Nonlinear impulse response

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]. 

FIGURE 5 Net impulse-response functions R. for the atmospheric CO, concentration (left column), the global mean temperature R, (center column), and the amplitude R, of the first EOF of the sea-level rise (right column, proportional to the mean sea-level rise) for different magnitudes of a d-function CO, input at time t = 0. The differences in the responses in the three cases (from top to bottom, 15, too and 30% increase in initial CO, concentration, respectively) arise from nonlinearities in the generalized nonlinear impulse-response model of Hooss ft al.y 2001. (Adopted from this source.)... 

Hooss, G., Voss, R., Hasselmann, K., Maier-Reimer, E., and Joos, F. (2001). A nonlinear impulse response model of the coupled carbon cycle-ocean-atmosphere climate system. Clim. Dyn. (in press). 

Figure 6.1 Nonlinear optical responses, (a) Second-order SF generation, the transition probability is enhanced when the IR light is resonant to the transition from the ground state g to a vibrational excited state V. CO is the angular frequency of the vibration, (b) Third-order coherent Raman scheme, the vibrational coherence is generated via impulsive stimulated...

Early applications of MPC took place in the 1970s, mainly in industrial contexts, but only later MPC became a research topic. One of the first solid theoretic formulations of MPC is due to Richalet et al. [53], who proposed the so-called Model Predictive Heuristic Control (MPHC). MPHC uses a linear model, based on the impulse response and, in the presence of constraints, computes the process input via a heuristic iterative algorithm. In [23], the Dynamic Matrix Control (DMC) was introduced, which had a wide success in chemical process control both impulse and step models are used in DMC, while the process is described via a matrix of constant coefficients. In later formulations of DMC, constraints have been included in the optimization problem. Starting from the late 1980s, MPC algorithms using state-space models have been developed [38, 43], In parallel, Clarke et al. used transfer functions to formulate the so-called Generalized Predictive Control (GPC) [19-21] that turned out to be very popular in chemical process control. In the last two decades, a number of nonlinear MPC techniques has been developed [34,46, 57],... 

Experimental NMR data are typically measured in response to one or more excitation pulses as a function of the time following the last pulse. From a general point of view, spectroscopy can be treated as a particular application of nonlinear system analysis [Bogl, Deul, Marl, Schl]. One-, two-, and multi-dimensional impulse-response functions are defined within this framework. They characterize the linear and nonlinear properties of the sample (and the measurement apparatus), which is simply referred to as the system. The impulse-response functions determine how the excitation signal is transformed into the response signal. A nonlinear system executes a nonlinear transformation of the input function to produce the output function. Here the parameter of the function, for instance the time, is preserved. In comparison to this, the Fourier transformation is a linear transformation of a function, where the parameter itself is changed. For instance, time is converted to frequency. The Fourier transforms of the impulse-response functions are known to the spectroscopist as spectra, to the system analyst as transfer functions, and to the physicist as dynamic susceptibilities. 

Fig. 4.2.1 A system transforms an input signal x t) into an output signal y(t). A linear system is described by the linear impulse-response function fc) (t). A nonlinear system is described by multi-dimensional impulse-response functions fe (ri > T2 > > r ).

The functions written in capital letters in (4.2.13)-(4.2.16) are the Fourier transforms of the functions written in small letters in (4.2.5)-(4.2.8). The superscript s indicates that the nonlinear transfer functions K (o)i,. .., o> ) in (4.2.15) and (4.2.16) are the Fourier transforms of impulse-response functions with indistinguishable time arguments, where the causal time order ti > >r is not respected. These transfer functions are invariant against permutation of frequency arguments. Equivalent expressions for the Fourier transforms of impulse-response functions with time-ordered arguments cannot readily be derived. 

Nonlinear cross-correlation of the system response y(t) (4.2.4) with different powers of a white-noise excitation x(t) yields multi-dimensional impulse-response functions hn (tTl,. . . , CTn),... 

Transformation of step and impulse response of tracer concentrations are used to compare mixing models 29). Other analytical procedures can be cited. Douglas and Eagleton 30) have given analytical solutions for the dynamics of adiabatic unpacked reactors. Douglas 31) has developed in detail an analytical procedure for determining the frequency response of a simple nonlinear reactor. 

The linear impulse-response model has recently been generalized by Hooss et al. (2001) to include some of the dominant non-linearities of the climate system. The net response curves shown in Fig. 5 were computed using this generalized model, showing the impact of nonlinearities in the lower two-panel rows. The princi-... 

Phenomenological reactor models are capable of predicting tracer impulse responses. Thus, they can predict reactor performance for linear kinetics exactly and offer a good starting point for assessment of performance of nonlinear systems. Much more effort should be spent on their systematic development for commonly used reactor types. 

The only violation of the central volume principle was reported by Awasthi and Vasudeva (54) who unfortunately did not demonstrate the linearity of their tracer responses and who dealt with two moving phases and two inlets and outlets. Their anomalous results could have been caused by poorly executed experiments, by nonlinearity of tracer responses (since the symmetry of W(t) and F(t) and the integrability of the impulse response E(t) to obtain F(t) were not checked), or by the movement of tracer in and out of the system by two flowing phases. 

Herrman, W. and Nunziato, J.W., Nonlinear Constitutive Equations, in Dynamic Response of Materials to Intense Impulsive Loading (edited by Chou, P.C. and Hopkins, A.K.), Air Force Materials Laboratory, Wright-Paterson AFB, 1973, Chap. 5. 

In impulsive multidimensional (1VD) Raman spectroscopy a sample is excited by a train of N pairs of optical pulses, which prepare a wavepacket of quantum states. This wavepacket is probed by the scattering of the probe pulse. The electronically off-resonant pulses interact with the electronic polarizability, which depends parametrically on the vibrational coordinates (19), and the signal is related to the 2N + I order nonlinear response (18). Seventh-order three-dimensional (3D) coherent Raman scattering, technique has been proposed by Loring and Mukamel (20) and reported in Refs. 12 and 21. Fifth-order two-dimensional (2D) Raman spectroscopy, proposed later by Tanimura and Mukamel (22), had triggered extensive experimental (23-28) and theoretical (13,25,29-38) activity. Raman techniques have been reviewed recently (12,13) and will not be discussed here. 

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