Backward Fourier transform

Equations (40.3) and (40.4) are called the Fourier transform pair. Equation (40.3) represents the transform from the frequency domain back to the time domain, and eq. (40.4) is the forward transform from the time domain to the frequency domain. A closer look at eqs. (40.3) and (40.4) reveals that the forward and backward Fourier transforms are equivalent, except for the sign in the exponent. The backward transform is a summation because the frequency domain is discrete for finite measurement times. However, for infinite measurement times this summation becomes an integral. 

The expressions for the forward and backward Fourier transforms of a data array of 2N+ 1 data points with the origin in the centre point are [3] ... 

Filtering and smoothing are related and are in fact complementary. Filtering is more complicated because it involves a forward and a backward Fourier transform. However, in the frequency domain the noise and signal frequencies are distinguished, allowing the design of a filter that is tailor-made for these frequency characteristics. 

This form of the general Jacobian element allows for the straightforward solution of the SSOZ equation for molecules of arbitrary symmetry. However, in the numerical solution using Gillan s methods, most of the computation time is involved in calculating the elements of the Jacobian matrix, rather than in the calculation of its inverse or in the calculation of transforms. Indeed, as the forward and backward Fourier transforms can be carried out using a fast Fourier transform routine, the time-limiting step is the double summation over / and j in Eq. (4.3.36). With this restriction in mind, it is... 

Fig. 2a displays the ion time-of-flight (TOF) distribution obtained when (n) = 1.6 104 Xe clusters interacted with a Fourier Transform-limited 100 fs 800 nm, 1015 W cm 2 laser pulse. The TOF displays a number of peaks corresponding to ions up to Xe1,+. The peaks in the TOF are quite broad, and even display a double peak structure due to the fact that ions are emitted in forward-backward directions with respect to the detector. Both the charge state reached and the kinetic energy of the ions are signatures of collective effects in the cluster ionisation. For example, when only atoms were present in the atomic beam, the maximum charged state reached was 4+. 

Fourier channel k. The phase is known up to a multiple of 27t(since only exp /( ),(/")) is known). Time-scale modifications also require the knowledge of the instantaneous frequency G) (f ). 0), Uua) can also be estimated from successive short-time Fourier transforms for a given value of k, computing the backward difference of the short-time Fourier transform phase yields... 

The Fourier transform infra-red spectrophotometer incorporates an interferometer (Fig. 3). Its basic components are a beam splitter and two mirrors, perpendicular to each other, one fixed and one which can be moved backwards and forwards at right angles to its plane. Approximately half of the radiation from the source is reflected to the fixed mirror where it is reflected back to the beam splitter which transmits about half (i.e. a quarter of the original) to the detector. The other half of the original radiation passes through the beam splitter to the movable mirror where it is reflected back to the beam splitter and about half (i.e. a quarter of the original) is reflected to the detector. When the two mirrors are equidistant from the beam splitter, the two beams... 

In words, E rp) is the backward finite Fourier transform of the product of —ifc, the forward finite Fourier transform of the mesh based charge density Pm and the so-called optimal influence function ( opt) given by... 

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