Finite Fourier transform

Expressing/// by using time convolution (DuhameTs integral) of/(f) and g t) under the condition that/(f) = 0 for 0, 

It is possible to estimate the error of the approximate time solution //f) defined by Equation 2.42 in comparison with the accurate one/(f) in Equation 2.41, which cannot be evaluated by numerical integration. When/(w) changes suddenly, noticeable oscillation called Gibbs oscillation appears in//f). This is the error caused by the finite Fourier transform. A countermeasure to this is to take an average for the time region [f - a, f + a] in the following form  

Substituting Equation 2.42 into the previous equation, and rearranging it, the following formula is obtained [38,39]  

By adopting Equation 2.49 rather than Equation 2.42 as a finite Fourier transform, the Gibbs oscillation due to the finite interval in a numerical calculation of Fourier transform is reduced. 

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The method of Finite Fourier Transform (9) is applied to solve Eq. (5) with boundary conditions (5a), (5b), and (5c). The resulting dimensionless concentration profile of reactant A is ... 

Equation 22 can be solved with the boundary conditions 28, 29, and 30 by the finite Fourier transform methods by assuming a solution of the form... 

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... 

Although the canonical transformation (67) has the very simple form of the finite Fourier transformation, the connection between the conventional number states and the radiation phase states (72) is not simple ... 

It was clear in the previous example that cosine functions occurred in the natural course of analysis. In fact, the transformation we performed there is often called the finite Fourier transform. However, the broad category of such finite transforms are called Sturm-Liouville. 

To determine the convolution for a discrete sample we follow Eq. (26.43) and find the product of two finite Fourier transforms Z = X T and take the inverse of the result, according to... 

The method proposed by Papoulis [7] to determine h(t) as a function of its Fourier transform within a band, is a non-linear adaptive modification of a extrapolation method.[8] It takes advantage of the finite width of impulse responses in both time and frequency. 

Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements (see Ref. 106). Spectral methods employ Chebyshev polynomials and the Fast Fourier Transform and are quite useful for nyperbohc or parabohc problems on rec tangular domains (Ref. 125). 

The funetion G uj) is the exponential Fourier transform of F t) and is a funetion of the eireular frequeney uj. In praetiee the funetion F t) is not given over the entire time domain but is known from time zero to some finite time T, as shown in Figure 16-2. The time span T may be divided into K equal inerements of At eaeh. For eomputational reasons, let K = 2 where p is an integer. Also, let the eireular frequeney span lu be divided into N parts where N = 2 . (In praetiee, N is often set equal to K.) By setting / = K/NT, the frequeney interval Alu beeomes... 

Comparison of the previous equations with Equations (16-6) and (16-7) reveal that the Fourier transform is really just a Fourier series constructed over a finite interval. 

Often the actions of the radial parts of the kinetic energy (see Section IIIA) on a wave packet are accomplished with fast Fourier transforms (FFTs) in the case of evenly spaced grid representations [24] or with other types of discrete variable representations (DVRs) [26, 27]. Since four-atom and larger reaction dynamics problems are computationally challenging and can sometimes benefit from implementation within parallel computing environments, it is also worthwhile to consider simpler finite difference (FD) approaches [25, 28, 29], which are more amenable to parallelization. The FD approach we describe here is a relatively simple one developed by us [25]. We were motivated by earlier work by Mazziotti [28] and we note that later work by the same author provides alternative FD methods and a different, more general perspective [29]. 

Equation (6a) implies that the scale (dilation) parameter, m, is required to vary from - ac to + =. In practice, though, a process variable is measured at a finite resolution (sampling time), and only a finite number of distinct scales are of interest for the solution of engineering problems. Let m = 0 signify the finest temporal scale (i.e., the sampling interval at which a variable is measured) and m = Lbe coarsest desired scale. To capture the information contained at scales m > L, we define a scaling function, (r), whose Fourier transform is related to that of the wavelet, tf/(t), by... 

A light pulse of a center frequency Q impinges on an interface. Raman-active modes of nuclear motion are coherently excited via impulsive stimulated Raman scattering, when the time width of the pulse is shorter than the period of the vibration. The ultrashort light pulse has a finite frequency width related to the Fourier transformation of the time width, according to the energy-time uncertainty relation. 

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. 

In single-scale filtering, basis functions are of a fixed resolution and all basis functions have the same localization in the time-frequency domain. For example, frequency domain filtering relies on basis functions localized in frequency but global in time, as shown in Fig. 7b. Other popular filters, such as those based on a windowed Fourier transform, mean filtering, and exponential smoothing, are localized in both time and frequency, but their resolution is fixed, as shown in Fig. 7c. Single-scale filters are linear because the measured data or basis function coefficients are transformed as their linear sum over a time horizon. A finite time horizon results infinite impulse response (FIR) and an infinite time horizon creates infinite impulse response (HR) filters. A linear filter can be represented as... 

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