Mixed time-frequency representation

The development of a mixed time-frequency representation in which both characteristics of the field and the response function are highlighted is currently receiving considerable attention. This activity is triggered by the rapid progress in pulse-shaping techniques, which made it possible to control the temporal profiles as well as the phases of optical fields with a remarkable accuracy [1-4]. These developments have further opened up the possibility of coherent control of dynamics in condensed phases [5-7]. 

The variables T ax and represent the hmits of the time interval the second or indirect dimension is recorded in. The index inc refers to the indirect spectral dimension. As wiU later be seen, the direct dimension can originate from two different data sets, then A B, or from the same data set, hence A = B. In the latter case, one data set is merely the transpose of the other one. Within a representation such as Eq. (5.5), the two data sets— which contain mixed time—frequency data before and frequency—frequency data after transformation—are correlated by a shared indirect dimension. The common feature can be understood as a perturbation, the dimension hence called perturbation dimension. 

The equality of transformations of the mixed time—frequency data and the completely Fourier transformed data is a consequence of Parseval s theorem (5.4) and has been previously discussed [10,11]. It can also be understood by taking into account that the spectral reconstruction is achieved by relating two direct dimensions via an indirect dimension, which in turn is discarded. Whether two frequency domains are correlated via a common time domain or a common frequency domain is therefore equivalent. It is also equivalent to Noda s model of relating two IR wavenumber dimensions via a common perturbation stemming from either time or sample space. From the matrix representation, it can be seen that Noda s synchronous matrix O, Eqs. (5.6) and (5.17) corresponds to the covariance map according to Eq. (5.16), if the data matrices yielding O are composed of... 

As a consequence ofEq. (5.16), four covariance NMR types can be defined. The use of symbols is the same as above C is the covariance matrix, that is, the spectrum, S is the mixed time-frequency 2D data representation, F and G are frequency-frequency data matrices, that is, spectra G eg is a regularized data matrix according to Eq. (5.26). 

The matrix representation of the matrix formalism, in general, is more illustrative than the notation using sums and multiple indices. The equivalence between a 2-dimensional data matrix and a 2D spectrum or a mixed time-frequency data set is evident. While the covariance literature provides numerous examples of spectra obtained from covariance transformations of experimental data sets, which will be discussed further below, few sim-phfied models have been designed and applied for instruction purposes [14,17,18,32—34]. Thoroughly elaborated cases are described in Refs. [6,11,12,14,35-39]. 

Q/I = [Ci,/I> > Cn,/r] are supposed to be as sparse as possible and approximate the targeted sparse time-frequency representations of the monotone modal responses. The obtained mixing matrix is therefore the estimated normal mode, i.e., = A = W. Once the normal modes are estimated, the time-domain modal responses can be recovered using the demixing matrix... 

Experiments described in this section are suited to investigate ultraslow motion with correlation times in the millisecond-to-second range. Here, the NMR spectra are given by their rigid-lattice limit and one correlates the probability to find given NMR frequencies at two different times separated by the so-called mixing time tm [11,72]. A two-dimensional (2D) spectrum results, being a function of two NMR frequencies at t = 0 and t = tm, respectively. Since the NMR frequency reflects the orientation of the molecule, 2D spectra provide a visual representation of the reorientational process. Time- and frequency-domain... 

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