Total discrete Fourier transform

In practical applications, x(t) is not a continuous function, and the data to be transformed are usually discrete values obtained by sampling at intervals. Under such circumstances, I hi discrete Fourier transform (DFT) is used to obtain the frequency function. Let us. suppose that the time-dependent data values are obtained by sampling at regular intervals separated by [Pg.43]

We discovered in Chapter 9 that the spatial function as given by the discrete Fourier transform (DFT) is a discrete Fourier series. Letting u(k) denote the (known) series consisting of only low-frequency terms and v(k) the series consisting of only high-frequency terms, we want to determine the unknown coefficients in v(k) that best satisfy the constraints. Expressing deviations of the total function forbidden by the constraints as some function of u(k) + v(k), we shall try to determine the coefficients of v(k) that minimize these deviations. Sum-of-squares expressions for these measures of the error have been found to result in the most efficient computational schemes. 

The discrete Fourier transform, if performed efficiently, takes N3 log N. This must be performed three times, but this does not increase the order of magnitude of the calculation. The pointwise multiplication takes N3. Thus the total runtime takes N3 log N + N3, which is of the same order as N3 log N. In contrast without the transform in real space the calculation would be of order N6. As N is of the order of 100, the discrete Fourier transform decreases the time required by more than a factor of 10 000. 

Descriptors for objects represented by their boundaries may be generated using the representations already described. Simple descriptors, such as perimeter, length, and orientation of the major axis, shape number, and eccentricity, may be readily computed from the boundary data. The ordered set of points on the boundary having two-dimensional coordinates (xt,yt), where 1 = 1,..., NandNis the total number of points on the boundary, can be treated as a one-dimensional complex function xt + iyk- The coefficients of the discrete Fourier transform applied to this function can also be used as a shape descriptor. 

Standard methods are used to propagate each Om in time. For the z and Z coordinates we make use of the fast fourier transform [99], and for the p coordinate we use the discrete Bessel transform [100]. The molecular component of asymptotic region at each time step, and projected onto the ro-vibrational eigenstates of the product molecule, for a wide range of incident energies included in the incident wave packet [82]. The results for all ra-components are summed to produce the total ER reaction cross section, a, and the internal state distributions. 

Each of the two noise-methods mentioned uses a periodic excitation by noise to produce a periodic response, apparently, so that signal-averaging techniques may be used to (a) diminish the total quantity of data stored, and (b) permit the computation of crosscorrelation and Fourier-transform functions as discrete sums of a reasonably finite number of terms.162,163... 

In fact, the interferogram is never totally symmetric about x = 0 and to recover the full spectral information, it is necessary to take the complex rather than the cosine Fourier transform. The interferogram is recorded to a finite path difference L rather than infinity. It is actually recorded by sampling it at discrete intervals At. 

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