Response frequency-domain

Figure 3. The acquired NMR signal (a), Free Induction Decay (FID) in the time domain response.Two transient signals each 90° out of phase comprise the real and imaginary part of the FID (b), its frequency domain response using Fburier transformation.

Figure 8 Linear response. (A) Analogous models. (B) Time-domain response. (C) Frequency-domain response.

Before proceeding to the frequency-domain response, we remark that it is sometimes useful to consider the force to be a function of the displacement (and not the other way round). This is particularly true for mechanical applications, and analogous developments to those outlined above show that stress may be related to strain by an equation of the same form as equation (16). 

Frequency-domain response It is often convenient to characterize the linear response of systems in the frequency rather than in the time domain. If one applies a periodic loading with angular frequency co [circular frequency / = co/(27r)], one will obtain a periodic response of the same frequency, but the response will generally be out of phase. 

Mathematically, the time and frequency-domain responses are connected via Fourier-iaplace transforms, but the same conclusions may be reached by inserting the complex force from equation (18) in equation (16). In this manner, one obtains the expression... 

It is interesting to contrast low-frequency dispersion with direct currents. To elucidate the effect of a nonzero direct-current (dc) conductivity (Tq on the frequency-domain response, let us assume that a driving force T = is applied, as in... 

In the previous sections, we have utilized Green s function techniques to eliminate some of the summations involved in the calculations of nonlinear susceptibilities. The general expression for R(t3,t2,t1) [Eq. (49) or (60)], involves four summations over molecular states a, b, c, d. In Eq. (80) we carried out two of these summations for harmonic molecules. It should be noted that for this particular model it is possible to carry out formally all the summations involved, resulting in a closed time-domain expression for R(t3,t2,t1). This expression, however, cannot be written in terms of simple products of functions of , r2 and t3. Therefore, calculating the frequency-domain response function / via Eq. (30) requires the performing of a triple Fourier transform (rather than three one-dimensional transforms). This formula is, therefore, useful for extremely short pulses when a time-domain expression is needed. Otherwise, it is more convenient to use the expressions of Section VI, whereby only two of the four summations were carried out, but the transformation to... 

To summarize, all of the information of the system is available from either means of exciting the resonance, driving it and sweeping the frequency, or hitting it with an impulse for its time response. The first experiment is performed in the frequency domain and the second in the time domain. The mathematical transformation of one representation into the other is the Fourier transform. The time domain response and the frequency domain response are called Fourier transform... 

Whenever the same parameters are available from two different curves (e.g., wq aiid t from Figure 1 or Figure 4a), there is some mathematical relation between the curves. For the "linear" system we have considered (i.e., displacement is proportional to driving amplitude Fq) the time-domain and frequency-domain responses are connected by a Fourier transform. Similarly, absorption and dispersion spectra both yield the same information, and are related by a Hilbert transform (see Chapter 4). In this Chapter, we will next develop some simple Fourier transform properties for continuous curves such as Figures 1-4, and then show the advantages of applying similar relations to discrete data sets consisting of actual physical responses sampled at equally-spaced intervals. 

In the previous section, we established a correspondence between the transient time-domain response (exponentially damped cosine wave) to a sudden "impulse" excitation and the steady-state frequency-domain response (Lorentzian absorption and dispersion spectra) to a continuous excitation. The Fourier transform may be thought of as the mathematical recipe for going from the time-domain to the frequency-domain. In this section, we shall introduce the mathematical forms of the transforms, along with pictorial examples of several of the most important signal shapes. 

Figures 33.15 and 33.16 show the experimental results of the time-domain and frequency-domain responses. The red line and the black line show the aerogel ultrasonic transducer and conventional ultrasonic transducer. An acoustic matching layer of...

Figure 33.16. Experimental results of frequency-domain response.

Let us look now at an example of an electromechanical system and implement the systematic process outlined herein with the automated computer-generated model shown below. Let us try a piezoelectric sensor and setup as an objective to find the frequency domain response to find out on which frequency range the sensor will measure accurately this means in which frequencies the relation between input and output should be one. 

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