FFT method

Precise comparison of the two methods of computing a convolution requires careful attention to details such as whether aliasing, computing the ends of the function, matching array lengths to powers of 2, or whatever other FFT base is employed. It is apparent, however, that when Na = Nb, the FFT method is superior. When Na Nb, the FFT method involves considerable unnecessary computation. In instrumental resolution studies, one of the two functions typically has a considerably smaller extent than the other that is, the response function is usually narrow... 

This Fourier transform process was well known to Michelson and his peers but the computational difficulty of making the transformation prevented the application of this powerful interferometric technique to spectroscopy. An important advance was made with the discovery of the fast Fourier transform algorithm by Cooley and Tukey 29) which revived the field of spectroscopy using interferometers by allowing the calculation of the Fourier transform to be carried out rapidly. The fast Fourier transform (FFT) has been discussed in several places 30,31). The essence of the technique is the reduction in the number of computer multiplications and additions. The normal computer evaluation requires n(n — 1) additions and multiplications whereas the FFT method only requires (n logj n) additions and multiplications. If we have a 4096-point array to Fourier transform, it would require (4096) (4095) or 16.7 million multiplications. The FFT allows us to reduce this to... 

Figure 2 shows the results of the IF analysis, revealing strong modulation at 50 on1 and other low frequency modes, similar to observations by Champion et al. [10][11] For comparison, the IF was also calculated using a sliding window FFT method, yielding similar results as shown in Figure 2. The observation of these low frequency modes is perhaps the most important result of the study. The 50 cm 1 mode in particular has been identified with the doming motion of the heme [12] and the lower frequencies can be correlated to the globin...

From an experimental point of view, it may be worth mentioning that the artificial membrane is reasonably stable, though only for a short time. It is therefore also a clear example of an object that should be studied by means of a quick method. The (small-amplitude) potential step method has been applied successfully [113], as has the FFT method described in Sect. 2.5.5. With the latter, the analysis in terms of an equivalent circuit like in Fig. 29 was demonstrated quite spectacularly [114]. 

A good computation of the response is now available based on extensive Monte-Carlo simulations of the COMPTEL instrument for various assumed input spectra. The convolution in equation (1) is time-consuming it can be done by FFT methods but only when the field studied is not too large (see discussion on large-scale imaging below). 

In this paper, based on the measured wave data within one year at a fixed point in Andaman sea, Myanmar, the wave frequency spectrum and the directional spectrum are calculate by Fast Fourier Transform (FFT) method, the wave spectrum characteristics of this engineering sea area are obtained, it can provide scientific basis for engineering construction. 

Phillips JR, White JA (1997) Piecorrected-FFT method for electrostatic analysis of complicated 3D structures. IEEE Trans Comput Aided Des Integr Circ Syst 16(10) 1059-1072... 

In fully periodic boundary conditions, the Ewald method can be accelerated to a computation time scaling of NlogN using fast Fourier transformation (FFT) methods by replacing the charges with a regular mesh. Various mesh-based methods exist, such as P M, PME, or SPME, but P M is known to be the computationally optimal variant [10]. 

The maximum entropy method (MEM) is developed to obtain the maximum spectrum information from the limited number of data. It enables us to estimate the power spectrum without an EFT (fast Fourier transform) using a distinct Fourier transform (DFT). The main problems in the FFT method are the so-called spectrum leaks from other frequencies, i.e., in addition to the true range of frequencies, the power spectrum also contains components at other unwanted frequencies, which leads to errors in spectral analysis. To demonstrate the spectrum leaks associated with FFT, suppose that an original continuous signal is the one shown in Fig. 37a. Its Fourier transform power spectrum is shown in Fig. 37b and has one sharp peak. From the limited number of data (c) the FFT is obtained as (d), which is still similar. However, from another set of limited data but half a period longer than the data (c), we obtain a spectrum with a few small peaks. This is the spectrum leak. To cope with this, a windowed Fourier transform shown in (g) with a lenslike window has to be applied to improve the spectrum to (h). 

An attractive feature of spectral methods is the fact that they exhibit exponential convergence when the spatial grid spacing is sufficiently small. In addition, one can make use of fast Fourier transform (FFT) methods. However, there are also a number of disadvantages. First, the traditional spectral methods are limited to very simple geometries. Karniadakis and Henderson [135,136] have alleviated this problem by developing spectral element methods in which spectral methods are combined with finite element methods. 

Equation (3.12) is then solved without further approximation and within a model including all multiphoton processes. The solution of (3.8) is obtained by the second-order differencing fast Fourier transform (SOD-FFT) method [287, 288, 299]. The relevant PESs for the multiphoton ionization process considered here are the X, A, b, (2) 77g, and ion states. The PESs and the transition dipole moments are obtained from ab-initio data (Sect. 3.1.5 and [328]). 

The exact dHvA oscillation contains many dHvA frequencies F, (i- 1, 2, 3,. ..) or cross-sectional areas F, and becomes a sum of their contributions, which are analyzed by the fast Fourier transformation (FFT) method. The amplitude At corresponds to the amplitude in the FFT spectrum. Figure 6 shows the dHvA oscillation and its FFT spectrum for a field along the (111) direction of the cubic crystal LaSn3 at 0.5 K (Umehara et al. 1991a). From the FFT spectrum we can see many dHvA oscillations due to harmonics or sums and differences of the several dHvA frequencies. 

We also involved in the data collected from MRS with FFT method (Fast Fourier Transform) on Matlab. In these studies, we saw that MRS performed like frequencies and divided the signal of MRS into each epoch (1 epoch =60 seconds in real time). During the signal processing, we found that MRS signals have several special frequencies. These frequencies appeared randomly in our subjects when they sleep. 

To evaluate HRV, several measures have been proposed. These measures are roughly classifiable into time domain analysis [5], frequency domain analysis, and nonlinear and fractal analysis [5]. Time domain analysis includes tone-entropy method [6]. Nonlinear and fractal analysis include de-trended fluctuation analysis (DFA) [7]. Frequency domain analysis is based on estimation of the power spectrum of RRI series. Depending on the estimation method of power spectrum, frequency domain analysis is classified into FFT method [8], AR model method [9], maximum entropy method [10], and complex de-modulation method [11]. Akselrod et al. [3] investigated the relation between spectral component of HRV and ANS activity [12]. They classified spectral component of HRV into a high-frequency (HF) band of 0.14-0.4 Hz, a low-frequency (LF) band of 0.04-0.14 Hz, a very-low-frequency (VLF) band of 0.003-0.04 Hz, and a ultra-low-frequency (ULF) band under 0.003 Hz. They further show that LF and HF components are affected from both sympathetic and parasympathetic nervous system activity and the parasympathetic nervous system activity, respectively [12]. Furthermore, VLF and ULF components are affected by the thermoregulation system [13],... 

Complete program for computing the DLT of Equation 2.54 by the FFT method-subroutine FLT. 

Further reading section for details of Fourier spectroscopy, including its application in NQR. To illustrate the FFT method in pulsed spectroscopy. Figure 5 shows a typical NQR spectrum of powder RDX (C3H O N ), obtained from a free induction decay (FID). 

Since the potential matrix V is diagonal in the grid representation, the multiplication with the exponential operators involving the potential is straightforward. The operation with the exponential kinetic energy operator is, as we have seen, equally as straightforward if we use the FFT method. Another time-propagation method is based upon the Lanczos iterative scheme where each time-step requires a number of operations with the hamiltonian operator on the wavefunction... 

---