Multi-dimensional impulse response functions

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 ).

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]. 

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),... 

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See also in sourсe #XX -- [ ]

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