Response function

If the impulse response function g(x) of a system is known, the output signal y(x) of the system is given for any input signal u(x). The integral equation, which is called superposition integral. 

Often an unit impulse is not available as a signal to get the impulse response function g(x). Therefore an other characteristic signal, the unit step, is be used. 

The step response function h(x) is the response of a system to an unit step s(x) at the input. 

The unit step function s(x) is defined as a step from 0 to /. The function s(x) is shown in fig. 1 (centre) together with an example of a step response function h(x). 

The first example presents the importance of the impulse response function for the comparison of several sensors with the same arrangement from chapter 3.1.. 

The determined eddy-eurrent parameter is the inductance of the eomplex impedance measured by impedance analyzer at j=100 kHz. Therefore the impulse response function from chapter 4.2.1. is used for calculation. The depth of the cracks is big in comparison to coil size. For presentation the measured and pre-calculated data are related to their maxima (in air). The path X is related to the winding diameter dy of the coil. 

So, a comparison of different types of magnetic field sensors is possible by using the impulse response function. High amplitude and small width of this bell-formed function represent a high local resolution and a high signal-to-noise-characteristic of a sensor system. On the other hand the impulse response can be used for calculation of an unknown output. In a next step it will be shown a solution of an inverse eddy-current testing problem. 

Due to its importance the impulse-pulse response function could be named. .contrast function". A similar function called Green s function is well known from the linear boundary value problems. The signal theory, applied for LLI-systems, gives a strong possibility for the comparison of different magnet field sensor systems and for solutions of inverse 2D- and 3D-eddy-current problems. 

Fluctuations of observables from their average values, unless the observables are constants of motion, are especially important, since they are related to the response fiinctions of the system. For example, the constant volume specific heat of a fluid is a response function related to the fluctuations in the energy of a system at constant N, V and T, where A is the number of particles in a volume V at temperature T. Similarly, fluctuations in the number density (p = N/V) of an open system at constant p, V and T, where p is the chemical potential, are related to the isothemial compressibility iCp which is another response fiinction. Temperature-dependent fluctuations characterize the dynamic equilibrium of themiodynamic systems, in contrast to the equilibrium of purely mechanical bodies in which fluctuations are absent. 

Because this current is given by a conmuitator, its equilibrium expectation value is zero. Using die first expression in (A3.2.43). the response function is given by... 

Keller G 1986 Random-phase-approximation study of the response function describing optical second-harmonic generation from a metal selvedge Rhys. Rev. B 33 990-1009... 

Figure C1.5.12.(A) Fluorescence decay of a single molecule of cresyl violet on an indium tin oxide (ITO) surface measured by time-correlated single photon counting. The solid line is tire fitted decay, a single exponential of 480 5 ps convolved witli tire instmment response function of 160 ps fwiim. The decay, which is considerably faster tlian tire natural fluorescence lifetime of cresyl violet, is due to electron transfer from tire excited cresyl violet (D ) to tire conduction band or energetically accessible surface electronic states of ITO. (B) Distribution of lifetimes for 40 different single molecules showing a broad distribution of electron transfer rates. Reprinted witli pennission from Lu andXie [1381. Copyright 1997 American Chemical Society.

Most response functions of poiymers obey a time-temperature or frequency-temperature superjDosition [43, 44], A change in temperature is equivaient to a shift of tire iogaritiimic frequency axis ... 

The relationship between output variables, called the response, and the input variables is called the response function and is associated with a response surface. When the precise mathematical model of the response surface is not known, it is still possible to use sequential procedures to optimize the system. One of the most popular algorithms for this purpose is the simplex method and its many variations (63,64). 

The distribution of residence times of reactants or tracers in a flow vessel, the RTD, is a key datum for determining reactor performance, either the expected conversion or the range in which the conversion must fall. In this section it is shown how tracer tests may be used to estabhsh how nearly a particular vessel approaches some standard ideal behavior, or what its efficiency is. The most useful comparisons are with complete mixing and with plug flow. A glossary of special terms is given in Table 23-3, and major relations of tracer response functions are shown in Table 23-4. 

The unsteady material balances of tracer tests are represented by linear differential equations with constant coefficients that relate an input function Cj t) to a response function of the form... 

The algorithm is based both on a mathematical simulation of a spectmm of secondary radiation emitted by a sample to be studied and the detector response function. The detection limit is given by criteria 3s ... 

This gives a step response function of the form shown in Figure 3.16. 

Provides criteria for evaluating existing emergency response functions ... 

I FIGUftE I 1.33 Typical response function to a unit pulse of the temperature or heat flux boundary. 

Role Responsibility Function in centralized companies may be Function in decentralized companies may be ... 

These response functions are equivalent to tlie functional aimexes of a multihazard emergency operations plan described in tlie Guide for Development of State and Local Emergency Operations Plans. prepared by the Federal Emergency Management Association, March, 1987, Wasliington, DC. 

These workers used binary solvent systems over a range of mole fractions to determine, for each solute, the constants a and b of equation (8.2). For methyl and phenacyl esters, TLC was used, while overpressured layer chromatography (OPLC) was used for dansyl amino acids. Nurok and co-workers (11) also evaluated how the quality of a simulated separation varies with changing solvent strength by using the inverse distance function (IDF) or planar response function (PRF), as follows ... 

FIGURE 2.16 Effects of successive rectangular hyperbolae on receptor stimulus, (a) Stimulus to three agonists, (b) Three rectangular hyperbolic stimulus-response functions in series. Function 1 ((3 = 0.1) feeds function 2 ((3 = 0.03), which in turn feeds function 3 ((3 = 0.1). (c) Output from function 1. (d) Output from function 2 (functions 1 and 2 in series), (e) Final response output from function 3 (all three functions in series). Note how all three are full agonists when observed as final response. 

FIGURE 3.5 Major components of classical receptor theory. Stimulus is the product of intrinsic efficacy (s), receptor number [R], and fractional occupancy as given by the Langmuir adsorption isotherm. A stimulus-response transduction function f translates this stimulus into tissue response. The curves defining receptor occupancy and response are translocated from each other by the stimulus-response function and intrinsic efficacy. 

FIGURE 3.10 Constitutive activity due to receptor overexpression visualization through binding and function, (a) Constitutive activity observed as receptor species ([RaG]/[RL0J) and cellular function ([RaG]/ ([RaG] + 3), where P = 0.03. Stimulus-response function ([RaG]/([RaG] + p)) shown in inset. The output of the [RaG] function becomes the input for the response function. Dotted line shows relative amounts of elevated receptor species and functional response at [R]/KG= 1. (b) Effects of an inverse agonist in a system with [R]/ Kq= 1 (see panel a) as observed through receptor binding and cellular function. 

For each EA spectrum, the transmission T was measured with the mechanical chopper in place and the electric field off. The differential transmission AT was subsequently measured without the chopper, with the electric field on, and with the lock-in amplifier set to detect signals at twice the electric-field modulation frequency. The 2/ dependency of the EA signal is due to the quadratic nature of EA in materials with definite parity. AT was then normalized to AT/T, which was free of the spectral response function. To a good approximation [18], the EA signal is related to the imaginary part of the optical third-order susceptibility ... 

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