Dirac delta function — Fourier transform

We may also evaluate the Fourier transform <5( ) of the Dirac delta function... 

Fourier transforms boxcar function 274 Cauchy function 276 convolution 272-273 Dirac delta function 277-279 Gaussian function 275-276 Lorentzian function 276-277 shah function 277-279 triangle function 275 fraction, rational algebraic 47 foil width at half maximum (FWHM) 55, 303... 

The Dirac delta function may be represented as a Fourier transform over time ... 

To connect the equations in (B.l) through the Fourier-Laplace transform, we need to define suitable complex contours to make the transforms convergent. Specifically we identify the contours C by the lines in upper and lower complex planes defined by CU ( id — oo — id + oo), where d > 0 may be arbitrary. Using the Heaviside function, 0(f), and the Dirac delta function, 5(f), we can characterize positive and negative times (with respect to f = 0) as linked with appropriate contours C as... 

The Kubo relation (25) of section 2.1 is obtained as the Fourier transform of (70). The term linear in V is the retarded two-time Green s function, first introduced in this context by Bogoliubov and Tyablikov [30]. The identification with Green s functions stems from the presence of the Heaviside step function that in part were introduced to allow integration over the full time interval and whose time derivative gives a Dirac delta function. For instance, t — to)U(tfo) is a solution of the inhomogeneous equation [31]... 

In electrotechnics, it is preferred to test the response of a system by a needle-like pulse of voltage or current, which is mathematically represented by the Dirac delta function. As can be seen from Eq. (16.9), since the Fourier transform of the Dirac delta function is 1, so Fj(t >) = 1, the Fourier transform of the system characteristics, Gitco) is directly the Fourier transform of the output response, Foiico). Of course this implies that also the corresponding functions in the time domains are equal. 

If the injection of a single occurs instantly, which corresponds to an initial profile of a Dirac delta function the peak should leave the column as a Dirac delta function delayed by the retention time tR. We denote the concentration profile now with c(t), thus the injected concentration profile as c,(f) and the eluted, outputted concentration profile as Co(t). The corresponding Fourier transforms we denote with capital letters, as before. Due to the needle-like shape of the injected amount we have Ci(t) = 8(t). Equation (16.9) now will be read as... 

The Forman phase correction algorithm, presented in Chap. 2, is shown in Fig. 3.6. Initially, the raw interferogram is cropped around the zero path difference (ZPD) to get a symmetric interferogram called subset. This subset is multiplied by a triangular apodization function and Fourier transformed. With the complex phase obtained from the FFT a convolution Kernel is obtained, which is used to filter the original interferogram and correct the phase. Finally the result of the last operation is Fourier transformed to get the phase corrected spectrum. This process is repeated until the convolution Kernel approximates to a Dirac delta function. 

Let us show how the standard diffusion equation (3.16) and anomalous diffusion equation (3.214) emerge as a result of long-time large-scale limit of a CTRW described by (3.53). If we use the Dirac delta-function as the initial condition, then Po(fc) = 1, and the Fourier-Laplace transform of p x, t), (3.53), is given by... 

F(k) and /(r) are said to be a Fourier transform pair. All the Fourier transform pairs we shall need here are given in Table A.l. In particular, the Fourier representation of the three-dimensional Dirac delta function is... 

The simplest approach to describe the ion dynamics in disordered materials is to assume completely uncorrelated, random ion movements [42]. In this case, the jump of an ion moving in a forward direction is only correlated to itself, thus the velocity autocorrelation function is proportional to a Dirac Delta function at = 0 (see Fig. la). The complex conductivity obtained by Fourier transform is then independent of frequency. This means that the real part of the conductivity shows no dispersion and at all frequencies the ac conductivity < (6 ) can be identified with the dc conductivity. By contrast, conductivity spectra of most ion-conducting materials show that o (o) varies with frequency. This is schematically illustrated... 

Fourier transforms provide a way of representing the Dirac delta function, 5 x), which is defined by the conditions (x) = 0 if x 0, and... 

It is a property of Fourier transform mathematics that multiplication in one domain is equivalent to convolution in the other. (Convolution has already been introduced with regard to apodization in Section 2.3.) If we sample an analog interferogram at constant intervals of retardation, we have in effect multiplied the interferogram by a repetitive impulse function. The repetitive impulse function is in actuality an infinite series of Dirac delta functions spaced at an interval 1 jx. That is,... 

The Dirac delta comb and its Fourier transform are referred to in Section 4.4.2. An infinite train of the Dirac delta functions at intervals of a on the x axis is called the Dirac delta comb and denoted by dJ (x). This function is defined as... 

The Fourier transform of is proportional to the Dirac delta function. Argue that the integral... 

Now, we may recall the representation III of the autocorrelation function because its Fourier transform leads to the well-known Franck-Condon progression of delta Dirac peaks appearing in the pioneering work of Marechal and Witkowski [7]. In this representation III, the general autocorrelation function (2) takes the form... 

On the other hand, the undamped autocorrelation function (17) we have obtained within the standard approach avoiding the adiabatic approximation must lead after Fourier transform to spectral densities involving very puzzling Dirac delta peaks given by... 

The Fourier transform of an infinite short pulse function h(t) = Kb(t), where 5(f) is Dirac s delta function, equals//(jco) = K, that is, it contains all the frequencies with the same amplitude K. Such a function caimot be realized in practice and must be substituted by a pulse of a short duration At. However, such a function does not have uniform response in the Fourier (i.e., frequency) space. The Fourier transform of such a function, defined as h(t) = 1 for r = 0 to To and h(t) = 0 elsewhere, equals... 

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