-film substrate is modeled as a colored Gaussian-Markov random field (MRF) with a polynomial mean-value function. Unsupervised-model parameter-estimation is accomplished by determining the segmentation and model parameter values that maximize the likelihood of the data or a more general Bayesian performance functional. Agglomerative clustering is used for this purpose. >

Colored noise in dynamical systems

Abstract: We consider the problem of escape of a particle activated by small amplitude Gaussian colored noise, from a potential well. We consider various ranges of the parameters representing the bandwidth of the noise, its spectral height, and the dissipation coefficient. We employ singular perturbation methods to derive explicit analytical approximations for the stationary density of fluctuations about the deterministically meta-stable state at the bottom of the well, and for the mean first passage time to overcome the potential barrier and escape the well. The latter leads to a formula for the escape (activation) rate. Among other results we find that the mean first passage time is exponentially larger than in the white noise case.

White-noise analysis in visual neuroscience

Abstract: In 1827, plant biologist Robert Brown discovered what is known as Brownian motion, a class of chaos. Formal derivative of Brownian motion is Gaussian white-noise. In 1938, Norbert Wiener proposed to use the Gaussian white-noise as an input probe to identify a system by a series of orthogonal functionals known as the Wiener G-functionals. White-noise analysis is uniquely suited for studying the response dynamics of retinal neurons because (1) white-noise light stimulus is a modulation around a mean luminance, as are the natural photic inputs, and it is a highly efficient input; and (2) the analysis defines the response dynamics and can be extended to spike trains, the final output of the retina. Demonstrated here are typical examples and results from applications of white-noise analysis to a visual system.

Perturbation analysis of an SVD-based linear prediction method for estimating the frequencies of multiple sinusoids

Abstract: A linear prediction approach is studied for estimating the frequencies of sinusoids in white noise. It is shown that in the first step, the continuity of the generalized inverse and the concept of angle between subspaces play an important role. The continuity concept helps explain the need for a low rank approximation, and the quality of the approximation is appraised by using the notion of angle between subspaces. For the second step, the sensitivity of the zeros of the predictor polynomial becomes an important consideration and is examined. It is shown that increasing the order of the predictor polynomial and computing the minimum norm solution provides a mechanism to reduce parameter sensitivity. >

Harmonic decomposition methods in cumulant domains (array processing)

Abstract: The authors address the bearing estimation problem of sources from array data (snapshots) in the presence of Gaussian color (spatially correlated) noises of unknown autocorrelation matrix They demonstrate that the harmonic decomposition methods (signal and noise subspace) can easily be reformulated using fourth-order cumulant matrices instead of autocorrelations Simulation results are presented and comparisons are made to show that the performance of the fourth-order cumulant-based methods (beamforming, MUSIC) is superior to that of their equivalent autocorrelation-based methods when the additive noise sources are colored Gaussian with unknown correlation matrix >

Confidence intervals for the signal-to-noise ratio when a signal embedded in noise is observed over repeated trials

Abstract: The problem of estimating the signal-to-noise ratio (SNR) when repeated measurements are made of a deterministic signal embedded in random noise is considered. An estimator is described, its asymptotic distribution is derived, and a method for constructing confidence intervals is proposed. The performance of the method is evaluated using simulated evoked potential data, and an application to real evoked potential data is presented. >

On the use of autoregressive order determination criteria in multivariate white noise tests

Abstract: Testing the hypothesis of multivariate white noise is seen as the selection of the order of a multivariate autoregressive model for the observed time series. Therefore, multivariate white noise tests can be carried out by applying autoregressive order-determination criteria such as AIC, BIC, etc. It is known, for example, that the BIC criterion estimates consistently the order of an autoregression. An order-determination criterion with this property leads to a white noise test with a significance level approaching zero as n, the number of observations, increases. The order of an autoregressive moving-average model is proposed to be determined by applying this kind of white noise test. The resulting model building procedure is a generalization of the procedure proposed by G.E.P. Box and G.M. Jenkins (1970). >

Adaptive matched filtering for QRS detection

Abstract: The authors have investigated the use of an adaptive filter to whiten the noise in the ECG (electrocardiogram) signal and adjust the matched filter response accordingly. They applied a simple QRS detection strategy to the filtered signal and evaluated the QRS detector with ECG data containing severe motion artifact and muscle noise. Preliminary results indicate that in the presence of severe motion artifact, adaptive matched filtering for QRS detection represents a significant improvement over application of a matched filter with a white noise assumption. >

Dirichlet forms and white noise analysis

Abstract: We use the white noise calculus as a framework for the introduction of Dirichlet forms in infinite dimensions. In particular energy forms associated with positive generalized white noise functionals are considered and we prove criteria for their closability. If the forms are closable, we show that their closures are Markovian (in the sense of Fukushima).

Spectral density of fluctuations of a double-well Duffing oscillator driven by white noise

Abstract: The power spectrum of the archetypal fluctuating bistable system, the underdamped double-well Duffing oscillator, is investigated both experimentally, with use of an electronic circuit, and theoretically. The experiment confirms previous analytic results for the structure of the spectrum, including the existence of three distinct peaks within a certain parameter range. The theory is extended to describe analytically the shape of the peak due to intrawell fluctuations for arbitrary noise intensities as well as certain other features of the spectrum. Good quantitative agreement of theory and experiment is demonstrated.

White-noise analysis of nonlinear behavior in an insect sensory neuron: kernel and cascade approaches

Abstract: A functional expansion was used to model the relationship between a Gaussian white noise stimulus current and the resulting action potential output in the single sensory neuron of the cockroach femoral tactile spine A new precise procedure was used to measure the kernels of the functional expansion Very similar kernel estimates were obtained from separate sections of the data produced by the same neuron with the same input noise power level, although some small time-varying effects were detectable in moving through the data Similar kernel estimates were measured using different input noise power levels for a given cell, or when comparing different cells under similar stimulus conditions The kernels were used to identify a model for sensory encoding in the neuron, comprising a cascade of dynamic linear, static nonlinear, and dynamic linear elements Only a single slice of the estimated experimental second-order kernel was used in identifying the cascade model However, the complete second-order kernel of the cascade model closely resembled the estimated experimental kernel Moreover, the model could closely predict the experimental action potential train obtained with novel white noise inputs