Fourier analyzer

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. 

If we Fourier analyze the Maxwell equations (2.1)-(2.4), with = 0, and assume that the operations of integration and differentiation may be interchanged, we obtain... 

Our fundamental task is to construct solutions to the Maxwell equations (3.1)—(3.4), both inside and outside the particle, which satisfy (3.7) at the boundary between particle and surrounding medium. If the incident electromagnetic field is arbitrary, subject to the restriction that it can be Fourier analyzed into a superposition of plane monochromatic waves (Section 2.4), the solution to the problem of interaction of such a field with a particle can be obtained in principle by superposing fundamental solutions. That this is possible is a consequence of the linearity of the Maxwell equations and the boundary conditions. That is, if Ea and Efc are solutions to the field equations,... 

There is one more conceptual step involved in the formal treatment. The perturbation <[> is Fourier analyzed, which means that it is constructed from the Fourier components 4>(k,t) with wavelength A = 2-n/k. 8cv is transformed in the same way. Explicitly,... 

Equations of type (12.30) can be used to describe the kinetics of the spinodal decomposition process. If an arbitrary, spontaneous concentration fluctuation is Fourier analyzed, one finds that for... 

The direct method (DM) for solution of this set of equations was proposed by Atherton et al. [5], and in a somewhat a modified form by Dickinson and Gelinas [4] who solved r sets of equations each of size In consisting of Eq. (1) coupled with a particular j—value of Eq. (2). Shuler and coworkers [5] took an alternative approach in the Fourier Amplitude method in which a characteristic periodic variation is ascribed to each a, and the resulting solution of (1) is Fourier analyzed for the component frequencies. These authors estimate that 1.2r2 5 solutions of Eq. (1) together with the appropriate Fourier analyses are required for the complete determination of the problem. Since even a modest reaction mechanism (e.g. in atmospheric chemistry or hydrocarbon cracking or oxidation) may easily involve 100 reactions with several tens of species, it is seen that a formidable amount of computation can result. 

Fourier analyze all physical quantities parallel to the surface, in the x-y plane. For example, a Fourier component of the induced charge density becomes... 

Single-frequency Fourier analyzers meike use of the orthogonality of sines and cosines to determine the complex impedance representing the ratio of the response to a single-frequency input signal. A brief outline of the approach is presented in this section. 

Through the use of more than one diode and tuning them to several analyte lines, multielement determinations and the use of an internal standard become simple. In AAS work the latter also enables instrumental drift to be overcome and/ or the short-term precision to be improved as well. For the detection, all that is required is pulsing of the primary source and the use of lock-in amplification or a Fourier analyzer. 

Integration is relatively insensitive to noise, but is very sensitive to bias or offset, such as may result from an incorrect zero setting of the measuring instrument, or from some other phenomenon affecting the baseline. That integration is relatively insensitive to noise is readily seen when we consider noise in terms of its Fourier-analyzed components, i.e.,... 

The heterodyne measurement is performed as follows. The dye laser excites the trapped ion with frequency (Ol while the fluorescence is observed in a direction of about 54 to the exciting laser beam (see Fig. 2). However, both the observation direction and the laser beam are in a plane perpendicular to the symmetry axis of the trap. Before reaching the ion, a fraction of this laser radiation is removed with a beamsplitter and then frequency shifted (by 137 MHz with an acousto-optic modulator (AOM)) to serve as the local oscillator. The local oscillator and fluorescence radiations are then overlapped and simultaneously focused onto the photodiode where the initial frequency mixing occurs. The frequency difference signal is amplified by a narrow band amplifier and then frequency down-converted to 1 kHz so that it could be analyzed by means of a fast Fourier analyzer (FFT). The intermediate frequency for this mixing of the signal was derived from the same frequency-stable synthesizer which was used to drive the accousto-optic modulator producing the sideband of the laser radiation so that any synthesizer fluctuations are canceled out. 

Table 2 Muon precession frequencies in 4.4M 2,3-dimethyl-2-butene in cyclohexane. 1024 bins of 2.61 ns width were Fourier analyzed using an exponential filter with an 0.5 time constant.

Figure 11.1 Experimental arrangement used for analyzing the current noise in potentiostatic regime. Nj and N2 are the parasitic noises of the measurement channels. The figure shows the Fourier analyzer, amplifiers and filters. Reprinted from Ref [89] with kind permission from Springer Science+Business Media.

Let b fixed to some value of 0(1), and integrated numerically by suitably discretizing x and t. The numerical results for (x, t) obtained are then Fourier-analyzed according to... 

It is customary to proceed by Fourier analyzing S(r) into its Fourier components... 

Since Q(r) is a symmetric, traceless tensor, only five of its nine components are independent, viz. Qn(r) - Q22(r), Qss( )f Qi2(f), Qisir), and Q23(r). As before we Fourier analyze each independent component. 

The force and vibration signals have been recorded at a speed of 30 in/s on a high precision instrument magnetic tape recorder. The power density of the vibration signals and the resultant cutting force were obtained using a HP Fourier Analyzer. 

A block diagram of the experiment set-up is shown in Fig. 2.1. Both the acceleration and cutting force signals are recorded on a magnetic tape and then processed in the HP Fourier Analyzer, the output of which is plotted on the plotter or other peripheral devices. These acceleration and cutting force signals were digitised by an A/DC and then stored on the computer disc. 

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