Fast Fourier Transform

It was shown in Section I. l(i) that the system impedance is defined as the ratio of Laplace transforms [Eq. (6)], of potential and current. In general, the transformation parameter is complex, 5 = v -i- jto. The imaginary Laplace transform 

In general, the perturbing signal may have an arbitrary form. However, in practice, the most often used perturbation signals are ° (l) pulse, (2) noise, and (3) sum of sine waves. 

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 

White noise, that is, noise consisting of a continuous spectrum of frequencies (or a computer-generated pseudo-random white noise), may be used as a perturbation signal in practical impedance measurements. However, single-frequency components obtained by the FFT have relatively low amplitudes and a long data acquisition time is necessary to 

This technique was introduced and used extensively by D. E. Smith In it, the perturbation signal is composed of a 

As the number of points to be calculated increases beyond about 10,000, the calculation time for a spectmm can become prohibitive, even for very fast present-day computers, for which that computation can take many hours. Prior to the development of fast, readily available computers, this problem was especially annoying and did not appear to be resolvable until about 1966. At that time, Forman [2] published a paper on the application of the fast Fourier transform technique to Fourier spectrometry. This technique had been described in the literature by Cooley and Tukey [3] one year earlier. This algorithm extended the use of Fourier transform spectrometry to encompass high-resolution data in all regions of the infrared spectmm. It is described in the next section. 

The fast Fourier transform (FFT) is an algorithm that was described by Cooley and Tukey [3] in which the number of necessary computations is drastically reduced when compared to the classical Fourier transform. To understand the algorithm, the interferogram and the spectmm should be regarded as the complex pair [4] 

The point of departure for the EFT from the continuous Fourier transform is the discrete Fourier transform (DFT). The DFT of an interferogram of N points to produce a spectmm of N points may be written as  

From this equation it can be seen that each discrete value of B r) requires N complex multiplications and N complex additions. Since there are N terms of B r),N complex multiplications are required, as stated in Section 4.1. The FFT is based on the idea that Eq. 4.13 can be expressed in general matrix form, and that matrix can be factored in a manner that will reduce the overall number of computations. 

The Cooley-Tukey algorithm is general and can be applied to any Fourier transform, but the computation is greatly simplified if AT is a base 2 number, that is, AT = 2 , where a is a positive integer. The factorization procedure may be demonstrated easily for the case when AT = 4 = 2. In this example, Eq. 4.13 becomes 

In 1965, Cooley and Tukey proposed an algorithm in which N is factored to some power N = 2 or N = 4, p and q being the powers or exponents. This 

In addition to the Fourier transform, there are the Laplace transform and the Melhn transform. They are all called integral transforms. The general formula is 

Given a second-order differential equation (d y/dbc ) + ay = 0, a suitably restricted solution may be found in the form of a power series/(jc) = n n where a and a are both coefficients. In 1807, Fourier proposed a theorem that the power series can be expressed as a combination of JZ cosm and 

He presented a paper to the French Academy and expanded it further by saying that the function/(x) can be expressed in terms of sines and cosines. The Academy 

The Development of Mathematics, New York McGraw-Hill Book Co., 

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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) ... 

Pig. 4. Photo dissociation of ArHCl. Left hand side the number of force field evaluations per unit time. Right hand side the number of Fast-Fourier-transforms per unit time. Dotted line adaptive Verlet with the Chebyshev approximation for the quantum propagation. Dash-dotted line with the Lanczos iteration. Solid line stepsize controlling scheme based on PICKABACK. If the FFTs are the most expensive operations, PiCKABACK-like schemes are competitive, and the Lanczos iteration is significantly cheaper than the Chebyshev approximation. 

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. 

The Fourier Series, Fourier Transform and Fast Fourier Transform... 

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. 

This process is continued until there is only one component. For this reason, the number N is taken as a power of 2. The vector [yj] is filled with zeroes, if need be, to make N = 2 for some p. For the computer program, see Ref. 26. The standard Fourier transform takes N operations to calculation, whereas the fast Fourier transform takes only N log2 N. For large N, the difference is significant at N = 100 it is a factor of 15, but for N = 1000 it is a factor of 100. 

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. 

The phrase full Fast Fourier Transform (FFT) signature is usually applied to the vibration spectrum that uniquely identifies a machine, component, system, or subsystem at a specific operating condition and time. It provides specific data on every frequency component within the overall frequency range of a machine-train. The typical frequency range can be from 0.1 to 20,000 Hz. 

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