Stochastic excitation

The sinc fiinction describes the best possible case, with often a much stronger frequency dependence of power output delivered at the probe-head. (It should be noted here that other excitation schemes are possible such as adiabatic passage [9] and stochastic excitation [fO] but these are only infrequently applied.) The excitation/recording of the NMR signal is further complicated as the pulse is then fed into the probe circuit which itself has a frequency response. As a result, a broad line will not only experience non-unifonn irradiation but also the intensity detected per spin at different frequency offsets will depend on this probe response, which depends on the quality factor (0. The quality factor is a measure of the sharpness of the resonance of the probe circuit and one definition is the resonance frequency/haltwidth of the resonance response of the circuit (also = a L/R where L is the inductance and R is the probe resistance). Flence, the width of the frequency response decreases as Q increases so that, typically, for a 2 of 100, the haltwidth of the frequency response at 100 MFIz is about 1 MFIz. Flence, direct FT-piilse observation of broad spectral lines becomes impractical with pulse teclmiques for linewidths greater than 200 kFIz. For a great majority of... 

A home-built solid-state NMR spectrometer with stochastic excitation has been described. An overview of the instrument has been given and the control unit and the module for the pulse generation have been described. A static probe with crossed coils for the transmitter and receiver circuits and the data processing part of the spectrometer software have been presented. Several examples of NMR measurements have been shown, including selective excitation in solids and the acquisition of static solid-state NMR spectra with a spectral width of up to 185 kHz. 

Impulse-response and transfer functions can be measured not only by pulse excitation, but also by excitation with monochromatic, continuous waves (CW), and with continuous noise or stochastic excitation. In general, the transformation executed by the system can be described by an expansion of the acquiired response signal in a series of convolutions of the impulse-response functions with different powers of the excitation [Marl, Schl]. Given the excitation and response functions, the impulse-response functions can be retrieved by deconvolution of the signals. For white noise excitation, deconvolution is equivalent to cross-correlation [Leel]. 

Fig. 4.1.1 Interrelationship between excitation (left) and response (right) in spectroscopy (a) Excitation with continuous waves (CW excitation) directly produces the spectrum, (b) For pulsed excitation, the spectrum is obtained by Fourier transformation of the impulse response, (c) For stochastic excitation, the impulse response is derived by cross-correlation of excitation and response signals.

Equation (4.2.11) describes the response to three delta pulses separated by ti =oi — 02 >0, t2 = 02 — 03 > 0, and t3 = 03 > 0. Writing the multi-pulse response as a function of the pulse separations is the custom in multi-dimensional Fourier NMR [Eml ]. Figure 4.2.3 illustrates the two time conventions used for the nonlinear impulse response and in multi-dimensional NMR spectroscopy for n = 3. Fourier transformation of 3 over the pulse separations r, produces the multi-dimensional correlation spectra of pulsed Fourier NMR. Foinier transformation over the time delays <7, produces the nonlinear transfer junctions known from system theory or the nonlinear susceptibilities of optical spectroscopy. The nonlinear susceptibilities and the multi-dimensional impulse-response functions can also be measured with multi-resonance CW excitation, and with stochastic excitation piul]. 

Fig. 4.3.1 [Bliil] The cross-correlation function C (ct) of linear response yx (t) and stochastic excitation x t) is proportional to the impulse-response function, and depends on the time delay a between excitation and response.

Stochastic excitation with m sequences and the use of the Hadamard transformation have been investigated in NMR spectroscopy [Kail, Zie2] as well as in NMR imaging [Chal]. Processing of the nonlinear response to m sequences by Hadamard and Fourier transformation results in signal distortions reminiscent of noise, which are caused by the nonlinear parts of the response [B1U2]. 

Stochastic excitation in NMR denotes excitation of the spin system with noise [Em3, Kail], which is, in general, white within the spectral region of the linear response. White means that the average of the power spectral density is independent of frequency. This is automatically fulfilled for a delta pulse. For noise this is valid only if successive values... 

Here we reduced to stationary noise sources. Without loss of generality the mean is set to zero. For the later on considered types of noise this formulation is sufficient to obtain general answers for ensembles and their averages of the stochastic excitable system. Thus we can formulate evolution laws for the probability densities and the other moments. We note that the generalization to cases with more than one noise sources is straightforward and crosscorrelations between the noise source have to be defined. 

If energy transport is stochastic, excitation is always localized at an individual chromophore. Energy transfer can be resonant or nonresonant. 

The aforementioned methods can be applied to evaluate the reliability of engineering systems subjected to stochastic input with a given mathematical model. On the other hand, if a parametric model of the underlying system is available and the probability density function of these parameters is obtained by Bayesian methods, the uncertain parameter vector can be augmented to include the model parameters and the uncertain input components. Then, robust reliability analysis can proceed for stochastic excitation with an uncertain mathematical model. This allows for more realistic reliability evaluation in practice so that the modeling error and other types of uncertainty of the mathematical model can be taken into account. 

For example, in reliability analysis, the quantity Q is considered as the probability that the structure with parameter vector 0 would fail, i.e., Q(0) = P(F 0, C). Then, the updated integral becomes the updated robust probability of failure for the structure, when it is subjected to some stochastic excitation [198] ... 

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... 

Adrezin, R., and Benaroya, H., Dynamic modelling of tension leg platfwms, in Stochastically Excited Nonlinear Ocean Structures, World Scientific, Singapore, 1998. 

Ching, J., Au, S-K. Beck, J.L. 2005a. Reliability estimation for dynamical systems subject to stochastic excitation using Subset Simulation with splitting. Computational Methods in Applied Mechanics and Engineering 194 1557-1579. 

Igusa, T Kiurghian, AD. 1988. Response of uncertain systems to stochastic excitation . J Engng Mech, ASCE 114 812-32. 

Let the vectors y, y, / = l,...,n, 0, 0 , i=l,..., ris and z, z i=l,..., t represent the vector of design variables, uncertain structural parameters, and random variables that specify the stochastic excitation, respectively. The uncertain system parameters 0 and the random variables z are modeled using prescribed probability density functions q 9 ) and p( z ), respectively. These functions indicate the relative plausibility of the possible values of the uncertain parameters 0 e c R" and random variables z e c R" , respectively. The structural synthesis problem considered... 

T. Igusa and A. Der Kiureghian, Response of Uncertain Systems to Stochastic Excitation, Journal of Engineering Mechanics, to appear. 

It may be noted that the expressions in Appendix A given by Papanicolaou and Kohler give rise to the same drift and diffusion terms as those derived from Eq. (6) by the usual averaging procedure of Khasminskii However, it is easier to make use of the formulas of Appendix A since in the calculations the terms that explicitly contain the stable modes in Eq. (5) (for e.g., z) are made identically zero. In the sequel we shall apply these results to study bifurcation behavior of stochastically excited nonlinear nonconservative problem. 

Sri Namachchivaya, N. Hopf bifurcation in the presence of both parametric and external stochastic excitation, to appear. Journal of Applied Mechanics (ASME). 

Advanced simulation techniques First excursion probability Isolation systems Rubber bearings Stochastic excitation... 

The performance of base-isolated structural systems is characterized by means of a set of response functions/i,(f, y, 6), i Ih,t [0, T], where y is the vector of controllable system parameters 6 is the vector of uncertain variables that characterizes the stochastic excitation model, i.e., the white noise sequence z and the... 

A reliability-based characterization of base-isolated systems under stochastic excitation has been proposed. The characterization allows to evaluate the effect of important design isolator... 

Ceccoli C, Mazzotti C, Savoia M (1999) Non-linear seismic analysis of base-isolated RC frame structures. Earthq Eng Struct Dyn 28(6) 633-653 Ching J, Au SK, Beck JL (2005) Reliability estimation for dynamical systems subject to stochastic excitation using subset simulation with splitting. Comput Method Appl Mech Eng 194(12-16) 1557-1579 Chopra AK (2001) Dynamics of structures theory and applications to earthquake engineering. Prentice Hall, Englewood Cliffs, Upper Saddle River, NJ 07458. USA... 

Au SK, Beck JL (2001) Estimation of small failure probabilities in high dimensions by subset simulation. Probab Eng Mech 16 263-277 Caughey TK, Payne HJ (1967) On the response of a class of self-excited oscillators to stochastic excitation, hit J Non Linear Mech 2 125-151... 

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