Noise Gaussian random

Additive Gaussian Noise Charmed.17 An example of the use of these bounds will now be helpful. Consider a channel for which tire input is an arbitrary real number and the output is the sum of the input and an independent gaussian random variable of variance a3. Thus,... 

The vector nk describes the unknown additive measurement noise, which is assumed in accordance with Kalman filter theory to be a Gaussian random variable with zero mean and covariance matrix R. Instead of the additive noise term nj( in equation (20), the errors of the different measurement values are assumed to be statistically independent and identically Gaussian distributed, so... 

Since Eq. (49) takes into account only the term of order Dt, the term of order in Eq. (51) is meaningless and the term linear in t in vanishes exactly. For T = 0, our result equals the well-known Smoluchowski rate. The main conclusion we can draw is that the activation rates for non-Markovian processes like Eq. (44) decrease as t increases the exact result of ref. 44 can thus be extended to the case of Gaussian random forces of finite correlation time as well. However, if we take Eq. (50) seriously, we obtain an Arrhenius factor, exp(A /Z)), of T(x) which does not exhibit a dependence on T. This is in contrast to the result found for telegr hic noises, where the Arrhenius factor increases with increasing autocorrelation time r (see ref. 44). The result of a numerical simulation for J(x) based on the bi-... 

The fluctuation-regression hypothesis, rephrased in modern language may now be stated as follows. To describe the dynamical fluctuations just mentioned, it is sufficient to use Onsager s purely macroscopic eqs. (A.38) and (A.40) modified to account for microscopic effects solely by the inclusion of random forces of the standard Brownian motion type namely, zero mean white noise Gaussian forces that obey fluctuation-dissipation relations that ensure recovery of Eq. (A.45) as r -> oc [2]. 

To account for the measurement noise, consider four Gaussian random variables yi, >>2, ys and y4 with zero mean and the covariance matrix ... 

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].

Since R(t) is Gaussian white noise, Bj (At) is a Gaussian random variable with simple statistics. Specifically [3.47]... 

Schematic diagrams of two receiver front ends that can be used to compute the coordinates of the data vector are shown in Fig. 12.60. The first is called the conelatorimplementation, and the second is called the matched filter implementation. Because the noise components are linear transformations of a Gaussian random process, they are also Gaussian, and can be shown to have zero means and covariances...

Another example of a model based system is the one proposed by Ephraim and Malah (1984) in which the spectral components of speech and noise in a given vector are modeled as statistically independent Gaussian random variables. The variance of a given spectral component of the speech signal varies from frame to frame and is estimated from the noisy signal. Under these assumptions, the minimum MSE estimate of the short-time spectral amplitude is derived. This approach outperforms spectral subtraction as it provides enhanced signals with similar clarity but with drastically reduced musical noise. 

Gaussian noise Noise in which the distribution of amplitude follows a Gaussian model that is, the noise is random but distributed about a reference voltage of zero. 

Equation (3-1) governs a dynamic system under parametric excitations In the spring term as well as the damping term. In addition to the external excitation on the right hand side. Each physical white noise j(t) is considered to be the limit of a stationary Gaussian random process as the correlation time approaches to zero It Is difficult from the formal derivative of a Wiener process known as an Ideal white noise. 

As noted before, the Brownian force n t) may be modeled as a white noise stochastic process. White noise is a zero mean Gaussian random process with a constant power spectrum given in (72). Thus,... 

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... 

If the detector noise originates from stationary, Gaussian random processes and the instrument is calibrated with zero-offset, this matched Alter output is a zero-mean Gaussian random variable with variance equal to the power signal-to-noise ratio of the known signal ... 

A MATLAB script file saved as NIRMAIN.M that generates data using c(l) = 2 and c(2) = 0.5, adds Gaussian random noise to these data (to make things a bit more realistic), calls fminsearch, and prints results is shown below ... 

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