Gaussian noise sequence

It can be shown [4] that the innovations of a correct filter model applied on data with Gaussian noise follows a Gaussian distribution with a mean value equal to zero and a standard deviation equal to the experimental error. A model error means that the design vector h in the measurement equation is not adequate. If, for instance, in the calibration example the model was quadratic, should be [1 c(j) c(j) ] instead of [1 c(j)]. In the MCA example h (/) is wrong if the absorptivities of some absorbing species are not included. Any error in the design vector appears by a non-zero mean for the innovation [4]. One also expects the sequence of the innovation to be random and uncorrelated. This can be checked by an investigation of the autocorrelation function (see Section 20.3) of the innovation. 

In Eq. (13), the vector q denotes a set of mass-weighted coordinates in a configuration space of arbitrary dimension N, U(q) is the potential of mean force governing the reaction, T is a symmetric positive-definite friction matrix, and , (/) is a stochastic force that is assumed to represent white noise that is Gaussian distributed with zero mean. The subscript a in Eq. (13) is used to label a particular noise sequence For any given a, there are infinitely many... 

One powerful technique is Maximum Likelihood Estimation (MLE) which requires the derivation of the Joint Conditional Probability Density Function (PDF) of the output sequence [ ], conditional on the model parameters. The input e n to the system shown in figure 4.25 is assumed to be a white Gaussian noise (WGN) process with zero mean and a variance of 02. The probability density of the noise input is ... 

The performance of each information hiding scheme was tested under three levels of noise power (variance). Random noise sequences were generated from i.i.d. samples of a Gaussian... 

Figure 2.10 A sequence of independent Gaussian random variables with zero mean and unit variance (Gaussian noise) observed at every fourth time step, i.e. at intervals of length 4t. (a) Independent random steps of the particle , (b) Position of the particle along the x axis. The time is in units of the atomistic time r between steps. Modified from [10].

The simple cross-correlation estimator is used extensively in the form of a matched filter implementation to detect a finite number of known signals (in other words, simultaneous acquisition of multiple chaimels of known signals). When these deterministic signals are embedded in white Gaussian noise, the matched filter (obtained from cross-correlation estimate at zero lag, k = 0, between the known signal sequence and the observed noisy signal sequence) gives the optimum detection performance (in the Bayes sense ). 

In this section, the a posteriori algorithms for solving the problems of detection and separation of waveforms presented by a quasi-periodic sequence and distorted by Gaussian noise are justified. We consider two variants of waveforms in a quasi-periodic sequence, both identical and different. To solve the problem, the following model of data for analysis is proposed. Let the vector components X = xq,. ..,xjv i) e form the sequence... 

Remark. The white noise limit is not sufficiently defined by just saying rc 0. We have to construct a sequence of processes which in this limit reduce to Gaussian white noise. For that purpose take a long time interval (0, T) and a Poisson distribution of time points Ta in it with density v. To each Ta attach a random number ca they are independent and identically distributed, with zero mean. Consider the process... 

We give two examples of fractal time series. The first is fractal Gaussian intermittent noise characterized by a long-time correlated waiting-time sequence, and the second is a Levy-walk intermittent noise. These examples were developed in an environmental context to explain the observed distribution of earthquakes in California [66]. 

The Gaussian intermittent noise has an autocorrelation function given by Eq. (20) and power-law index given by 277 2. The statistics of this sequence is... 

Corollary 2.5 Define an IHS in the following way The covertext sequence is a realization ofn i.i.d. r.v. distributedQ). The mean distortion constraint is based on the quadratic distortion measure d x,y) = x — s). The attack channel is memoryless and stationary, with output Y = X Y Z, where Z A/ (0, N) is independent of S and of X. Then, the capacity of this IHS is equal to the capacity of the additive Gaussian channel with signal to noise ratio of P (without distortion constraint), i.e. log2(l + ). 

Here 0 is a p x 1 vector of unknown system parameters, Xk and / are the r x 1 state and externally applied force vectors in the time discretized form, (ftk is a nonlinear state transition vector, and Wk is the x 1 process noise which represents the error in arriving at mathematical model for the vibrating system, modeled as a sequence of zero-mean Gaussian random variables with known covariance, i.e.,... 

Here is an s x 1 vector of measurements from the s sensors Vk is the /iv x 1 measurement noise term, modeled as a sequence of zero-mean Gaussian random variables with (vkV = where is riy x riy covariance matrix and Hk is a nmilinear function that relates the measurements to system states through a pertinent mathematical model. The measurement noise arises inherently... 

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