Single-Frequency Fourier Analysis

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. 

Consider that a periodic function of time can be expressed as a Foxuier series, 

The trigonometric fimctions can be expressed in terms of exponentials following equations (1.82) and (1.83) to yield 

Equation (7.30) provides the basis for single-frequency Fourier analysis for impedance measurement. 

Linear sinusoidal input and output signals can be expressed in terms of equation (7.22) with n = 1. For example, for an input potential 

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Calculation of mean crystallite size, lattice strain and frequency distributions of crystallite sizes from the same XRD line-profiles used for crystallinity determinations. In addition to the application of the Scherrer equation, two single-line methods were used the variance method of Wilson (1963) (Akai and To th 1983 Nieto and Smchez-Navas 1994), and the Voigt method of Langford (1978) in combination with single-line Fourier analysis (Akai et al. 1996, 1997, 2000 Warr 1996 Jiang et al. 1997 Li et al. 

The results of the analysis are shown in Table 3.4. It is seen that, although some of the periodicities are extracted, some which appear are difference or beat frequencies and do not correspond to a single layer. The conclusion is that Fourier analysis, with the above procedure, is a powerful aid to a skilled researcher, but it is not yet appropriate for automated analysis. 

Gangwall et al. [47] were the first to apply Fourier analysis for the evaluation of the transport parameters of the Kubin-Kucera model. Gunn et al. applied the frequency response [80] and the pulse response method [83] in order to determine the coefficients of axial dispersion and internal diffusion in packed beds from experiments performed at various Reynolds numbers. Bashi and Gunn [83] compared the methods based on the analytical properties of the Fourier and the Laplace transforms for the calculation of transport coefficients. MacDonnald et al. [84] discussed the applications of the method of moments to the analysis of the profiles of skewed chromatographic peaks. When more than two parameters have to be determined from one single run, the moment analysis method is less suitable, because only the first and second moments are reliable (see Figure 6.9). Therefore, only two parameters can be determined accurately. 

In Fig. 7.4-5 the cosine wave fits exactly eight times, and this shows in its transform, which exhibits a single point at f = 8 X (1/128) = 0.0625. On the other hand, the cosine wave in Fig. 7.4-6 does not quite form a repeating sequence, and its frequency, 77/(0.4 X 128) —0.06136, likewise does not fit any of the frequencies used in the transform. Consequently the Fourier transform cannot represent this cosine as a single frequency (because it does not have the proper frequency to do so) but instead finds a combination of sine and cosine waves at adjacent frequencies to describe it. This is what is called leakage the signal at an in-between (but unavailable) frequency as it were leaks into the adjacent (available) analysis frequencies. 

FIGURE 7.4 Fourier analysis of trumpet playing single note of frequency Vo 523 Hz, one octave above middle C. Upper curve represents the signal in the time domain and lower curve in the.frequency domain. 

A number of other operational problems exist when using the FFT algorithm. The most important of these, as far as electrochemistry is concerned, is due to the inherently nonlinear nature of the system. When Eq. (56) is used to measure the impedance with an arbitrary time domain input function (i.e. not a single-frequency sinusoidal perturbation), then the Fourier analysis will incorrectly ascribe the harmonic responses due to system nonlinearity, to input signal components which may or may not be present at higher frequencies. As a consequence, the measured impedance spectrum may be seriously in error. 

For some tasks in ultrahigh-resolution spectroscopy, the residual finite linewidth AyL, which may be small but nonzero, still plays an important role and must therefore be known. Furthermore, the question why there is an ultimate lower limit for the linewidth of a laser is of fundamental interest, since this leads to basic problems of the nature of electromagnetic waves. Any fluctuation of amplitude, phase, or frequency of our monochromatic wave results in a finite linewidth, as can be seen from a Fourier analysis of such a wave (see the analogous discussion in Sects. 3.1,3.2). Besides the technical noise caused by fluctuations of the product nd, there are essentially three noise sources of a fundamental nature, which cannot be eliminated, even by an ideal stabilization system. These noise sources are, to a different degree, responsible for the residual linewidth of a single-mode laser. 

In the present state-of-the-art equipment, it is possible to measure and plot the electrochemical impedance automatically. Electronic circuitry is designed to generate the frequency sweep of a desired resolution over the range of interest. The generator can be programmed to sweep from a maximum to a minimum frequency in a number of required frequency steps. The commonly used modem equipment AC impedance measurement techniques can be subdivided into two main groups - single sine (lock-in amplification and frequency response analysis) and multiple sine techniques such as fast Fourier transforms (FFT). 

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