Response function time domain

Owing to the coherence, we need to consider the macroscopic evolution of the field in a medium that shows a macroscopic polarization induced by the field-matter interaction. This will be done in three steps. First, the polarization induced by an arbitrary field will be calculated and expanded in power series in the field, the coefficients of the expansion being the material susceptibilities (frequency domain) or response function (time domain) of wth-order. Nonlinear Raman effects appear at third order in this expansion. Second, the perturbation theory derivation of the third-order nonlinear susceptibility in terms of molecular eigenstates and transition moments will be outlined, leading to a connection with the spontaneous Raman scattering tensor components. Last, the interaction of the initial field distribution with the created polarization will be evaluated and the signal expression obtained for the relevant techniques of Table 1. 

Mathematically,/(l) can be determined from F t) or W t) by differentiation according to Equation (15.7). This is the easiest method when working in the time domain. It can also be determined as the response of a dynamic model to a unit impulse or Dirac delta function. The delta function is a convenient mathematical artifact that is usually defined as... 

Example 15.1 shows how it can be detennined in the time domain as the response to a delta function input. 

Equation (15.38) gives the Laplace transform of the outlet response to an inlet delta function i.e., a utik) = k[f t)]- In principle. Equation (15.38) could be inverted to obtain/(r) in the time domain. This daunting task is avoided by... 

Example 2.10 What is the time domain response C (t) in Eq. (2-27) if the change in inlet concentration is (a) a unit step function, and (b) an impulse function ... 

Instead of spacing out in the Laplace-domain, we can (as we are taught) guess how the process behaves from the pole positions of the transfer function. But wouldn t it be nice if we could actually trace the time profile without having to do the reverse Laplace transform ourselves Especially the response with respect to step and impulse inputs Plots of time domain dynamic calculations are extremely instructive and a useful learning tool.1... 

Note In the text, we emphasize the importance of relating pole positions of a transfer function to the actual time-domain response. We should get into the habit of finding what the poles are. The time response plots are teaching tools that reaffirm our confidence in doing analysis in the Laplace-domain. So, we should find the roots of the denominator. We can also use the damp () function to find the damping ratio and natural frequency. 

The LTI Viewer was designed to do comparative plots, either comparing different transfer functions, or comparing the time domain and (later in Chapter 8) frequency response properties of a transfer function. So a more likely (and quicker) scenario is to enter, for example,... 

There is significant debate about the relative merits of frequency and time domain. In principle, they are related via the Fourier transformation and have been experimentally verified to be equivalent [9], For some applications, frequency domain instrumentation is easier to implement since ultrashort light pulses are not required, nor is deconvolution of the instrument response function, however, signal to noise ratio has recently been shown to be theoretically higher for time domain. The key advantage of time domain is that multiple decay components can, at least in principle, be extracted with ease from the decay profile by fitting with a multiexponential function, using relatively simple mathematical methods. 

As you will see, several different approaches are used in this book to analyze the dynamics of systems. Direct solution of the differential equations to give functions of time is a time domain teehnique. Use of Laplace transforms to characterize the dynamics of systems is a Laplace domain technique. Frequency response methods provide another approaeh to the problem. 

At this point it might be useful to pull together some of the concepts that you have waded through in the last several chapters. We now know how to look at and think about dynamics in three languages time (English), Laplace (Russian) and frequency (Chinese). For example, a third-order, underdamped system would have the time-domain step responses sketched in Fig. 14.10 for two different values of the real TOOt. In the Laplace domain, the system is represented by a transfer function or by plotting the poles of the transfer function (the roots of the system s characteristic equation) in the s plane, as shown in Fig. 14.10. In the frequency domain, the system could be represented by a Bode plot of... 

Similarly to non-selective experiments, the first operation needed to perform experiments involving selective pulses is the transformation of longitudinal order (Zeeman polarization 1 ) into transverse magnetization or ly). This can be achieved by a selective excitation pulse. The first successful shaped pulse described in the literature is the Gaussian 90° pulse [1]. This analytical function has been chosen because its Fourier transform is also a Gaussian. In a first order approximation, the Fourier transform of a time-domain envelope can be considered to describe the frequency response of the shaped pulse. This amounts to say that the response of the spin system to a radio-frequency (rf) pulse is linear. An exact description of the... 

The relaxation rate R t) described by Eqs. (4.49)-(4.51) embodies our universal recipe for dynamically controlled relaxation [10, 21], which has the following merits (i) it holds for any bath and any type of interventions, that is, coherent modulations and incoherent interruptions/measurements alike (ii) it shows that in order to suppress relaxation, we need to minimize the spectral overlap of G( ), given to us by nature, and Ffo)), which we may design to some extent (iii) most importantly, it shows that in the short-time domain, only broad (coarse-grained) spectral features of G( ) and Ffa>) are important. The latter implies that, in contrast to the claim that correlations of the system with each individual bath mode must be accounted for, if we are to preserve coherence in the system, we actually only need to characterize and suppress (by means of Ffco)) the broad spectral features of G( ), the bath response function. The universality of Eqs. (4.49)-(4.51) will be elucidated in what follows, by focusing on several limits. 

All the system response curves in frequency and time domains were calculated numerically from equations that are much too involved to reproduce in detail here. Transfer functions in Laplace transform notation are easily defined for the potentiostat and cell of Figure 7.1. Appropriate combinations of these functions then yield system transfer functions that may be cast into time- or frequency-dependent equations by inverse Laplace transformation or by using complex number manipulation techniques. These methods have become rather common in electrochemical literature and are not described here. The interested reader will find several citations in the bibliography to be helpful in clarifying details. 

Frequently, it is required to determine the initial or final value of the system response to some forcing function. It is possible to evaluate this information without inverting the appropriate transform into the time domain. 

Only the special case of the impulse will be considered (Section 7.8.1). This is a particularly useful function for testing system dynamics as it does not introduce any further s terms into the analysis (equation 7.78). The determination of the response of any system in the time domain to an impulse forcing function is facilitated by noting that ... 

The response of the controlled variable to different types of perturbation (forcing function) in set point or load can be determined by inverting the appropriate transform (e.g. equation 7.112). This is possible only for simple loops containing low order systems. More complex control systems involving higher order elements require a suitable numerical analysis in order to obtain the time domain response. 

Thus, in order to simulate a perceptually convincing room reverberation, it is necessary to simulate both the pattern of early echoes, with particular concern for lateral echoes, and the late energy decay relief. The latter can be parameterized as the frequency response envelope and the reverberation time, both of which are functions of frequency. The challenge is to design an artificial reverberator which has sufficient echo density in the time domain, sufficient density of maxima in the frequency domain, and a natural colorless timbre. 

The phase response of the allpass filter is a non-linear function of frequency, leading to a smearing of the signal in the time domain. 

The dipole response in real time gives access to the response in frequency domain by Fourier transfrom D (a)), from which one can extract the strength function S(n>) = cA b yf and the power spectrum P( ) = I)(a/) 2. The strength function is the more suited quantity in the linear regime, where it can be related to the photoabsorption cross section [31], while the power spectrum better applies for spectral analysis in the non linear regime [24],... 

Multiple-point fluorescent deteclion has been proposed to enhance detection sensitivity. This method is based on the use of a detector function, such as the Shah function. The time-domain signals were first detected, and they were converted into a frequency-domain plot by Fourier transformation. Therefore, this technique was dubbed Shah convolution Fourier transform detection (SCOFT). As a comparison, the single-detection point time-domain response is commonly known as the electropherogram [698,699,701]. 

As used here, a DC model is characterized entirely in terms of dielectric constants (e) of the pure solvent (i.e., in the absence of the solute and its cavity) and the structure of the molecular cavity (size and shape) enclosing the solute [3], We confine ourselves to dipolar medium response, due either to the polarizability of the solvent molecules or their orientational polarization1 [15,16]. Within this framework, in its most general space and time-resolved form, one is dealing with the dielectric function s(k, >), where k refers to Fourier components of the spatial response of the medium, and oj. to the corresponding Fourier components of the time domain [17]. In the limit of spatially local response (the primary focus of the present contribution), in which the induced medium polarization (P) at a point r in the medium is specified entirely by the electric field (E) at the same point, only the Tong wavelength component of s is required (i.e., k = 0) [18,19]. 

The Laplace transform may be inverted to provide a tracer response in the time domain. In many cases, the overall transfer function cannot be analytically inverted. Even in this case, moments of the RTD may be derived from the overall transfer function. For instance, if Go and GJare the limits of the first and... 

FIGURE 10.12 Time-domain smoothing of the noisy data in Figure 10.1 with the impulse response function of Figure 10.7, processed from left to right in this spectrum. The true signal is shown as a dotted line. Note the significant filter lag in this example. 

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