Showing papers on "White noise published in 1994"

Attractors for random dynamical systems

Abstract: A criterion for existence of global random attractors for RDS is established. Existence of invariant Markov measures supported by the random attractor is proved. For SPDE this yields invariant measures for the associated Markov semigroup. The results are applied to reation diffusion equations with additive white noise and to Navier-Stokes equations with multiplicative and with additive white noise.

On lattice quantization noise

Abstract: We present several results regarding the properties of a random vector, uniformly distributed over a lattice cell. This random vector is the quantization noise of a lattice quantizer at high resolution, or the noise of a dithered lattice quantizer at all distortion levels. We find that for the optimal lattice quantizers this noise is wide-sense-stationary and white. Any desirable noise spectra may be realized by an appropriate linear transformation ("shaping") of a lattice quantizer. As the dimension increases, the normalized second moment of the optimal lattice quantizer goes to 1/2/spl pi/e, and consequently the quantization noise approaches a white Gaussian process in the divergence sense. In entropy-coded dithered quantization, which can be modeled accurately as passing the source through an additive noise channel, this limit behavior implies that for large lattice dimension both the error and the bit rate approach the error and the information rate of an additive white Gaussian noise (AWGN) channel.

Adaptive signal processing algorithms: stability and performance

Abstract: PART I. 1. Introduction. 2. Offline Analysis. 3. Iterative Minimization. 4. Algorithm Construction. 5. Algorithm Analysis: Gaussian White Noise Setting. 6. Algorithm Analysis: Deterministic Global Theory. PART II. 7. Deterministic Averaging: Single Time Scale. 8. Deterministic Averaging: Mixed Time Scale. PART III. 9. Stochastic Averaging: Single Time Scale. 10. Stochastic Averaging: Mixed Time Scale. APPENDICES. A. Matrix Analysis Review. B. Stochastic Signals and Systems Review. C. Deterministic Signals and Systems Review. D. Mathematical Analysis Review. E. Probability Review. Bibliography. Index.

Radon transformation of time-frequency distributions for analysis of multicomponent signals

Abstract: The Radon transform of a time-frequency distribution produces local areas of signal concentration that facilitate interpretation of multicomponent signals. The Radon-Wigner transform can be efficiently implemented with dechirping in the time domain, however, only half of the possible projections through the time-frequency plane can be realized because of aliasing. We show here that the frequency dual to dechirping exists, so that all of the time-frequency plane projections can be calculated efficiently. Both time and frequency dechirping are shown to warp the time-frequency plane rather rotating it, producing an angle dependent dilation of the Radon-Wigner projection axis. We derive the discrete-time equations for both time and frequency dechirping, and highlight some practical implementation issues. Discrete dechirping is shown to correspond to line integration through the extended-discrete, rather than the discrete, Wigner-Ville distribution. Computationally, dechirping is O(2N log 2N) instead of O(N/sup 3/) for direct projection, and the computation is dominated by the fast Fourier transform calculation. The noise and cross-term suppression of the Radon-Wigner transform are demonstrated by several examples using dechirping and using direct Radon-Wigner transformation. >

Bounds on the decoding error probability of binary linear codes via their spectra

Abstract: Bounds on the error probability of maximum likelihood decoding of a binary linear code are considered. The bounds derived use the weight spectrum of the code and they are tighter than the conventional union bound in the case of large noise in the channel. The bounds derived are applied to a code with an average spectrum, and the result is compared to the random coding exponent. The author shows that the bound considered for the binary symmetrical channel case coincides asymptotically with the random coding bound. For the case of AWGN channel the author shows that Berlekamp's (1980) tangential bound can be improved, but even this improved bound does not coincide with the random coding bound, although it can be very close to it. >

Further results in the fast estimation of a single frequency

Abstract: The author proposes a new frequency estimator for a single complex sinusoid in complex white Gaussian noise. The estimator is applicable to problems in communications requiring high speed, recursive frequency estimation. The estimator is computationally efficient yet obtains near optimum performance at moderate signal-to-noise ratios. >

The stochastic Burgers equation

Abstract: We study Burgers Equation perturbed by a white noise in space and time. We prove the existence of solutions by showing that the Cole-Hopf transformation is meaningful also in the stochastic case. The problem is thus reduced to the anaylsis of a linear equation with multiplicative half white noise. An explicit solution of the latter is constructed through a generalized Feynman-Kac formula. Typical properties of the trajectories are then discussed. A technical result, concerning the regularizing effect of the convolution with the heat kernel, is proved for stochastic integrals.

Neural networks and inversion of seismic data

Abstract: Neural networks can be viewed as applications that map one space, the input space, into some output space. In order to simulate the desired mapping the network has to go through a learning process consisting of an iterative change of the internal parameters, through the presentation of many input patterns and their corresponding output patterns. The training process is accomplished if the error between the computed output and the desired output pattern is minimal for all examples in the training set. The network will then simulate the desired mapping on the restricted domain of the training examples. We describe an experiment where a neural network is designed to accept a synthetic common shot gather (i.e., a set of seismograms obtained from a single source), as its input pattern and to compute the corresponding one-dimensional large-scale velocity model as its output. The subsurface models are built up of eight layers with constant layer thickness over a homogeneous half-space, 450 examples are used to train the network. After the training process the network never computes a subsurface model which perfectly fits the desired one, but the approximation of the network is sufficient to take this model as starting model for further seismic imaging algorithms. The trained network computes satisfactory velocity profiles for 80% of the new seismic gathers not included in the training set. Although the network gives results that are stable when the input is contaminated with white noise, the network is not robust against strong, i.e., correlated, noise. This application proves that neural networks are able to solve nontrivial inverse problems.

A recursive multiple model approach to noise identification

Abstract: Correct knowledge of noise statistics is essential for an estimator or controller to have reliable performance. In practice, however, the noise statistics are unknown or not known perfectly and thus need to be identified. Previous work on noise identification is limited to stationary noise and noise with slowly varying statistics only. An approach is presented here that is valid for nonstationary noise with rapidly or slowly varying statistics as well as stationary noise. This approach is based on the estimation with multiple hybrid system models. As one of the most cost-effective estimation schemes for hybrid system, the interacting multiple model (IMM) algorithm is used in this approach. The IMM algorithm has two desirable properties: it is recursive and has fixed computational requirements per cycle. The proposed approach is evaluated via a number of representative examples by both Monte Carlo simulations and a nonsimulation technique of performance prediction developed by the authors recently. The application of the proposed approach to failure detection is also illustrated. >

Performance of Mean-Frequency Estimators for Doppler Radar and Lidar

Abstract: The performance of mean-frequency estimators for Doppler radar and lidar measurements of winds is presented in terms of two basic parameters: Phi, the ratio of the average signal energy per estimate to the spectral noise level; and Omega, which is proportional to the number of independent samples per estimate. For fixed Phi and Omega, the Cramer-Rao bound (CRB) (theoretical best performance) for unbiased estimators of mean frequency (normalized by the spectral width of the signal), signal power, and spectral width are essentially independent of the number of data samples M. For large Phi, the estimators of mean frequency are unbiased and the performance is independent of M. The spectral domain estimators and covariance based estimators are bounded by the approximate period of M. The spectral domain estimators and covariance based estimators are bounded by the approximate periodogram CRB. The standard deviation of the maximum-likelihood estimator approaches the exact CRB, which can be more than a factor of 2 better than the performance of the spectral domain estimators or covariance-based estimators for typical Omega. For small Phi, the estimators are biased due to the effects of the uncorrelated noise (white noise), which results in uniformly distributed 'bad' estimates. The fraction of bad estimates is a function of Phi and M with weak dependence on the parameter Omega. Simple empirical models describe the standard deviation of the good estimates and the fraction of bad estimates. For Doppler lidar and for large Phi, better performance is obtained by using many low-energy pulses instead of one pulse with the same total energy. For small Phi, the converse is true.

Speech recognition using hidden Markov models with polynomial regression functions as nonstationary states

Abstract: Proposes, implements, and evaluates a class of nonstationary-state hidden Markov models (HMMs) having each state associated with a distinct polynomial regression function of time plus white Gaussian noise. The model represents the transitional acoustic trajectories of speech in a parametric manner, and includes the standard stationary-state HMM as a special, degenerated case. The authors develop an efficient dynamic programming technique which includes the state sojourn time as an optimization variable, in conjunction with a state-dependent orthogonal polynomial regression method, for estimating the model parameters. Experiments on fitting models to speech data and on limited-vocabulary speech recognition demonstrate consistent superiority of these nonstationary-state HMMs over the traditional stationary-state HMMs. >

Random and pseudorandom inputs for Volterra filter identification

Abstract: This paper studies input signals for the identification of nonlinear discrete-time systems modeled via a truncated Volterra series representation. A Kronecker product representation of the truncated Volterra series is used to study the persistence of excitation (PE) conditions for this model. It is shown that i.i.d. sequences and deterministic pseudorandom multilevel sequences (PRMS's) are PE for a truncated Volterra series with nonlinearities of polynomial degree N if and only if the sequences take on N+1 or more distinct levels. It is well known that polynomial regression models, such as the Volterra series, suffer from severe ill-conditioning if the degree of the polynomial is large. The condition number of the data matrix corresponding to the truncated Volterra series, for both PRMS and i.i.d. inputs, is characterized in terms of the system memory length and order of nonlinearity. Hence, the trade-off between model complexity and ill-conditioning is described mathematically. A computationally efficient least squares identification algorithm based on PRMS or i.i.d. inputs is developed that avoids directly computing the inverse of the correlation-matrix. In many applications, short data records are used in which case it is demonstrated that Volterra filter identification is much more accurate using PRMS inputs rather than Gaussian white noise inputs. >

A noise-whitening approach to multiple access noise rejection .I. Theory and background

Abstract: Minimum probability of bit error is difficult to achieve in a DS-CDMA receiver. Since multiple-access noise is the sum of many independent random processes, it is reasonable to approximate it by a Gaussian process of the same power spectral density. This leads to the criterion of maximizing signal-to-noise ratio (SNR). In this paper, receivers that maximize SNR in a particular DS-CDMA system model under various constraints are proposed and analyzed. The method proposed here does not require locking and despreading multiple arriving CDMA signals. The maximization of SNR is compared with the minimization of probability of error, when the receiver is constrained to operate bit-by-bit, in the absence of knowledge of the other users' spreading codes, timing, and phase. >

Iterative and sequential algorithms for multisensor signal enhancement

Abstract: In problems of enhancing a desired signal in the presence of noise, multiple sensor measurements will typically have components from both the signal and the noise sources. When the systems that couple the signal and the noise to the sensors are unknown, the problem becomes one of joint signal estimation and system identification. The authors specifically consider the two-sensor signal enhancement problem in which the desired signal is modeled as a Gaussian autoregressive (AR) process, the noise is modeled as a white Gaussian process, and the coupling systems are modeled as linear time-invariant finite impulse response (FIR) filters. The main approach consists of modeling the observed signals as outputs of a stochastic dynamic linear system, and the authors apply the estimate-maximize (EM) algorithm for jointly estimating the desired signal, the coupling systems, and the unknown signal and noise spectral parameters. The resulting algorithm can be viewed as the time-domain version of the frequency-domain approach of Feder et al. (1989), where instead of the noncausal frequency-domain Wiener filter, the Kalman smoother is used. This approach leads naturally to a sequential/adaptive algorithm by replacing the Kalman smoother with the Kalman filter, and in place of successive iterations on each data block, the algorithm proceeds sequentially through the data with exponential weighting applied to allow adaption to nonstationary changes in the structure of the data. A computationally efficient implementation of the algorithm is developed. An expression for the log-likelihood gradient based on the Kalman smoother/filter output is also developed and used to incorporate efficient gradient-based algorithms in the estimation process. >

An SNR estimation algorithm using fourth-order moments

Abstract: An algorithm is presented that allows an estimation of the SNR just by the observation of the noisy signal. For the estimation, shape factors of the signal's and the noise's probability density functions are used. The algorithm is based on the second and fourth order moments of the observed noisy signal. >

Adaptive filtering for non-Gaussian stable processes

Abstract: A large class of physical phenomena observed in practice exhibit non-Gaussian behavior. In the letter /spl alpha/-stable distributions, which have heavier tails than Gaussian distributions, are considered to model non-Gaussian signals. Adaptive signal processing in the presence of such a noise is a requirement of many practical problems. Since direct application of commonly used adaptation techniques fail in these applications, new algorithms for adaptive filtering for /spl alpha/-stable random processes are introduced. >

System for monitoring an industrial process and determining sensor status

Abstract: A method and system for monitoring an industrial process and a sensor. The method and system include generating a first and second signal characteristic of an industrial process variable. One of the signals can be an artificial signal generated by an auto regressive moving average technique. After obtaining two signals associated with one physical variable, a difference function is obtained by determining the arithmetic difference between the two pairs of signals over time. A frequency domain transformation is made of the difference function to obtain Fourier modes describing a composite function. A residual function is obtained by subtracting the composite function from the difference function and the residual function (free of nonwhite noise) is analyzed by a statistical probability ratio test.

Stochastic gradient adaptation under general error criteria

Abstract: Examines a family of adaptive filter algorithms of the form W/sub k+1/=W/sub k/+/spl mu/f(d/sub k/-W/sub k//sup t/X/sub k/)X/sub k/ in which f(/spl middot/) is a memoryless odd-symmetric nonlinearity acting upon the error. Such algorithms are a generalization of the least-mean-square (LMS) adaptive filtering algorithm for even-symmetric error criteria. For this algorithm family, the authors derive general expressions for the mean and mean-square convergence of the filter coefficients For both arbitrary stochastic input data and Gaussian input data. They then provide methods for optimizing the nonlinearity to minimize the algorithm misadjustment for a given convergence rate. Using the calculus of variations, it is shown that the optimum nonlinearity to minimize misadjustment near convergence under slow adaptation conditions is independent of the statistics of the input data and can be expressed as -p'(x)/p(x), where p(x) is the probability density function of the uncorrelated plant noise. For faster adaptation under the white Gaussian input and noise assumptions, the nonlinearity is shown to be x/{1+/spl mu//spl lambda/x/sup 2///spl sigma//sub k//sup 2/}, where /spl lambda/ is the input signal power and /spl sigma//sub k//sup 2/ is the conditional error power. Thus, the optimum stochastic gradient error criterion for Gaussian noise is not mean-square. It is shown that the equations governing the convergence of the nonlinear algorithm are exactly those which describe the behavior of the optimum scalar data nonlinear adaptive algorithm for white Gaussian input. Simulations verify the results for a host of noise interferences and indicate the improvement using non-mean-square error criteria. >

Multilevel polarization shift keying: optimum receiver structure and performance evaluation

Abstract: Multilevel digital coherent optical modulation schemes based on the state of polarization of a fully polarized lightwave are proposed and analyzed. Based on the complete statistical characterization of the Stokes parameters, extracted through appropriate signal processing in the presence of shot and additive Gaussian noise, the optimum maximum likelihood receiver operating symbol by symbol is derived. The exact performance in terms of the average symbol error probability is found. Optimum constellations for the case of equipower 4, 8, 16 and 32 signals are found on the basis of the minimization of the error probability for a given average power. Their performance turns out to be promising as compared to other standard modulation techniques. The spectral analysis of polarization modulated signals is presented. A new receiver structure, which solves the problem of the excess penalties incurred in the presence of channel dichroism, is proposed and analyzed. >

Temperature effects in a nonlinear model of monolayer Scheibe aggregates

Abstract: A nonlinear dynamical model of molecular monolayers arranged in Scheibe aggregates is derived from a proper Hamiltonian. Thermal fluctuations of the phonons are included. The resulting equation for the excitons is the two dimensional nonlinear Schr\"odinger equation with noise. Two limits of the complicated spectrum of the noise are considered: time independent, spatially white noise, simply corresponding to disorder in the arrangement of the molecules, and pure white noise. Parameter values are found by comparison with experiments by M\"obius and Kuhn [Isr. J. Chem. 18, 375 (1979)] and order of magnitude estimates given where experiments are not available. The temperature dependent coherence time is found from numerical simulations. Experiments show that the excitons stay coherent during their lifetime. This is in correspondence with the model at temperatures lower than 3 K. To increase this limiting temperature it is found that the dipole-dipole coupling and the exciton-phonon coupling must be decreased significantly.

A zero crossing algorithm for the estimation of the frequency of a single sinusoid in white noise

Abstract: Presents a new algorithm for the estimation of the frequency of a single sinusoid in white noise, based on the computation of the interval between zero crossings. It is shown that for a high signal-to-noise ratio, the output error spectrum is concentrated in the high-frequency region. Theoretically, the desired precision can be achieved using a proper low-pass filter. The spectrum of the error due to the interpolation of the zero crossing is computed. Simulation results show that a precision of 10/sup -7/-10/sup -8/ can be obtained with modest computational effort. >

White noise calculus and Fock space

Abstract: Prerequisites.- White noise space.- White noise functionals.- Operator theory.- Toward harmonic analysis.- Addendum.

Techniques of bounding the probability of decoding error for block coded modulation structures

Abstract: Two techniques for upper bounding the average probability of decoding error in coded modulation structures are presented. The first bound, which is applicable to the additive white Gaussian noise (AWGN) channel, is tighter than the well-known union bound and the minimum distance bound, especially for low signal to noise ratio. It is shown that for the Leech lattice this upper bound is very close to a sphere lower bound. For the second upper bound, which is applicable to any memoryless channel (not necessarily AWGN), a method of random coset coding is presented. For the AWGN channel, a tighter upper bound is obtained by employing the method of random coset coding for calculating the average spectrum of distances of the code, which is required for the computation of the first upper bound. >

Asymptotic minimax risk for sup-norm loss: Solution via optimal recovery

Abstract: We study the problem of estimating an unknown function on the unit interval (or itsk-th derivative), with supremum norm loss, when the function is observed in Gaussian white noise and the unknown function is known only to obey Lipschitz-β smoothness, β>k≧0. We discuss an optimization problem associated with the theory ofoptimal recovery. Although optimal recovery is concerned with deterministic noise chosen by a clever opponent, the solution of this problem furnishes the kernel of the minimax linear estimate for Gaussian white noise. Moreover, this minimax linear estimator is asymptotically minimax among all estimates. We sketch also applications to higher dimensions and to indirect measurement (e.g. deconvolution) problems.

Cumulant-based blind optimum beamforming

Abstract: Sensor response, location uncertainty, and use of sample statistics can severely degrade the performance of optimum beamformers. We propose blind estimation of the source steering vector in the presence of multiple, directional, correlated or coherent Gaussian interferers via higher order statistics. In this way, we employ the statistical characteristics of the desired signal to make the necessary discrimination, without any a-priori knowledge of array manifold and direction-of-arrival (DOA) information about the desired signal. We then improve our method to utilize the data in a more efficient manner. In any application, only sample statistics are available, so we propose a robust beamforming approach that employs the steering vector estimate obtained by cumulant-based signal processing. We further propose a method that employs both covariance and cumulant information to combat finite sample effects. We analyze the effects of multipath propagation on the reception of the desired signal. We show that even in the presence of coherence, cumulant-based beamformer still behaves as the optimum beamformer that maximizes the signal-to-interference-plus-noise ratio (SINR). Finally, we propose an adaptive version of our algorithm simulations demonstrate the excellent performance of our approach in a wide variety of situations. >