Noise white

The mean values of the. (t) are zero and each is assumed to be stationary Gaussian white noise. The linearity of these equations guarantees that the random process described by the a. is also a stationary Gaussian-... 

Since the f. are linearly related to the they are also stationary Gaussian white noises. This property is explicitly expressed by... 

Equation (A3.3.57) must be supplied with appropriate initial conditions describing the system prior to the onset of phase separation. The initial post-quench state is characterized by the order parameter fluctuations characteristic of the pre-quench initial temperature T.. The role of these fluctuations has been described in detail m [23]. Flowever, again using the renomialization group arguments, any initial short-range correlations should be irrelevant, and one can take the initial conditions to represent a completely disordered state at J = xj. For example, one can choose the white noise fomi (i /(,t,0)v (,t, 0)) = q8(.t -. ), where ( ) represents an... 

Amplifier noise. Can be of two kinds white noise results from random fluctuations of signal over a power spectrum that contains all frequencies equally over a specified bandwidth pink noise results when the frequencies diminish in a specified fashion over a specified range. 

Semiconductor devices ate affected by three kinds of noise. Thermal or Johnson noise is a consequence of the equihbtium between a resistance and its surrounding radiation field. It results in a mean-square noise voltage which is proportional to resistance and temperature. Shot noise, which is the principal noise component in most semiconductor devices, is caused by the random passage of individual electrons through a semiconductor junction. Thermal and shot noise ate both called white noise since their noise power is frequency-independent at low and intermediate frequencies. This is unlike flicker or ///noise which is most troublesome at lower frequencies because its noise power is approximately proportional to /// In MOSFETs there is a strong correlation between ///noise and the charging and discharging of surface states or traps. Nevertheless, the universal nature of ///noise in various materials and at phase transitions is not well understood. 

Environmentally Responsive Work-stations (ERWs). Workers in open-office areas have direct, individual control over both the temperature and air-flow. Radiant heaters and vents are built directly into their furniture and are controlled by a panel on their desks, which also provides direct control of task lighting and of white noise levels (to mask out nearby noises). A motion sensor in each ERW turns it off when the worker leaves the space, and brings it back on when he or she returns. 

Much of the regularity in classical systems can often be best discerned directly by observing their spatial power spectra (see section 6.3). We recall that in the simplest cases, the spectra consist of few isolated discrete peaks in more complex chaotic evolutions, we might get white noise patterns (such as for elementary additive rules). A discrete fourier transform (/ ) of a typical quantum state is defined in the most straightforward manner ... 

For classical evolutions, we merely substitute crj for p. Looking at plots of N fi, p vs. v/N, it is clear that although the quantum dynamics generally appears to preserve the characteristic structure of the classical spectrum, particular structural details tend to be washed-away [ilachSSbj. If high or low frequency components are heavily favored in the classical evolution, for example, they will similarly be favored in the quantum model discrete peaks, however, will usually disappear. White-noise spectra, of course, will remain so in the quantum model. 

Note that, while the random chain twists always decrease the hopping amplitudes (all ()/ , + are negative), // (a) can be both positive and negative, as it is the alternating part of the fluctuations. As in the FCM, we consider white noise disorder with a correlation function given by Eq. (3.22). This corresponds to independent random variations of the hopping amplitudes <5/ on different bonds. 

As an example of the use of Eq. (3-321), we shall calculate the output power density spectrum of an RC filter whose input consists of white noise. The RC filter in question is shown in Fig. 3-14. It is a simple matter to verify that the impulse response of this filter is given by... 

Fig. 1.1. Time-dependence of the components of angular momentum J, (Markovian process) and the torque M, (white noise) in the impact approximation.

The measurement of the electrode impedance has also been ealled Faradaie impedanee method. Since measurements are possible by applying either an electrode potential modulated by an AC voltage of discrete frequeney (which is varied subsequently) or by applying a mix of frequencies (pink noise, white noise) followed by Fourier transform analysis, the former method is sometimes called AC impedance method. The optimization of this method for the use with ultramicroelectrodes has been described [91Barl]. (Data obtained with these methods are labelled IP.)... 

Figure 21. Noise spectrum of detector amplifiers. Note that both axes have logarithmic scale. There are two main components of noise - the white noise which is present at all frequencies, and the 1// noise that is dominant at low frequencies. 1// noise has a fractal structure and is seen in many physical systems. The bandpass of a measurement decreases for slower readout, and the readout noise will correspondingly decrease. A limit to reduction in readout noise is reached at the knee of the noise spectrum (where white noise equals l/f noise) - reading slower than the frequency knee will not decrease readout noise.

Wiener inverse-filter however yields, possibly, unphysical solution with negative values and ripples around sharp features (e.g. bright stars) as can be seen in Fig. 3b. Another drawback of Wiener inverse-filter is that spectral densities of noise and signal are usually unknown and must be guessed from the data. For instance, for white noise and assuming that the spectral density of object brightness distribution follows a simple parametric law, e.g. a power law, then ... 

Unconstrained ML for Gaussian white noise. For Gaussian stationary noise, the covariance matrix is diagonal and proportional to the identity matrix ... 

The regularized solution is easy to obtain in the case of Gaussian white noise if we choose a smoothness prior measured in the Fourier space. In this case, the MAP penalty writes ... 

In our case, i.e. gaussian white noise and smoothness prior, CV and GCV have the same expression ... 

Ideally, any procedure for signal enhancement should be preceded by a characterization of the noise and the deterministic part of the signal. Spectrum (a) in Fig. 40.18 is the power spectrum of white noise which contains all frequencies with approximately the same power. Examples of white noise are shot noise in photomultiplier tubes and thermal noise occurring in resistors. In spectrum (b), the power (and thus the magnitude of the Fourier coefficients) is inversely proportional to the frequency (amplitude 1/v). This type of noise is often called 1//... 

Fig. 40.18. Noise characterisation in the frequency domain. The power spectrum IF(v)l of three types of noise, (a) White noise, (b) Flicker or 1//noise, (c) Interference noise.

In this case we assume that e is white noise, i.e., the e s are identically and independently distributed normally with zero mean and a constant variance cr. Thus, the model equation can be rewritten as... 

In this case we assume the disturbance term, e , is not white noise, rather it is related to through the following transfer function (noise filter)... 

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... 

---