Showing papers on "White noise published in 2009"

Ensemble empirical mode decomposition: a noise-assisted data analysis method

Abstract: A new Ensemble Empirical Mode Decomposition (EEMD) is presented. This new approach consists of sifting an ensemble of white noise-added signal (data) and treats the mean as the final true result. Finite, not infinitesimal, amplitude white noise is necessary to force the ensemble to exhaust all possible solutions in the sifting process, thus making the different scale signals to collate in the proper intrinsic mode functions (IMF) dictated by the dyadic filter banks. As EEMD is a time–space analysis method, the added white noise is averaged out with sufficient number of trials; the only persistent part that survives the averaging process is the component of the signal (original data), which is then treated as the true and more physical meaningful answer. The effect of the added white noise is to provide a uniform reference frame in the time–frequency space; therefore, the added noise collates the portion of the signal of comparable scale in one IMF. With this ensemble mean, one can separate scales naturall...

Stochastic Methods: A Handbook for the Natural and Social Sciences

Abstract: A Historical Introduction.- Probability Concepts.- Markov Processes.- The Ito Calculus and Stochastic Differential Equations.- The Fokker-Planck Equation.- The Fokker-Planck Equation in Several Dimensions.- Small Noise Approximations for Diffusion Processes.- The White Noise Limit.- Beyond the White Noise Limit.- Levy Processes and Financial Applications.- Master Equations and Jump Processes.- The Poisson Representation.- Spatially Distributed Systems.- Bistability, Metastability, and Escape Problems.- Simulation of Stochastic Differential Equations.

Spectrum-Sensing Algorithms for Cognitive Radio Based on Statistical Covariances

Abstract: Spectrum sensing, i.e., detecting the presence of primary users in a licensed spectrum, is a fundamental problem in cognitive radio. Since the statistical covariances of the received signal and noise are usually different, they can be used to differentiate the case where the primary user's signal is present from the case where there is only noise. In this paper, spectrum-sensing algorithms are proposed based on the sample covariance matrix calculated from a limited number of received signal samples. Two test statistics are then extracted from the sample covariance matrix. A decision on the signal presence is made by comparing the two test statistics. Theoretical analysis for the proposed algorithms is given. Detection probability and the associated threshold are found based on the statistical theory. The methods do not need any information about the signal, channel, and noise power a priori. In addition, no synchronization is needed. Simulations based on narrow-band signals, captured digital television (DTV) signals, and multiple antenna signals are presented to verify the methods.

Piezoelectric energy harvesting from broadband random vibrations

Abstract: Energy harvesting for the purpose of powering low power electronic sensor systems has received explosive attention in the last few years. Most works using deterministic approaches focusing on using the piezoelectric effect to harvest ambient vibration energy have concentrated on cantilever beams at resonance using harmonic excitation. Here, using a stochastic approach, we focus on using a stack configuration and harvesting broadband vibration energy, a more practically available ambient source. It is assumed that the ambient base excitation is stationary Gaussian white noise, which has a constant power-spectral density across the frequency range considered. The mean power acquired from a piezoelectric vibration-based energy harvester subjected to random base excitation is derived using the theory of random vibrations. Two cases, namely the harvesting circuit with and without an inductor, have been considered. Exact closed-form expressions involving non-dimensional parameters of the electromechanical system have been given and illustrated using numerical examples.

Multichannel Eigenspace Beamforming in a Reverberant Noisy Environment With Multiple Interfering Speech Signals

Abstract: In many practical environments we wish to extract several desired speech signals, which are contaminated by nonstationary and stationary interfering signals. The desired signals may also be subject to distortion imposed by the acoustic room impulse responses (RIRs). In this paper, a linearly constrained minimum variance (LCMV) beamformer is designed for extracting the desired signals from multimicrophone measurements. The beamformer satisfies two sets of linear constraints. One set is dedicated to maintaining the desired signals, while the other set is chosen to mitigate both the stationary and nonstationary interferences. Unlike classical beamformers, which approximate the RIRs as delay-only filters, we take into account the entire RIR [or its respective acoustic transfer function (ATF)]. The LCMV beamformer is then reformulated in a generalized sidelobe canceler (GSC) structure, consisting of a fixed beamformer (FBF), blocking matrix (BM), and adaptive noise canceler (ANC). It is shown that for spatially white noise field, the beamformer reduces to a FBF, satisfying the constraint sets, without power minimization. It is shown that the application of the adaptive ANC contributes to interference reduction, but only when the constraint sets are not completely satisfied. We show that relative transfer functions (RTFs), which relate the desired speech sources and the microphones, and a basis for the interference subspace suffice for constructing the beamformer. The RTFs are estimated by applying the generalized eigenvalue decomposition (GEVD) procedure to the power spectral density (PSD) matrices of the received signals and the stationary noise. A basis for the interference subspace is estimated by collecting eigenvectors, calculated in segments where nonstationary interfering sources are active and the desired sources are inactive. The rank of the basis is then reduced by the application of the orthogonal triangular decomposition (QRD). This procedure relaxes the common requirement for nonoverlapping activity periods of the interference sources. A comprehensive experimental study in both simulated and real environments demonstrates the performance of the proposed beamformer.

Quantitative Error Assessment in Pattern Matching: Effects of Intensity Pattern Noise, Interpolation, Strain and Image Contrast on Motion Measurements

Abstract: Basic concepts in probability are employed to develop analytic formulae for both the expectation (bias) and variance for image motions obtained during subset-based pattern matching. Specifically, the expectation and variance in image motions in the presence of uncorrelated Gaussian intensity noise for each pixel location are obtained by optimising a least squares intensity matching metric. Results for both 1D and 2D image analyses clearly quantify both the bias and the covariance matrix for image motion estimates as a function of: (a) interpolation method, (b) sub-pixel motion, (c) intensity noise, (d) contrast, (e) level of uniaxial normal strain and (f) subset size. For 1D translations, excellent agreement is demonstrated between simulations, theoretical predictions and experimental measurements. The level of agreement confirms that the analytical formulae can be used to provide a priori estimates for the ‘quality’ of local, subset-based measurements achievable with a given pattern. For 1D strain with linear interpolation, theoretical predictions are provided for the expectation and co-variance matrix for the local displacement and strain parameters. For 2D translations with bi-linear interpolation, theoretical predictions are provided for both the expectation and the co-variance matrix for both displacement components. Theoretical results in both cases show that the expectations for the local parameters are biased and a function of: (a) the interpolation difference between the translated and reference images, (b) magnitude of white noise, (c) decimal part of the motion and (d) intensity pattern gradients. For 1D strain, the biases and the covariance matrix for both parameters are directly affected by the strain parameter p1 as the deformed image is stretched by (1 + p1). For 2D rigid body motion case, the covariance matrix for measured motions is shown to have coupling between the motions, demonstrating that the directions of maximum and minimum variability do not generally coincide with the x and y directions.

On detecting the large separation in the autocorrelation of stellar oscillation times series

Abstract: Context. The observations carried out by the space missions CoRoT and Kepler provide a large set of asteroseismic data. Their analysis requires an efficient procedure first to determine if a star reliably shows solar-like oscillations, second to measure the so-called large separation, third to estimate the asteroseismic information that can be retrieved from the Fourier spectrum. Aims. In this paper we develop a procedure based on the autocorrelation of the seismic Fourier spectrum that is capable of providing measurements of the large and small frequency separations. The performance of the autocorrelation method needs to be assessed and quantified. We therefore searched for criteria able to predict the output that one can expect from the analysis by autocorrelation of a seismic time series. Methods. First, the autocorrelation is properly scaled to take into account the contribution of white noise. Then we use the null hypothesis H0 test to assess the reliability of the autocorrelation analysis. Calculations based on solar and CoRoT time series are performed to quantify the performance as a function of the amplitude of the autocorrelation signal. Results. We obtain an empirical relation for the performance of the autocorrelation method. We show that the precision of the method increases with the observation length, and with the mean seismic amplitude-to-background ratio of the pressure modes to the power 1.5 ± 0.05. We propose an automated determination of the large separation, whose reliability is quantified by the H0 test. We apply this method to analyze red giants observed by CoRoT. We estimate the expected performance for photometric time series of the Kepler mission. We demonstrate that the method makes it possible to distinguish � = 0 from � = 1 modes. Conclusions. The envelope autocorrelation function (EACF) has proven to be very powerful for the determination of the large separation in noisy asteroseismic data, since it enables us to quantify the precision of the performance of different measurements: mean large separation, variation of the large separation with frequency, small separation and degree identification.

Scale invariance and noise in natural images

Abstract: Natural images are known to have scale invariant statistics. While some eariler studies have reported the kurtosis of marginal bandpass filter response distributions to be constant throughout scales, other studies have reported that the kurtosis values are lower for high frequency filters than for lower frequency ones. In this work we propose a resolution for this discrepancy and suggest that this change in kurtosis values is due to noise present in the image. We suggest that this effect is consistent with a clean, natural image corrupted by white noise. We propose a model for this effect, and use it to estimate noise standard deviation in corrupted natural images. In particular, our results suggest that classical benchmark images used in low-level vision are actually noisy and can be cleaned up. Our results on noise estimation on two sets of 50 and a 100 natural images are significantly better than the state-of-the-art.

A Foundation in Digital Communication

Abstract: This intuitive yet rigorous introduction derives the core results of digital communication from first principles. Theory, rather than industry standards, motivates the engineering approaches, and key results are stated with all the required assumptions. The book emphasizes the geometric view, opening with the inner product, the matched filter for its computation, Parseval's theorem, the sampling theorem as an orthonormal expansion, the isometry between passband signals and their baseband representation, and the spectral-efficiency optimality of quadrature amplitude modulation (QAM). Subsequent chapters address noise, hypothesis testing, Gaussian stochastic processes, and the sufficiency of the matched filter outputs. Uniquely, there is a treatment of white noise without generalized functions, and of the power spectral density without artificial random jitters and random phases in the analysis of QAM. This systematic and insightful book, with over 300 exercises, is ideal for graduate courses in digital communication, and for anyone asking 'why' and not just 'how'.

Design of robust superdirective beamformers as a convex optimization problem

Abstract: Broadband data-independent beamforming designs aiming at constant beamwidth often lead to superdirective beamformers for low frequencies, if the sensor spacing is small relative to the wavelengths. Superdirective beamformers are extremely sensitive to spatially white noise and to small errors in the array characteristics. These errors are nearly uncorrelated from sensor to sensor and affect the beamformer in a manner similar to spatially white noise. Hence the White Noise Gain (WNG) is a commonly used measure for the robustness of beamformer designs. In this paper, we present a method which incorporates a constraint for the WNG into a least-squares beamformer design and still leads to a convex optimization problem that can be solved directly, e.g. by Sequential Quadratic Programming. The effectiveness of this method is demonstrated by design examples.

A SURE Approach for Digital Signal/Image Deconvolution Problems

Abstract: In this paper, we are interested in the classical problem of restoring data degraded by a convolution and the addition of a white Gaussian noise. The originality of the proposed approach is twofold. First, we formulate the restoration problem as a nonlinear estimation problem leading to the minimization of a criterion derived from Stein's unbiased quadratic risk estimate. Secondly, the deconvolution procedure is performed using any analysis and synthesis frames that can be overcomplete or not. New theoretical results concerning the calculation of the variance of the Stein's risk estimate are also provided in this work. Simulations carried out on natural images show the good performance of our method with respect to conventional wavelet-based restoration methods.

The Number of Magnetic Null Points in the Quiet Sun Corona

Abstract: The coronal magnetic field above a particular photospheric region will vanish at a certain number of points, called null points These points can be found directly in a potential field extrapolation or their density can be estimated from the Fourier spectrum of the magnetogram The spectral estimate, in which the extrapolated field is assumed to be random and homogeneous with Gaussian statistics, is found here to be relatively accurate for quiet Sun magnetograms from SOHO’s MDI The majority of null points occur at low altitudes, and their distribution is dictated by high wavenumbers in the Fourier spectrum This portion of the spectrum is affected by Poisson noise, and as many as five-sixths of null points identified from a direct extrapolation can be attributed to noise The null distribution above 1500 km is found to depend on wavelengths that are reliably measured by MDI in either its low-resolution or high-resolution mode After correcting the spectrum to remove white noise and compensate for the modulation transfer function we find that a potential field extrapolation contains, on average, one magnetic null point, with altitude greater than 15 Mm, above every 322 Mm2 patch of quiet Sun Analysis of 562 quiet Sun magnetograms spanning the two latest solar minima shows that the null point density is relatively constant with roughly 10% day-to-day variation At heights above 15 Mm, the null point density decreases approximately as the inverse cube of height The photospheric field in the quiet Sun is well approximated as that from discrete elements with mean flux 〈|φ|〉=10×1019 Mx distributed randomly with density n=0007 Mm−2

Mapping stochastic processes onto complex networks

Abstract: We introduce a method by which stochastic processes are mapped onto complex networks. As examples, we construct the networks for such time series as those for free-jet and low-temperature helium turbulence, the German stock market index (the DAX), and white noise. The networks are further studied by contrasting their geometrical properties, such as the mean length, diameter, clustering, and average number of connections per node. By comparing the network properties of the original time series investigated with those for the shuffled and surrogate series, we are able to quantify the effect of the long-range correlations and the fatness of the probability distribution functions of the series on the networks constructed. Most importantly, we demonstrate that the time series can be reconstructed with high precision by means of a simple random walk on their corresponding networks.

Modeling Multiscale Subbands of Photographic Images with Fields of Gaussian Scale Mixtures

Abstract: The local statistical properties of photographic images, when represented in a multi-scale basis, have been described using Gaussian scale mixtures. Here, we use this local description as a substrate for constructing a global field of Gaussian scale mixtures (FoGSM). Specifically, we model multi-scale subbands as a product of an exponentiated homogeneous Gaussian Markov random field (hGMRF) and a second independent hGMRF. We show that parameter estimation for this model is feasible, and that samples drawn from a FoGSM model have marginal and joint statistics similar to subband coefficients of photographic images. We develop an algorithm for removing additive Gaussian white noise based on the FoGSM model, and demonstrate denoising performance comparable with state-of-the-art methods.

A Class of Sparseness-Controlled Algorithms for Echo Cancellation

Abstract: In the context of acoustic echo cancellation (AEC), it is shown that the level of sparseness in acoustic impulse responses can vary greatly in a mobile environment When the response is strongly sparse, convergence of conventional approaches is poor Drawing on techniques originally developed for network echo cancellation (NEC), we propose a class of AEC algorithms that can not only work well in both sparse and dispersive circumstances, but also adapt dynamically to the level of sparseness using a new sparseness-controlled approach Simulation results, using white Gaussian noise (WGN) and speech input signals, show improved performance over existing methods The proposed algorithms achieve these improvement with only a modest increase in computational complexity

A theory of 1/f noise in human cognition

Abstract: The brain is probably the most interesting example of a complex network having 1 / f variability as determined through the analysis of EEG time series and magnetoencephalogram recordings Herein we develop a theory of 1 / f noise of human cognition to explain the recent experimental observations that increasing the difficultly of cognitive tasks accelerates the transition from observed 1 / f noise to white noise in decision-making time series

Generalized Fokker-Planck equation: Derivation and exact solutions

Abstract: We derive the generalized Fokker-Planck equation associated with the Langevin equation (in the Ito sense) for an overdamped particle in an external potential driven by multiplicative noise with an arbitrary distribution of the increments of the noise generating process. We explicitly consider this equation for various specific types of noises, including Poisson white noise and Levy stable noise, and show that it reproduces all Fokker-Planck equations that are known for these noises. Exact analytical, time-dependent and stationary solutions of the generalized Fokker-Planck equation are derived and analyzed in detail for the cases of a linear, a quadratic, and a tailored potential.

Unbiased FIR Filtering of Discrete-Time Polynomial State-Space Models

Abstract: We address an unbiased finite impulse response (FIR) filter for discrete-time state-space models with polynomial representation of the states. The unique l-degree polynomial FIR filter gain and the estimate variance are found for a general case. The noise power gain (NG) is derived for white Gaussian noises in the model and in the measurement. The filter does not involve any knowledge about noise in the algorithm. It is unstable at short horizons, 2 les N les l, and inefficient (NG exceeds unity) in the narrow range l 1, the estimate noise becomes negligible and the filter thus optimal in the sense of zero bias and zero noise. Having such properties, the proposed unbiased FIR filter fits well slowly changing with time models. An example is given for a two-state system.

Empirical mode decomposition: applications on signal and image processing

Abstract: In this paper, we propose some recent works on data analysis and synthesis based on Empirical Mode Decomposition (EMD). Firstly, a direct 2D extension of original Huang EMD algorithm with application to texture analysis, and fractional Brownian motion synthesis. Secondly, an analytical version of EMD based on PDE in 1D-space is presented. We proposed an extension in 2D-case of the so-called "sifting process" used in the original Huang's EMD. The 2D-sifting process is performed in two steps: extrema detection (by neighboring window or morphological operators) and surface interpolation by splines (thin plate splines or multigrid B-splines). We propose a multiscale segmentation approach by using the zero-crossings from each 2D-intrinsic mode function (IMF) obtained by 2D-EMD. We apply the Hilbert–Huang transform (which consists of two parts: (a) Empirical mode decomposition, and (b) the Hilbert spectral analysis) to texture analysis. We analyze each 2D-IMF obtained by 2D-EMD by studying local properties (amplitude, phase, isotropy, and orientation) extracted from the monogenic signal of each one of them. The monogenic signal proposed by Felsberg et al. is a 2D-generalization of the analytic signal, where the Riesz transform replaces the Hilbert transform. These local properties are obtained by the structure multivector such as proposed by Felsberg and Sommer. We present numerical simulations of fractional Brownian textures. Recent works published by Flandrin et al. relate that, in the case of fractional Gaussian noise (fGn), EMD acts essentially as a dyadic filter bank that can be compared to wavelet decompositions. Moreover, in the context of fGn identification, Flandrin et al. show that variance progression across IMFs is related to Hurst exponent H through a scaling law. Starting with these results, we proposed an algorithm to generate fGn, and fractional Brownian motion (fBm) of Hurst exponent H from IMFs obtained from EMD of a White noise, i.e., ordinary Gaussian noise (fGn with H = 1/2). Delechelle et al. proposed an analytical approach (formulated as a partial differential equation (PDE)) for sifting process. This PDE-based approach is applied on signals. The analytical approach has a behavior similar to that of the EMD proposed by Huang.

Pathwise Numerical Approximations of SPDEs with Additive Noise under Non-global Lipschitz Coefficients

Abstract: We consider the pathwise numerical approximation of nonlinear parabolic stochastic partial differential equations (SPDEs) driven by additive white noise under local assumptions on the coefficients only. We avoid the standard global Lipschitz assumption in the literature on the coefficients by first showing convergence under global Lipschitz coefficients but with a strong error criteria and then by applying a localization technique for one sample path on a bounded set.

Lévy stable noise-induced transitions: stochastic resonance, resonant activation and dynamic hysteresis

Abstract: A standard approach to analysis of noise-induced effects in stochastic dynamics assumes a Gaussian character of the noise term describing interaction of the analyzed system with its complex surroundings. An additional assumption about the existence of timescale separation between the dynamics of the measured observable and the typical timescale of the noise allows external fluctuations to be modeled as temporally uncorrelated and therefore white. However, in many natural phenomena the assumptions concerning the above mentioned properties of 'Gaussianity' and 'whiteness' of the noise can be violated. In this context, in contrast to the spatiotemporal coupling characterizing general forms of non-Markovian or semi-Markovian Levy walks, so called Levy flights correspond to the class of Markov processes which can still be interpreted as white, but distributed according to a more general, infinitely divisible, stable and non-Gaussian law. Levy noise-driven non-equilibrium systems are known to manifest interesting physical properties and have been addressed in various scenarios of physical transport exhibiting a superdiffusive behavior. Here we present a brief overview of our recent investigations aimed at understanding features of stochastic dynamics under the influence of Levy white noise perturbations. We find that the archetypal phenomena of noise-induced ordering are robust and can be detected also in systems driven by memoryless, non-Gaussian, heavy-tailed fluctuations with infinite variance.

Eddy covariance methane measurements at a Ponderosa pine plantation in California

Abstract: . Long term methane flux measurements have been mostly performed with plant or soil enclosure techniques on specific components of an ecosystem. New fast response methane analyzers make it possible to use the eddy covariance (EC) technique instead. The EC technique is advantageous because it allows continuous flux measurements integrating over a larger and more representative area including the complete ecosystem, and allows fluxes to be observed as environmental conditions change naturally without disturbance. We deployed the closed-path Fast Methane analyzer (FMA) from Los Gatos Research Ltd and demonstrate its performance for EC measurements at a Ponderosa pine plantation at the Blodgett Forest site in central California. The fluctuations of the CH4 concentration measured at 10 Hz appear to be small and their standard deviation is comparable to the magnitude of the signal noise (±5 ppbv). Consequently, the power spectra typically have a white noise signature at the high frequency end (a slope of +1). Nevertheless, in the frequency range important for turbulent exchange, the cospectra of CH4 compare very well with all other scalar cospectra confirming the quality of the FMA measurements are good for the EC technique. We furthermore evaluate the complications of combined open and closed-path measurements when applying the Webb-Pearman-Leuning (WPL) corrections (Webb et al., 1980) and the consequences of a phase lag between the water vapor and methane signal inside the closed path system. The results of diurnal variations of CH4 concentrations and fluxes are summarized and compared to the monthly results of process-based model calculations.

Invariant wideband spectrum sensing under unknown variances

Abstract: In this paper, we divide a wide frequency range into multiple subbands and in each subband detect whether in a primary user (PU) is active or not. We assume that PU signal at each subband and the additive noise are white zero-mean independent Gaussian random processes with unknown variances. We also assume that at least a minimum given number of subbands is vacant of PU signal and propose an invariant generalized likelihood ratio (GLR) detector. The concept of the grouping of subbands allows faster spectrum sensing of a subset of subbands which may be occupied by a specific PU. Also, we evaluate trade-offs involved in the proposed algorithms by simulation.