5-pulse response

The pulse response function is the output function/j(t) caused by the action of the input impulse function (Dirac function). It is applied for determination of the particular forms of the Laplace transmittance. It can be obtained by applying the Laplace inverse transformation to the transmittance Eq. (2.41)  

The pulse response function is a positive function of real argument t. It fulfils the condition given by Eq. (2.43)  

If we take advantage of the theorem of the initial value of the original, i.e. the function/if r), the initial value of the pulse response h(t) on the basis of Eq. (2.34) can be given by 

The pulse response can be obtained experimentally as the response of the calorimetric system  

With the measured values of the response Tu(t) at times to, t and t2, respectively 

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The difference in widths of the impulse responses are small. Especially visible the pulse response of the inductive sensors, curves (1) and... 

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. 

For quadnipolar nuclei, the dependence of the pulse response on Vq/v has led to the development of quadnipolar nutation, which is a two-dimensional (2D) NMR experiment. The principle of 2D experiments is that a series of FIDs are acquired as a fimction of a second time parameter (e.g. here the pulse lengdi applied). A double Fourier transfomiation can then be carried out to give a 2D data set (FI, F2). For quadnipolar nuclei while the pulse is on the experiment is effectively being carried out at low field with the spin states detemiined by the quadnipolar interaction. In the limits Vq v the pulse response lies at v and... 

For a large amount of dispersion or small value of Np, the pulse response is broad, and it passes the measurement point slowly enough for changes to occur in the shape of the tracer curve. This gives a non-symmetrical E-curve. 

The simple fitting procedure is especially useful in the case of sophisticated nonlinear spectroscopy such as time domain CARS [238]. The very rough though popular strong collision model is often used in an attempt to reproduce the shape of pulse response in CARS [239]. Even if it is successful, information obtained in this way is not useful. When the fitting law is used instead, both the finite strength of collisions and their adiabaticity are properly taken into account. A comparison of... 

Figure 7. Influence of the polarization preservation in a fibre interferometer. Right Standard fibre use leads to a complex pulse response associated with fringe degradations with polarization crosstalk. Left Using polarization preserving fibre, the two polarization modes give rise to contrasted fringes and can be separated using a polarizer.

Figure 3.51. CO chemisorption by pulse response of a reduced 5 wt% Pt/Al203 a. Thermal Conductivity Detector (TCD) signals after the CO pulses, b. Cumulative amount of CO chemisorbed. The monolayer capacity is 0.06 mmol/g Pt, corresponding with a dispersion of 24%.

Fig. 15. Waveforms used for in vivo electrochemical analysis. A = chronoamperometry, B = double chronoamperometry (response = SI — S2), C = linear sweep, D = differential pulse (response = S2 — SI), S = sample window...

Figure 3.27. Tracer pulse response of a tanks-in-series system with and without dead zones.

What is the pulse response in the effluent If we do not have the patience of 10 s and dump all the extra tracer in at one shot, what is the impulse response ... 

In the general case of axially dispersed plug flow with bidispersed particles, the first and second moment of the pulse response are [Haynes and Sarma, AIChEJ., 19,1043 (1973)] ... 

Goguet, A., Shekhtman, S.O., Burch, R., Hardacre, C., Meunier, F.C., and Yablonsky, G.S. 2006. Pulse-response TAP studies of the reverse water-gas shift reaction over a Pt/Ce02 catalyst../. Catal. 237 102-10. 

Pudjiono and Tavare (1993) used the pulse-response technique to study the residence time... 

Fig. 2 Pulse response curves for tracer at 3 recycle rates Fr=l, 5.5 and 10.

Evaluate the conversion for first-order reaction from a tracer pulse response curve using the method in example CSTRPULSE. Show that although the residence time distributions may be the same in the two cases, the overall chemical conversion is not, excepting for the case of first-order reaction. 

In other words, the fluorescence quantum yield is the ratio of the number of emitted photons (over the whole duration of the decay) to the number of absorbed photons. According to Eq. (3.4), the ratio of the d-pulse response tp(f) and the number of absorbed photons is given by... 

It is worth noting that integration of the (5-pulse response iF(t) of the fluorescence intensity over the whole duration of the decay (Eq. 3.8) yields... 

The d-pulse response of the fluorescence intensity can then be obtained by introducing the relation giving fci(t) into Eq. (4.14) and by analytical or numerical integration of this equation. 

Consequently, in the case of diffusion-limited quenching, the J-pulse response of the fluorescence intensity can be calculated by means of the following equation,... 

Under continuous illumination, the steady-state intensity can be easily calculated by considering a light of constant intensity as an infinite sum of infinitely short light pulses. It is thus simply obtained by integration of the b-pulse response. The ratio of the intensities in the absence and in the presence of quencher is then given by... 

Because formation ofexcimer E is a diffusion-controlled process, Eqs (4.11)-(4.13) apply to the diffusional rate constant ki for excimer formation. Under the approximation that ki is time-independent, the d-pulse responses, under the initial conditions (at t = 0), [M ] = [M ]o and [E ]o = 0, are... 

Knowledge of the dynamics of excited states is of major importance in understanding photophysical, photochemical and photobiological processes. Two time-resolved techniques, pulse fluorometry and phase-modulation fluorometry, are commonly used to recover the lifetimes, or more generally the parameters characterizing the S-pulse response of a fluorescent sample (i.e. the response to an infinitely short pulse of light expressed as the Dirac function S). 

Pulse fluorometry uses a short exciting pulse of light and gives the d-pulse response of the sample, convoluted by the instrument response. Phase-modulation fluorometry uses modulated light at variable frequency and gives the harmonic response of the sample, which is the Fourier transform of the d-pulse response. The first technique works in the time domain, and the second in the frequency domain. Pulse fluorometry and phase-modulation fluorometry are theoretically equivalent, but the principles of the instruments are different. Each technique will now be presented and then compared. 

The principles of pulse and phase-modulation fluorometries are illustrated in Figures 6.5 and 6.6. The d-pulse response I(t) of the fluorescent sample is, in the simplest case, a single exponential whose time constant is the excited-state lifetime, but more often it is a sum of discrete exponentials, or a more complicated function sometimes the system is characterized by a distribution of decay times. For any excitation function E(t), the response R(t) of the sample is the convolution product of this function by the d-pulse response ... 

Phase-modulation fluorometry The sample is excited by a sinusoidally modulated light at high frequency. The fluorescence response, which is the convolution product (Eq. 6.9) of the sinusoidal excitation function, is sinusoidally... 

Relationship between harmonic response and rt-pulse response It is worth demonstrating that the harmonic response is the Fourier transform of the d-pulse response. The sinusoidal excitation function can be written as... 

In practice, the phase shift and the modulation ratio M are measured as a function of co. Curve fitting of the relevant plots (Figure 6.6) is performed using the theoretical expressions of the sine and cosine Fourier transforms of the b-pulse response and Eqs (6.23) and (6.24). In contrast to pulse Jluorometry, no deconvolution is required. 

General relations for single exponential and multi-exponential decays For a single exponential decay, the b-pulse response is... 

For a multi-exponential decay with n components, the (5-pulse response is... 

Data analysis in phase fluorometry requires knowledge of the sine and cosine of the Fourier transforms of the b-pulse response. This of course is not a problem for the most common case of multi-exponential decays (see above), but in some cases the Fourier transforms may not have analytical expressions, and numerical calculations of the relevant integrals are then necessary. 

The maximum entropy method has been successfully applied to pulse fluorometry and phase-modulation fluorometry3- . Let us first consider pulse fluorometry. For a multi-exponential decay with n components whose fractional amplitudes are a , the d-pulse response is... 

The procedure in phase-modulation fluorometry is more straightforward. The sine and cosine Fourier transforms of the d-pulse response are, according to Eqs (6.30) and (6.31), given by... 

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