Fast fourier transform, use

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). 

The collection efficiencies are most easily determined by calibration experiments on redox systems. In practice, the deconvolution of (10) is carried out by FFT (fast Fourier transform) using the classical equation N((o) = Fourier transform... 

In fig. 2 an ideal profile across a pipe is simulated. The unsharpness of the exposure rounds the edges. To detect these edges normally a differentiation is used. Edges are extrema in the second derivative. But a twofold numerical differentiation reduces the signal to noise ratio (SNR) of experimental data considerably. To avoid this a special filter procedure is used as known from Computerised Tomography (CT) /4/. This filter based on Fast Fourier transforms (1 dimensional FFT s) calculates a function like a second derivative based on the first derivative of the profile P (r) ... 

This scheme requires the exponential only of matrices that are diagonal or transformed to diagonal form by fast Fourier transforms. Unfortunately, this matrix splitting leads to time step restrictions of the order of the inverse of the largest eigenvalue of T/fi. A simple, Verlet-like scheme that uses no matrix splitting, is the following ... 

Furthermore, one may need to employ data transformation. For example, sometimes it might be a good idea to use the logarithms of variables instead of the variables themselves. Alternatively, one may take the square roots, or, in contrast, raise variables to the nth power. However, genuine data transformation techniques involve far more sophisticated algorithms. As examples, we shall later consider Fast Fourier Transform (FFT), Wavelet Transform and Singular Value Decomposition (SVD). 

Fast Fourier Transformation is widely used in many fields of science, among them chemoractrics. The Fast Fourier Transformation (FFT) algorithm transforms the data from the "wavelength" domain into the "frequency" domain. The method is almost compulsorily used in spectral analysis, e, g., when near-infrared spectroscopy data arc employed as independent variables. Next, the spectral model is built between the responses and the Fourier coefficients of the transformation, which substitute the original Y-matrix. 

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 fast Fourier transform (FFT) is used to calculate the Fourier transform as well as the inverse Fourier transform. A discrete Fourier transform of length N can be written as the sum of two discrete Fourier transforms, each of length N/2. 

The Fourier analyzer is a digital deviee based on the eonversion of time-domain data to a frequeney domain by the use of the fast Fourier transform. The fast Fourier transform (FFT) analyzers employ a minieomputer to solve a set of simultaneous equations by matrix methods. 

From the time function F t) and the calculation of [IT], the values of G may be found. One way to calculate the G matrix is by a fast Fourier technique called the Cooley-Tukey method. It is based on an expression of the matrix as a product of q square matrices, where q is again related to N by = 2 . For large N, the number of matrix operations is greatly reduced by this procedure. In recent years, more advanced high-speed processors have been developed to carry out the fast Fourier transform. The calculation method is basically the same for both the discrete Fourier transform and the fast Fourier transform. The difference in the two methods lies in the use of certain relationships to minimize calculation time prior to performing a discrete Fourier transform. 

If further resolution is necessary one-third octave filters can be used but the number of required measurements is most unwieldy. It may be necessary to record the noise onto tape loops for the repeated re-analysis that is necessary. One-third octave filters are commonly used for building acoustics, and narrow-band real-time analysis can be employed. This is the fastest of the methods and is the most suitable for transient noises. Narrow-band analysis uses a VDU to show the graphical results of the fast Fourier transform and can also display octave or one-third octave bar graphs. 

Frequency-domain data are obtained by converting time-domain data using a mathematical technique referred to as Fast Fourier Transform (FFT). FFT allows each vibration component of a complex machine-train spectrum to be shown as a discrete frequency peak. The frequency-domain amplitude can be the displacement per unit time related to a particular frequency, which is plotted as the Y-axis against frequency as the X-axis. This is opposed to time-domain spectrums that sum the velocities of all frequencies and plot the sum as the Y-axis against time... 

Most of the early vibration analysis was carried out using analog equipment, which necessitated the use of time-domain data. The reason for this is that it was difficult to convert time-domain data to frequency-domain data. Therefore, frequency-domain capability was not available until microprocessor-based analyzers incorporated a straightforward method (i.e.. Fast Fourier Transform, FFT) of transforming the time-domain spectmm into its frequency components. 

The frequency-domain format eliminates the manual effort required to isolate the components that make up a time trace. Frequency-domain techniques convert time-domain data into discrete frequency components using a mathematical process called Fast Fourier Transform (FFT). Simply stated, FFT mathematically converts a time-based trace into a series of discrete frequency components (see Figure 43.19). In a frequency-domain plot, the X-axis is frequency and the Y-axis is the amplitude of displacement, velocity, or acceleration. 

Most predictive-maintenance programs rely almost exclusively on frequency-domain vibration data. The microprocessor-based analyzers gather time-domain data and automatically convert it using Fast Fourier Transform (FFT) to frequency-domain data. A frequency-domain signature shows the machine s individual frequency components, or peaks. 

When performing optical simulations of laser beam propagation, using either the modal representation presented before, or fast Fourier transform algorithms, the available number of modes, or complex exponentials, is not inhnite, and this imposes a frequency cutoff in the simulations. All defects with frequencies larger than this cutoff frequency are not represented in the simulations, and their effects must be represented by scalar parameters. 

The most notable advance in computational crystallography was the availability of methods for rehning protein structures by least-squares optimization. This developed in a number of laboratories and was made feasible by the implementation of fast Fourier transform techniques [32]. The most widely used system was PROLSQ from the Flendrickson lab [33]. 

In general, the topology of interprocessor communication reflects both the structure of the mathematical algorithms being employed and the way that the wave packet is distributed. For example, our very first implementation of parallel algorithms in a study of planar OH - - CO [47] used fast Fourier transforms (FFTs) to compute the action of 7, which also required all-to-all communication but in a topology that is very different from the simple ring-like structure shown in Fig. 5. 

Fast Fourier transform (FFT) A procedure for carrying out Fourier transformation at high speed, with a minimum of storage space being used. Field frequency lock The magnitude of the field Bq is stabilized by locking onto the fixed frequency of a resonance in the solution, usually of the solvent. 

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. 

This is not a fundamental mathematics bookt nor is it intended to serve a textbook for a specific course, but rather as a reference for students in chemistry and physics at all university levels. Although it is not computer-based, I have made many references to current applications - in particular to try to convince students that they should know more about what goes on behind the screen when they do one of their computer experiments. As an example, most students in the sciences now use a program for the fast Fourier transform. How many of them have any knowledge of the basic mathematics involved ... 

---