Noise variance

Therefore, if A represents the spectrum, the various a represent convolution coefficients and Var(A) represents a noise source that gives a constant noise level to the spectral values, then equation 57-36 gives the noise variance expected to be found on the computed resultant value, whether that is a smoothed spectral value, or any order derivative computed from a Savitzky-Golay convolution. For a more realistic computation, an interested (and energetic) reader may wish to compute and use the actual noise that will occur on a spectrum, from the information determined in the previous chapters [6-7] instead of using a constant-noise model. But for our current purposes we will retain the constant-noise model then equation 57-36 can be simplified slightly ... 

As we mentioned, the two-point first derivative is equivalent to using the convolution function -1, 1. We also treated this in our previous chapter, but it is worth repeating here. Therefore the multiplying factor of the spectral noise variance is — l2 + l2 = 2,... 

These probabilities are collected in the vector pd,. The required conditional probabilities Py (ynj dnj = 0) and py (ynj dnj = 1) depend on the used watermarking scheme, but also on possible attacks. We designed our scheme for an AWGN attack of a certain noise variance, e.g. WNR = -3dB. This heuristic is useful since up to now little about possible statistical attacks on the watennarked stmcture data is known. The vectors Pdj and ij are the result of the detection process for the molecule Mj. 

We can thus expect from the short-time approximation that quantum noise does not significantly affect the classical solutions when the initial pump field is strong. We will return to this point later on, but now let us try to find the short-time solutions for the evolution of the quantum noise itself—let us take a look at the quadrature noise variances and the photon statistics. Using the operator solutions (94) and (95), one can find the solutions for the quadrature operators Q and P as well as for Q2 and P2. It is, however, more convenient to use the computer program to calculate the evolution of these quantities directly. Let us consider the purely SHG process, we drop the terms containing b and b+ after performing the normal ordering and take the expectation value in the coherent... 

The most demanding part of Table 7.4 is the a priori part, where a priori distributional knowledge is required (e.g. external knowledge of noise variance). This knowledge often comes from previous experiments and not from the data themselves. In multi-way analysis, levels 2 and 3 have not been studied extensively. 

P2(T) has to be minimized with respect to T. The normalized integrated peak area in case of a known peak shape is known, and we have derived an expression for the integrated noise variance. As usual, we can determine the derivative, setting it equal to zero and the desired result can be calculated. This procedure leads to ... 

To verify the result produced by the Kalman filter, the time histories of the displacement and the velocity estimated by the Kalman filter are compared to their actual values in Figures 2.30 and 2.31, respectively. Here, the actual values of the noise variances are used. The solid lines in both figures represent the actual response, whereas the dotted lines represent the response estimated by the Kalman filter. It is noted that the estimated velocity in Figure 2.31 is close to... 

As demonstrated previously the process noise and the measurement noise parameters directly affect the state vectors estimated by the Kalman filter. Furthermore, the covariance matrix of the state estimation is affected as well. Therefore, accurate estimation of the noise parameters is necessary for good performance of the filter. In this example, the Bayesian approach is applied to select a p and a. Figure 2.32 shows the contours of the likelihood function p V 0, C) together with the actual noise variances 0 = [cr, and its optimal estimate 6. The two contours correspond to 50% and 10% of the peak value. The optimal values of ap = 2.8N and a = 7.1 x 10 m /s are at reasonable distance to the actual values as the actual noise variances are located within the region with significant probability density. Therefore, the Bayesian approach is validated to give accurate estimation for both noise variances for the linear oscillator. 

Figure 2.33 (Figure 2.34) shows the comparison between the displacement (velocity) estimated with the actual noise variances and their corresponding actual curves of the 2nd and 9th floors. The solid lines represent the actual displacement or velocity, whereas the dotted lines represents its Kalman estimation but the two sets of curves are on top of each other. The observation is further supported by the small rms errors which are only 2.59% and 2.41% of the rms values of the displacements for the 2nd floor and the 9th floor, respectively. In addition, the estimated velocities also agree well with the actual velocities as can be seen in Figure 2.34. The rms errors are 9.44% and 5.47% of the rms values of the velocities for the 2nd floor and the 9th floor, respectively. In conclusion, the Kalman filter is validated to provide an accurate state estimation for the linear MDOF system.

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