Zero-mean White Gaussian Noise Research Articles

On the representation of fractional Brownian motion as an integral with respect to [formula omitted

Maruyama introduced the notation d b ( t ) = w ( t ) ( d t ) 1 / 2 where w ( t ) is a zero-mean Gaussian white noise, in order to represent the Brownian motion b ( t ) . Here, we examine in which way this notation can be extended to Brownian motion of fractional order a (different from 1/2) defined as the Riemann–Liouville derivative of the Gaussian white noise. The rationale is mainly based upon the Taylor’s series of fractional order, and two cases have to be considered: processes with short-range dependence, that is to say with 0 ⊲ a ≤ 1 / 2 , and processes with long-range dependence, with 1 / 2 ⊲ a ≤ 1 .

Noise response of detection filters: relation between detection space and completion space

An analysis is presented of the noise response of detection filters with zero-mean gaussian white noise. The major focus is placed on the relation between eigenvalues and error covariance. By similarity transformation it is shown that assigning the eigenvalues associated with a detection space does not affect the noise response of a completion space or other detection spaces. Based on this relationship, a brief optimisation procedure is given for an optimal problem in terms of the error covariance.

Optimum beamforming performance degradation in the presence of imperfect spatial coherence of wavefronts

Optimum-adaptive beamforming is generally analyzed and used assuming perfectly known wavefronts. There are several applications involving propagation media that are neither isotropic, nor homogeneous, nor deterministic. There are also applications where the arrays may be nondeterministic and/or nonrigid or, more simply, are in motion. Typical examples of such applications are radar, sonar, underwater and satellite and, in general, wireless communication systems. In these cases, the resulting wavefronts can be randomly distorted. We analyze the performance of an optimum-adaptive beamformer when the desired signal component or the interference are not fully coherent over the array aperture. The effects of reduced spatial coherence, for both signal and interference cases, are evaluated in terms of output signal-to-interference-plus-noise ratio (SINR) degradation with respect to the ideal case where the reception of a fully coherent signal is only affected by the presence of a zero-mean additive white Gaussian noise. For this purpose, we provide a theoretical analysis, deriving analytical expressions for general models of spatial coherence loss, supported by numerical results.

PASSIVE VIBRATION CONTROL OF AIRBORNE EQUIPMENT USING A CIRCULAR STEEL RING

Vibration isolation is needed to protect avionics equipment from adverse aircraft vibration environments. Passive isolation is the simplest means to achieve this goal. The system used here consists of a circular steel ring with a lump mass on top and exposed to base excitation. Sinusoidal and filtered zero-mean Gaussian white noise are used to excite the structure and the acceleration response spectra at the top of the ring are computed. An experiment is performed to identify the natural frequencies and modal damping of the circular ring. Comparison is made between the analytical and experimental results and good agreement is observed. The ring response is also evaluated with a concentrated mass attached to the top of the ring. The effectiveness of the ring in isolating the equipment from base excitation is studied. The acceleration response spectra of a single-degree-of-freedom (s.d.o.f.) system attached to the top of the ring are evaluated and the results are compared with those exposed directly to the base excitation. It is shown that a properly designed ring could effectively protect the avionics from possible damaging excitation levels.

NOISE EFFECT ON GUIDANCE AND CONTROL LOOP OF GUIDED BOMBS

Smart or homing guided bombs are used to destroy difficult-to-hit targets with minimum number of strikes and high accuracy. There are two basic types of smart bombs, laser or TV guided bombs. The guided bomb is formed by adding a seeker and a control section to the traditional unguided bombs. The seeker searches for and tracks the target. It provides the line-of-sight rate to the guidance module that generates and provides guidance commands to the control fins. The autopilot of the guided bomb is mainly used to improve the accuracy, maintain a near constant steady-state aerodynamic gain, and help in steering the bomb to the target with a very high accuracy. Undesirable behavior characteristics of the guidance and control may originate due to several sources of disturbances such as flight vibrations and oscillations due to external environmental effects and/or the airframe coupling mechanisms flutter or the propulsion system motor chamber pressure oscillations. The latter will produce phase-coherent quasi-sinusoidal vibration while the former is random and unpredicted and can only be evaluated using statistical methods of analysis. In this paper, the external noise presented by the atmospheric turbulence during the bomb flight is investigated through computer simulation. The noise process is modeled and generated to enable evaluation of the system performance. The zero-mean white Gaussian noise is generated and provided as a superimposed input with the acceleration input command of the bomb autopilot. The system performance is measured for different signal-to-noise ratios (SNR) of the input noise. The performance evaluation is made via a six-degrees-of-freedom (6-D0F) flight model of the bomb. The measured parameters are the trajectory, the final missdistance, and the autopilot acceleration response.

Direct implementation of stochastic linearization for SDOF systems with general hysteresis

The first and second moments of response variables for SDOF systems with hysteretic nonlinearity are obtained by a direct linearization procedure. This adaptation in the implementation of well-known statistical linearization methods, provides concise, model-independent linearization coefficients that are well-suited for numerical solution. The method may be applied to systems which incorporate any hysteresis model governed by a differential constitutive equation, and may be used for zero or non-zero mean random vibration. The implementation eliminates the effort of analytically deriving specific linearization coefficients for new hysteresis models. In doing so, the procedure of stochastic analysis is made independent from the task of physical modeling of hysteretic systems. In this study, systems with three different hysteresis models are analyzed under various zero and non-zero mean Gaussian White noise inputs. Results are shown to be in agreement with previous linearization studies and Monte Carlo Simulation.

Asymmetric motion in a double well under the action of zero-mean Gaussian white noise and periodic forcing

Residence times of a particle in both the wells of a double-well system, under the action of zero-mean Gaussian white noise and zero-averaged but temporally asymmetric periodic forcings, are recorded in a numerical simulation. The difference between the relative mean residence times in the two wells shows monotonic variation as a function of asymmetry in the periodic forcing and for a given asymmetry the difference becomes largest at an optimum value of the noise strength. Moreover, the passages from one well to the other become less synchronous at small noise strength as the asymmetry parameter (defined below) differs from zero, but at relatively larger noise strengths the passages become more synchronous with asymmetry in the field sweep. We propose that asymmetric periodic forcing (with zero mean) could provide a simple but sensible physical model for unidirectional motion in a symmetric periodic system aided by a symmetric Gaussian white noise.

White noise approach for estimating the passive electrical properties of neurons.

1. The passive electrical properties of whole cell patched dentate granule cells were studied with the use of zero-mean Gaussian white noise current stimuli. Transmembrane voltage responses were used to compute the first-order Wiener kernels describing the current-voltage relationship at the soma for six cells. Frequency domain optimization techniques using a gradient method for function minimization were then employed to identify the optimal electrical parameter values. Low-power white noise stimuli are presented as a favorable alternative to the use of short-pulse current inputs for investigating neuronal passive electrical properties. 2. The optimization results demonstrated that the lumped resistive and capacitive properties of the recording electrode must be included in the analytic input impedance expression to optimally fit the measured cellular responses. The addition of the electrode resistance (Re) and capacitance (Ce) to the original parameters (somatic conductance, somatic capacitance, axial resistance, dendritic conductance, and dendritic capacitance) results in a seven-parameter model. The mean Ce value from the six cells was 5.4 +/- 0.3 (SE) pF, whereas Re following formation of the patch was found to be 20 +/- 2 M omega. 3. The six dentate granule cells were found to have an input resistance of 600 +/- 20 M omega and a dendritic to somatic conductance ratio of 6.3 +/- 1.1. The electronic length of the equivalent dendritic cylinder was found to be 0.42 +/- 0.03. The membrane time constant in the soma was found to be 13 +/- 3 ms, whereas the membrane time constant of the dendrites was 58 +/- 5 ms. Incorporation of morphological estimations led to the following distributed electrical parameters: somatic membrane resistance = 25 +/- 4 k omega cm2, somatic membrane capacitance = 0.48 +/- 0.05 microF/cm2, Ri (input resistance) = 72 +/- 5 omega cm, dendritic membrane resistance = 59 +/- 4 k omega cm2, and dendritic membrane capacitance = 0.97 +/- 0.06 microF/cm2. On the basis of capacitive measurements, the ratio of dendritic surface area to somatic surface area was found to be 34 +/- 2. 4. For comparative purposes, hyperpolarizing short pulses were also injected into each cell. The short-pulse input impedance measurements were found to underestimate the input resistance of the cell and to overestimate both the somatic conductance and the membrane time constants relative to the white noise input impedance measurements.

Stochastic response analysis of nonlinear systems under Gaussian inputs

In this paper the extension of Itô's rule for the case of vector real valued functions of the response of nonlinear systems excited by zero-mean Gaussian white noise processes is presented. A suitable particularization of the vector function, in order to obtain the statistical moments of every order to the response, is treated, obtaining the differential equations of the response moments in an elegant and compact form. Polynomial expansion and closure schemes are framed in the context outlined here in order to obtain an effective procedure from a computational point of view. An application to a trigonometric nonlinear system, solved in the literature by the stochastic averaging method, is treated here by the moment equation approach using the polynomial expansion of the nonlinear terms in order to evidence the validity of this approach.

Response moments of dynamic systems with modal interactions

Statistical response moments of two-degree-of-freedom systems with quadratic and cubic non-linearities to zero-mean Gaussian white noise are determined. The cases of two-to-one, one-to-one and three-to-one autoparametric resonances are investigated. Ito stochastic calculus is used to derive general first order ordinary differential equations describing the evolution of the response moments. The response-moment equations are an infinite hierarchy set due to the non-linearities. Gaussian and non-Gaussian closure schemes are used to close this set, leading, respectively, to 14 and 69 equations for the lower order response moments. Guided by the narrowband nature of the response of lightly damped systems, the number of response-moment equations is reduced for the case of non-Gaussian closure by a factor of approximately two. The results obtained by using the reduced scheme are in good agreement with those obtained by using the full 69 equations. The results reveal major discrepancies between the Gaussian and non-Gaussian closure schemes. Close to autoparametric resonance, stationary bimodal solutions exist, and energy shifts between the two modes. However, the response does not display the saturation phenomenon for the case of a two-to-one autoparametric resonance. The results are applied to a shallow arch, a string and a hinged-clamped beam.

Maximum likelihood estimation of object location in diffraction tomography

The problem is formulated within the context of diffraction tomography, where the complex phase of the diffracted wavefield is modeled using the Rytov approximation and the measurements consist of noisy renditions of this complex phase at a single frequency. The log likelihood function is computed for the case of additive zero mean Gaussian white noise and shown to be expressible in the form of the filtered backpropagation algorithm of diffraction tomography. In this form however, the filter function is no longer the rho filter appropriate to least square reconstruction but is now the generalized projection (propagation) of the object (centered at the origin) onto the line(s) parallel to the measurement line(s), but passing through the origin. This result allows the estimation problem to be solved via a diffraction tomographic imaging procedure where the noisy data is filtered and backpropagated in a first step, and the point of maximum value of the resulting image is then the maximum likelihood (ML) estimate of the object's location. The authors include a calculation of the Cramer-Rao bound for the estimation error and a computer simulation study illustrating the estimation procedure. >

Performance of quasi-optimum digital FM demodulators for fading channels

This paper deals with the problem of digital demodulation of FM signals transmitted over Rayleigh and Rician fading channels. The Rayleigh and Rician fading channels are represented by two quadrature multiplicative non-zero mean white Gaussian processes in addition to an additive zero-mean white Gaussian noise. Quasi-optimum digital baseband demodulation algorithms using various nonlinear estimation techniques are derived. The digital demodulator structures are then simulated on a digital computer for an FM system with first order message spectrum for various values of the parameters for Rayleigh and Rician channels.

A note on some efficient estimates of the noise variance for first-order Reed-Muller codes (Corresp.)

The maximum-likelihood estimate based on order statistics for both a single-sample and an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> -sample estimate are derived for the first-order Reed-Muller code sent over a zero-mean white Gaussian noise channel. The estimation method is subject to an equipment-complexity constraint. The constraint imposed requires that the estimates be obtained from successive comparisons of the correlations in a serial system, without the use of arithmetic operations. The maximum-likelihood estimates obtained here are compared to that of Posner [1], Posner et aL [2], and also the unrestricted maximum-likelihood estimate.

Optimal state-vector estimation for non-Gaussian initial state-vector

The optimal estimate, in the mean-square-error sense, of state-vector of a linear system excited by zero-mean white Gaussian noise with non-Gaussian initial state-vector is obtained. Both the optimal estimate and the corresponding error covariance matrix are given. It is shown that the optimal estimator consists of two parts: a linear estimator that is obtained from a Kalman filter and a nonlinear estimator. In addition, the a posteriori probability p(x_{k}/\lambda_{k}) is also given.

Automatic frequency control via digital filtering

This paper presents a unique digital realization of an automatic frequency control (AFC) for tracking signals that are slowly drifting in frequency. The frequency discriminator consists of two stagger-tuned digital filters, whose squared outputs are differenced and averaged. The frequency tracking is performed by changing the centers of the digital filters and hence the discriminator characteristic. This can be done easily by only a few cosine calculations. The digital loop filter, operating at a much slower sampling rate than the discriminator filters, is designed to minimize probability of loss of lock. Zero mean white additive Gaussian noise is inserted in the linearized loop to account for both the additive channel noise and the short-term signal instabilities. The variance of the frequency estimate is then calculated. The only assumption made on the frequency drift process is that the drift rate be less than a given value. A worst-case analysis is performed to obtain the largest possible mean frequency error. Loss of lock is said to occur when the actual frequency error is greater than the width of the linear portion of the discriminator characteristic.