Fourier transform pair

These transform equations may be written more symmetrically by putting c k) (l/x/27i)g(fe). Then we have 

Two functions / and g related in the above way are called a pair of Fourier transforms that is, g is the Fourier transform of / and vice versa. The simplest 

Fourier transform pairs in mathematics are the sine transforms 

The Fourier transform pairs that we used in Chapters 16, 17, 19, and 20 may be summarized as follows. 

The inverse transform for connecting the frequency domain amplitude to the time domain signal is very similar  

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The interferogram and the spectmm are related by the Fourier-transform pair ... 

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 homogeneous turbulence, the velocity spectrum tensor is related to the spatial correlation function defined in (2.20) through the following Fourier transform pair ... 

Similar Fourier transform pairs relate the spatial correlation functions defined in (3.40) and (3.41) to corresponding cospectra t) and t), respectively. 

Scale factors can be used in various ways to define Fourier transform pairs. We adopt the symmetrical convention... 

An alternative symmetrical convention that has gained popularity specifies the Fourier transform pair g(x) and G( ) to be related by... 

The persistence of the fluctuating local fields before being averaged out by molecular motion, and hence their effectiveness in causing relaxation, is described by a time-correlation function (TCF). Because the TCF embodies all the information about mechanisms and rates of motion, obtaining this function is the crucial point for a quantitative interpretation of relaxation data. As will be seen later, the spectral-density and time-correlation functions are Fourier-transform pairs, interrelating motional frequencies (spectral density, frequency domain) and motional rates (TCF, time domain). 

The integral transforms given in Equation 10.9 can now be approximated by discrete sums, so that the Fourier transform pairs now are described by the equations... 

More commonly, the Fourier transform pair may be defined using two arbitrary constants a and b thus,... 

Equation 8 mathematically describes the interferogram (intensity versus optical retardation, which is a function of time) and that which is physically measured by the spectrometer. It represents one half of a cosine Fourier transform pair, the other being... 

The two functions f t) and F w) are said to comprise Fourier transform pairs. 

The NMR signal obtained from the resonating nuclei after the sample has been irradiated by the pulse is the so-called free-induction decay curve. This curve consists of peaks and valleys. The spectrometer samples the free-induction decay curve at set time intervals and records the data, which are in a time domain. NMR spectra, however, are normally given in terms of frequency and therefore the spectrum must be transformed by use of the Fourier transform pairs ... 

For s = 0 and for s - as, the interferogram given by Eq. (A 1.2) exhibits the properties quoted in Section 3.2. The functions I (v) and I (s) are a Fourier transform pair well known from the theory of forced vibrations and resonance of damped oscillators. One reason why they are very useful to demonstrate problems of Fourier transform spectroscopy is that the different contributions to I v) and to I (s) can be studied separately (cf. Fig. 12). 

The structure factor and the density autocorrelation function are Fourier transform pairs thus. 

Two further aspects of Fourier transformation with respect to NMR data must be mentioned. With quadrature detection a complex Fourier transformation must be performed, there is a 90° phase shift between the two detectors and the sine and cosine dependence of the sequential or simultaneous detected data points are different. In addition because the FID is a finite number of data points, the integral of the continuous Fourier transform pair must be replaced by a summation. 

Since the complex signal s-H(t) is proportional M+(t) the complex Fourier transform pair s+(t), detector 2... 

The two functionsand F(w) are said to comprise Fourier transform pairs. As discussed previously with regard to sampling theory, real analytical signals are band-limited. The Fourier equations therefore should be modified for practical use as we cannot sample an infinite number of data points. With this practical constraint, the discrete forward complex transform is given by... 

Make a plot of the Fourier transform pairs in the preceding problem. 

Verify the following Fourier transform pairs in three dimensions, where/(r) in every case is a function of the length r only. 

These two equations are interconvertible and are known as a Fourier-transform pair. The first shows the variation in power density as a function of difference in pathlength, which is an interference pattern. The second shows the variation in intensity as a function of wavenumber. Each can be converted into the other by the mathematical method of Fourier transformation. Do not worry if your knowledge of calculus is not up to these equations. It is not necessary to have a detailed knowledge of the mathematics involved in order to carry out experiments using an FT-IR spectrometer ... 

Figure 13.29 (a) A single space mask pattern and (b) its corresponding Fraunhofer diffraction pattern. These are Fourier transform pairs. (Reprinted with permission from Taylor Francis Group LLC. )... 

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