Fourier domain

Low and High frequency can be restored by use of a deconvolution algorithm that enhances the resolution. We operate an improvement of the spectral bandwidth by Papoulis deconvolution based essentially on a non-linear adaptive extrapolation of the Fourier domain. 

Since the integral is over time t, the resulting transform no longer depends on t, but instead is a function of the variable s which is introduced in the operand. Hence, the Laplace transform maps the function X(f) from the time domain into the s-domain. For this reason we will use the symbol when referring to Lap X t). To some extent, the variable s can be compared with the one which appears in the Fourier transform of periodic functions of time t (Section 40.3). While the Fourier domain can be associated with frequency, there is no obvious physical analogy for the Laplace domain. The Laplace transform plays an important role in the study of linear systems that often arise in mechanical, electrical and chemical kinetic systems. In particular, their interest lies in the transformation of linear differential equations with respect to time t into equations that only involve simple functions of s, such as polynomials, rational functions, etc. The latter are solved easily and the results can be transformed back to the original time domain. 

It may thus be necessary to calculate the Fourier transform of the measured signal to return to the domain of interpretation, here wavelength or wavenumber. In FTIR the signal is measured in the displacement domain 6 and transformed to the wavelength or wavenumber domain by a Fourier transform. Because the wavelength domain is the Fourier transform of the displacement domain, and vice versa, we say that the spectrum is measured in the Fourier domain. 

Instead of considering a spectrum as a signal which is measured as a function of wavelength, in this chapter we consider it measured as a function of time. The frequency scale in the Fourier domain is given in cycles (v) per second (Hz) or radians (to) per second (s" ). These units are related by 1 Hz = 2k s" . 

Fig. 40.9. Relationship between measurement time (2T ), digitization interval and the maximum and minimal observable frequencies in the Fourier domain.

From the convolution theorem it follows that the convolution of the two triangles in our example can also be calculated in the Fourier domain, according to the following scheme ... 

Fig. 40.17. Convolution in the time domain offlf) with h t) carried out as a multiplication in the Fourier domain, (a) A triangular signal (w, = 3 data points) and its FT. (b) A triangular slit function h(t) (wi/, = 5 data points) and its FT. (c) Multiplication of the FT of (a) with that of (b). (d) The inverse FT of (c).

These four steps are illustrated in Fig. 40.17 where two triangles (array of 32 data points) are convoluted via the Fourier domain. Because one should multiply Fourier coefficients at corresponding frequencies, the signal and the point-spread function should be digitized with the same time interval. Special precautions are needed to avoid numerical errors, of which the discussion is beyond the scope of this text. However, one should know that when J(t) and h(t) are digitized into sampled arrays of the size A and B respectively, both J(t) and h(t) should be extended with zeros to a size of at least A + 5. If (A -i- B) is not a power of two, more zeros should be appended in order to use the fast Fourier transform. 

As a consequence, a moving average in the time domain is a multiplication in the Fourier domain, namely ... 

Hadamard transform [17], For example the IR spectrum (512 data points) shown in Fig. 40.31a is reconstructed by the first 2, 4, 8,. .. 256 Hadamard coefficients (Fig. 40.38). In analogy to spectrometers which directly measure in the Fourier domain, there are also spectrometers which directly measure in the Hadamard domain. Fourier and Hadamard spectrometers are called non-dispersive. The advantage of these spectrometers is that all radiation reaches the detector whereas in dispersive instruments (using a monochromator) radiation of a certain wavelength (and thus with a lower intensity) sequentially reaches the detector. 

Normally, discrete convolution involves shifting, adding, and multiplying —a laborious and time-consuming process, even in a large digital computer. The convolution theorem presents us with an alternative. It reveals the possibility of computing in the Fourier domain. What are the trade-offs between the two methods ... 

The principal thrust of the present chapter is to describe methods that facilitate superior restoration through the use of bounds. Sometimes bounds are implemented in the object domain, for example, positivity or finite extent, and sometimes in the Fourier domain, for example, band limitedness. The effectiveness of this type of prior knowledge was apparent from the early work with constraints. A number of researchers therefore focused their efforts on making use of all possible combinations of constraints and partial data. What could be more suggestive than direct application of bounds to trial solutions by truncation, first in the object domain and then in the Fourier domain ... 

Blass (1976a) and Blass and Halsey (1981) discuss data acquisition for a continuous scanning spectrometer in detail. The principal concept is that as a system scans a spectral line at some rate, the resulting time-varying signal will have a distribution of frequency components in the Fourier domain. 

Filter out any fast ripple of period Aw/2 in V(z), due to interference with internal reverberations in the lens (Fig. 8.5(b)). This may be achieved most simply by convolving with a rectangular function of length Aw/2. This is known as a moving average filter it is equivalent to a sine filter in the Fourier domain, but is computationally somewhat more efficient. Because of its period the ripple removed at this stage is sometimes called water ripple. 

One of the fundamental theorems of Fourier transforms states that multiplying two functions in one Fourier domain is equivalent to convoluting the two functions in the other domain [60], The FT spectrum thus has a lineshape corresponding to the Fourier transformation of >(<5), which is the sine function... 

Fast Quantitative Docking Using Fourier Domain Correlation Techniques. 

A critical difference between the Fourier transform defined in Equation 10.9 and the wavelet transform defined in Equation 10.22 is the fact that the latter permits localization in both frequency and time that is, we can use the equation to determine what frequencies are active at a specific time interval in a sequence. However, we cannot get exact frequency information and exact time information simultaneously because of the Heisenberg uncertainty principle, a theorem that says that for a given signal, the variance of the signal in the time domain a2, and the variance of the signal in the frequency (e.g., Fourier) domain c2p are related... 

Betty, K.R. and Horlick, G., A simple and versatile Fourier domain digital filter, Appl. 

A widely used approach to extract information on protein secondary structure from infrared spectra is linked to computational techniques of Fourier deconvolution. These methods decrease the widths of infrared bands, allowing for increased separation and thus better identification of overlapping component bands present under the composite wide contour in the measured spectra [705]. Increased separation can also be achieved by calculating the nth derivative of the absorption spectrum, either in the frequency domain or though mathematical manipulations in the Fourier domain [114], An example is the method of Susi [775] which uses second derivative FT-IR spectra recorded in D20 for comparison with similar spectra derived from proteins with known structure. These methods have not yielded quantitative results that are more accurate than those obtained with methods that do not use deconvolution. 

The Fourier transform is an important image processing tool that is used to decompose an image into its sine and cosine components. The output of the transformation represents the image in the Fourier or frequency domain, while the input image is the spatial domain equivalent. In the Fourier domain image, each point represents a particular frequency contained in the spatial domain image. Loren et al. 

The analysis of outlet peaks is based on the model of processes in the column. Today the Kubi n - Kucera model [14,15], which accounts for all the above-mentioned processes, as long as they can be described by linear (differential) equations, is used nearly exclusively. Several possibilities exist for obtaining rate parameters of intracolumn processes (axial dispersion coefficient, external mass transfer coefficient, effective diffusion coefficient, adsorption/desorption rate or equilibrium constants) from the column response peaks. The moment approach in which moments of the outlet peaks are matched to theoretical expressions developed for the system of model (partial) differential equations is widespread because of its simplicity [16]. The today s availability of computers makes matching of column response peaks to model equations the preferred analysis method. Such matching can be performed in the Laplace- [17] or Fourier-domain [18], or, preferably in the time-domain [19,20]. 

Fig. 12. Radiofrequency and gradient pulse sequence for velocity exchange spectroscopy (VEXSY) in which successive PGSE pulse pairs (G, and G2X separated by a delay time t , are applied. For an unambiguous correlation, Gj and Gj are required to be collinear. Note the two orthogonal Fourier domains represented (schematically) by r, and tj- Fourier transformation with respect to the acquisition time leads to a third spectral dimension.

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