Time domain approach

J.-C. Bolomey, D. Lesselier, C. Pichot and W. Tabbara, Spectral and time domain approaches to some inverse scattering problems, 1981, / Trans. Antennas Propagat., 29, pp. 206-212. 

Binsch [6] provided the standard way of calculating these lineshapes in the frequency domain, and implemented it in the program DNMR3 [7], Fonnally, it is the same as the matrix description given in section (B2.4.2.3). The calculation of the matrices L, R and K is more complex for a coupled spin system, but that should not interfere witii the understanding of how the method works. This work will be discussed later, but first the time-domain approach will be developed. 

The noise can be described by means of two approaches, namely, in the time domain and in the frequency domain. For the time domain approach the characteristic value is the well-known mean square deviation from the value over the time T,... 

It is possible that a skin, which is moist and cool gives exactly the same electrical response to measurements made at a single frequency as a skin, which is dry and warm. To separate and specify potentially confounding influences such as water content, temperature change, and sweat gland activity, it is necessary to use some form of electrical spectroscopic technique, that is, stimulation at three or more different frequencies, or a time-domain approach followed by Fourier transformation.44-46... 

As can be deduced, for m > 2, expression (2.67) leads to cross derivatives by x and y, whose evaluation is rather cumbersome. To alleviate this difficulty, only one fictitious point can be considered at each side of the interface and hence only the zero- and first-order jump conditions are implemented. While this notion gives reliable solutions, an alternative quasi-fourth-order strategy has been presented in [28] for the consideration of higher order conditions and crossderivative computation. A fairly interesting feature of the derivative matching method is that it encompasses various schemes with different orders that permit its hybridization with other high-accuracy time-domain approaches. 

Other CRAMPS-based time domain approaches... 

Besides the aforementioned time-domain approaches, many frequency-domain methods have also been developed and widely used. Examples are the complex curve fitting method [153], the maximum entropy method [4,263], the pole/zero assignment technique [271], the simultaneous frequency-domain approach [62], the rational fraction polynomial approach [219], the orthogonal polynomial approach [264], the polyreference frequency-domain approach [73], the multi-reference simultaneous frequency-domain approach [64] and the best-fit reciprocal vectors method [173]. 

The Bayesian time-domain approach presented in this chapter addresses this problem of parametric identification of linear dynamical models using a measured nonstationary response time history. This method has an explicit treatment on the nonstationarity of the response measurements and is based on an approximated probability density function (PDF) expansion of the response measurements. It allows for the direct calculation of the updated PDF of the model parameters. Therefore, the method provides not only the most probable values of the model parameters but also their associated uncertainty using one set of response data only. It is found that the updated PDF can be well approximated by an appropriately selected multi-variate Gaussian distribution centered at the most probable values of the parameters if the problem is... 

The additional measurement of y3 improves only slightly the prediction for y but the additional measurement of y4 has virtually no effect on the prediction. This example is useful to demonstrate the approximation used in the Bayesian time-domain approach. The random variable yi is predicted by the measurements of y2. y3 and 4, which are 0.5,1 and 1.5 periods apart from yi, respectively. It turns out that including the data points within one period is sufficient. Furthermore, in a usual situation, the sampling time step is much less than half of... 

Figure 4.4 shows the conditional PDF of the natural frequency and the damping ratio C with the spectral intensity and the prediction-error variance fixed at their optimal values. The conditional PDFs by the Bayesian time-domain approach and the Gaussian approximation are plotted with solid lines and dashed lines, respectively. The two groups of curves are on top on each other, indicating that the Gaussian approximation is accurate when the number of data points is sufficiently large. This can be used to represent the posterior PDF, e.g., for statistical moments computation. 

Another case is investigated with a very short period of measurement, namely T = 5 s, so it contains less than four fundamental periods of the oscillator. The Bayesian time-domain method is used for the identification and Figure 4.5 shows the conditional PDF of and f with all other parameters fixed at their optimal values. The solid lines show the conditional posterior PDF obtained by the Bayesian method and the dashed lines show the Gaussian approximation. It is clear that the posterior PDF is non-Gaussian. This confirms that the Bayesian time-domain approach is capable to offer the correct inference without assuming the type of the posterior PDF. In the case of a non-Gaussian posterior PDF, statistical moments, such as the variances of the estimates, can be computed by direct Monte Carlo simulation. The results are shown in Table 4.2 in the same fashion as Table 4.1. The computed uncertainty obtained here is reasonable by judging the normalized distance of the estimates. 

The solid lines are obtained by using the Bayesian time-domain approach without assuming the type of posterior distributions and the dashed lines are obtained by using the Gaussian approximation. It can be seen that the two sets of curves are on top of each other, implying that the Gaussian approximation is accurate. 

In this chapter, the Bayesian time-domain approach was introduced for identification of the model parameters and stochastic excitation parameters of linear multi-degree-of-freedom systems using noisy stationary or nonstationary response measurements. The direct exact formulation was presented but it turned out to be computationally prohibited for a large number of data points. Then, an approximated likelihood function expansion was proposed to resolve this obstacle. For a globally identifiable case with a large number of data points, the updated PDF... 

Comparison of Spectral Density Approach and Time-domain Approach... 

Chapter 3 and Chapter 4 presented the Bayesian spectral density approach and Bayesian time-domain approach. The comparison can be summarized as follows ... 

The Bayesian time-domain approach utilizes the Bayes theorem repeatedly to factorize the likelihood function into the product of a joint PDF and conditional PDFs ... 

Even though the spectral density approach requires computation of the inverse and determinant of a number of matrices, the size of these matrices is only No y. No. They are significantly smaller than the NgNp x NgNp matrix Eyj j (= E22) required in the time-domain approach. Comparison of the computational efficiency between the two methods depends on the number of the elements in the frequency index set and the number of data points in a fundamental period. The ratio of the computations required by the Bayesian spectral density approach and the Bayesian time-domain approach can be approximated by ... 

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