Uniformly distributed noise

In equation 46-79, Twu represents the mean computed transmittance for Uniformly distributed noise and the parenthesized (1) in both the numerator and the denominator is a surrogate for the actual voltage difference between successive values represented by the A/D steps essentially a normalization factor for the actual physical voltages involved. In any case, if the actual voltage difference were used in equation 46-79, it would be factored out of both the numerator and the denominator integrals, and the two would then cancel. Since the denominator is unity in either case, equation 46-79 now simplifies to... 

Here again, our task is simplified by the two facts we have mentioned above first, that we can reuse many of the results we obtained previously for the case of Normally distributed noise, and second, that the nature of uniformly distributed noise characteristics simplify the mathematical analysis. Our first step in this analysis starts with equation 44-71, that we derived previously in Chapter 44 referenced as [5] as a general description of noise behavior ... 

In our previous development, we presented a family of curves, corresponding to different values of SD(A ). In the case of uniformly distributed noise, which is of necessity contained within a limited range of values, the well-known fact that the standard deviation of the noise equals the range/ f 2 helps us, in that it requires only one curve to display, rather than a family of curves. ([7], p. 146). For this case, then, equation 44-71 becomes equation (46-84) ... 

The figure corresponding to Figure 45-9 (which appeared in Chapter 45 [6]) that was calculated for Normally distributed noise is Figure 46-15, which presents the results of calculating the variance of the two terms of equation 44-77 for uniformly distributed noise instead. We note that while these terms follows the same trends as the Normally distributed errors, these errors do not become appreciable until Er has fallen below 0.6, which corresponds to the point where values occur close to or less than zero. For values of Er below 0.6 the values of both terms of equation 44-77 become very large and erratic. 

Statistics plays a crucial role in any data analysis, and accordingly, the statistical aspects are mentioned and appropriate equations/code are supplied. E.g. examples are given for the least-squares analysis of data with white noise as well as y2-analyses for data with non-uniformly distributed noise. However, the statistical background for the appropriate choice of the two methods and more importantly, the effects of wrong assumptions about the noise structure are not included. 

Chemometrics in Spectroscopy Variances for uniformly distributed noise... 

Let the input signal u for identification be a uniformly distributed noise in the interval [-2, 2], the number of sample dataiV= 1001, and assume the number of fuzzy rules is not known... 

On the other hand, if no data processing is involved and one is seeking a very weak signal, the absorbance scale can be expanded to 1/3 the signal-to-noise ratio rather than the highest signal. Of course, a linear optical signal requires that the infrared sample have no residual orientation, no voids or holes, and a uniform distribution of material. Careful control of the sample preparation procedure must be achieved in order that reproducible samples are obtained for the infrared examination. 

A well known result states that the values of the discrete Fourier transform of a stationary random process are normally distributed complex variables when the length of the Fourier transform is large enough (compared to the decay rate of the noise correlation function) [Brillinger, 1981], This asymptotic normal behavior leads to a Rayleigh distributed magnitude and a uniformly distributed phase (see [McAulay and Malpass, 1980, Ephraim andMalah, 1984] and [Papoulis, 1991]). 

The SNR from do measured on an Fourier transform instrument is more complicated. The signal is given in Eq. 17. The noise is given by the square root of the signal, but is uniformly distributed throughout the whole spectrum, thus making the noise in do dependent on all of the other spectral elements. We approximate the shot noise due to the whole spectrum as,... 

Figure 4 shows a further test of the effect of imperfect information on the inferred source profiles. This time, random noise is added to the concentration profile before carrying out the inversion with Eq. (30), to test the ability of the method to cope with random errors in concentration measurement. Again, three scenarios are used the first is identical with scenario (1) above. The other two use (n, m) = (10, 5) and the full Dij as in scenario (2), but also add to each C a random error, uniformly distributed over the interval [ i and Fj, but the larger one leads to significant degradation in both profiles. 

To test the inverse predictions of i and Spi from C and S , random noise was added to the S, profile predicted by the forward calculation. This simulates a random error in the measurement of S. The random noise was uniformly distributed over the interval [ — (r,5, where ag is 0, 0.01, 0.03, or 0.1%. The perturbed, noisy profiles of 5 used for the inverse analysis are shown in Figure 7d. (No noise was added to C, because ihe effect of such noise has already been tested in Figure 4.) When a direct inversion of Eq. (38) was used to calculate random noise even at very small assumed random measurement errors (not shown). To obtain an acceptably smooth profile of 5p, the least-squares error minimization procedure of Eq. (30) was used with (n, m) = (10,... 

The performance of various rectification methods is compared for the noise-free underlying signal represented as a uniform distribution, non-stationary stochastic process, and data with deterministic features. 

Uniform distribution. The data used for this illustration are similar to those used by Johnston and Kramer [24]. The noise-free measurements for the flowrates, F and F4, are uniformly distributed in the intervals [1,5,15,40], respectively. The flowrates F through F5 are contaminated by independent... 

The primary sources of noise in this region are classified as thermal or Johnson noise together with shot noise due to the particulate nature of photons and electrons. Additionally there is non-equilibrium or inverse frequency noise, often termed pink noise to distinguish it from the other two that have a uniform white noise frequency distribution. 

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