Pulse or Step Excitation

The FT of an infinite short pulse, h(t) = KS t), where 5(t) is Dirac s delta function, equals H(jco) = K, that is, it contains all frequencies with the same amplitude K. Such a method is used in FFT nuclear magnetic resonance. An ideal pulse function caimot be realized in practice because of the limitations of the electronics, and in practice it must be substituted by a pulse of a short duration Af. However, such a function does not have a uniform response in the Fourier (i.e., frequency) domain. The applied pulse function is defined as h f) = 1 for f = 0 to Af and h t) = 0 elsewhere. Its FT is 

It is presented in Fig. 3.7. Clearly, at y 1/Af, aU the frequencies are presented in the excitation function, but their amplitude decreases quickly at higher frequencies. By decreasing Af, a wider frequency window is obtained. However, in 

Because it is difhcult to apply a very short pulse function, there is another way of obtaining this effect one can apply a step function and take its derivative, which is Dirac s delta function, 6(t) (Fig. 3.8). 

Such a method was used in the literature for the determination of impedances [99-101], and a commercial apparatus [102] applying a current step was described. Taking the FT of the derivative of the potential and current versus time gives the impedance as a function of frequency. However, some authors [100, 101] tried to extrapolate the obtained results to low frequencies beyond the experimental values. If the data are acquired during time T = NAt, the information in the measured signal is obtained for frequencies from l/T up to the Nyquist frequency l/2At, where At is the sampling time. It has been shown [103] that extrapolating impedances to frequencies lower than l/T introduces artifacts. In addition, the measured 

Although the methods using pulse or step excitation are correct from a mathematical point of view, their FT contains all frequencies between /low = 1/T = 1/NAt and /high = l/2Af, and the amplitude for each frequency is quite weak. This causes large sensitivity to noise (Fig. 3.9a). In addition, the measured response signals decay fast with time, and at longer times the signals are hidden in noise. 

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