Linear system response

If x(f) is white noise with a zero mean value, and yi(t) is the linear system response (4.2.6), then the cross-correlation function is proportional to the memory function k (a) (Fig. 4.3.1),... 

Fig. 5.3.3 [Houl Relationships between excitation spectrum X (w) (a) of the excitation transfer function K (a>) (b), and spectrum K (a>) (c) of the linear system response M, ) =, Vi (/) (d) for nonselective excitation (left) and selective excitation (right).

The other is to attenuate a linear system response by an exponentially increasing attenuation. 

Linearity a system is linear when its response to a sum of individual input signals is equal to the sum of the individual respmises Hafi + bf ) = aE(/i) + bL f ). This condition implies that the system is described by a system of linear differential equations. Electrochemical systems are usually highly nonlinear, and the impedance is obtained by linearization of equations using small amplitudes of the applied perturbation. In addition, for linear systems responses are independent of the amplitude. This condition can be easily verified experimentally (Mie should decrease the applied amplitude twice and compare the results. If the obtained impedance is the same, the system is linear. 

It should be a symmetrical form in the impulse response of a linear system. 

A dynamic system is linear if the Principle of Superposition can be applied. This states that The response y t) of a linear system due to several inputs x t),... 

The steady-state response of a linear system will be... 

Avdeef and Bucher [24] investigated the use of universal buffers in potentiomet-ric titrations. Recently, such a buffer system, formulated with several of the Good components, has been designed specifically for robotic applications, where automated pH control in 96-well microtiter plates is required, with minimal interference to the UV measurement [48]. This universal buffer has a nearly perfectly linear pH response to additions of standard titrant in the pH 3-10 region [8, 48]. 

Here (Oj is the excitation energy ErE0 and the sum runs over all excited states I of the system. From equation (5-37) we immediately see that the dynamic mean polarizability a(co) diverges for tOj=co, i. e has poles at the electronic excitation energies 0)j. The residues fj are the corresponding oscillator strengths. Translated into the Kohn-Sham scheme, the exact linear response can be expressed as the linear density response of a non-interacting... 

In this study the linearized equations were used to determine the control strategy, but the non-linear equations were used to test this strategy. For small deviations from the steady state (5% or less) there is very little difference between the responses of the non-linear and the linearized system. 

For linear systems the relative response to a pulse input is equal to the derivative of the relative response to a step input. Illustration 11.1 indicates how the response of a reactor network to a pulse input can be used to generate an F(t) curve. 

For a linear system like a probe circuit, the response fq(f) (RF field created by the coil) to the excitation v(t) (RF voltage produced by the spectrometer) is expressed by a convolution relation... 

The basic principle of heat-flow calorimetry is certainly to be found in the linear equations of Onsager which relate the temperature or potential gradients across the thermoelements to the resulting flux of heat or electricity (16). Experimental verifications have been made (89-41) and they have shown that the Calvet microcalorimeter, for instance, behaves, within 0.2%, as a linear system at 25°C (41)-A. heat-flow calorimeter may be therefore considered as a transducer which produces the linear transformation of any function of time f(t), the input, i.e., the thermal phenomenon under investigation]] into another function of time ig(t), the response, i.e., the thermogram]. The problem is evidently to define the corresponding linear operator. 

The combination of Eqs. (28) and (22) gives the Laplace transform of the impulse response H(p) which allows us to solve Eq. (21). By the inverse transformation, the relation which gives the output of the linear system g(t) (the thermogram) to any input/(0 (the thermal phenomenon under investigation) is obtained. This general equation for the heat transfer in a heat-flow calorimeter may be written (40, 46) ... 

The ratio data were normalized by assuming that the highest ratio measured was i0Be/9Be — 10(In fact, the 9Be(n,y/0Be cross section is only known to 10%.) The diagonal line represents the response of a perfectly linear system, and the dashed horizontal line gives the present limit of sensitivity. 

It follows that a small periodic perturbation applied to a system, the eigenstates of which are densely distributed in energy, leads to a power dissipation quadratic in the perturbation. For such a linear system it is possible to define an impedance Z(co), the ratio of the force V to the response Q, where all quantities are now assumed to be in standard complex notation, V = Z(u>)Q. The instantaneous power is VQR u )/ Z oj), where R(co), the resistance, is the real part of Z(lu). 

The simplest response of a linear system is described mathematically by the following standard form of first-order differential equation... 

The data in Figures 2.17 and 2.18 are displayed in terms of the dimensionless centre-to-centre separation of particles, i.e. r/2a — (2a + h)/2a. This has been done to illustrate another important point the range of linear elastic response. In a concentrated system, which is showing solid-like or elastic responses, the structure has to be able... 

These two mathematical Equations (4.59) and (4.60) illustrate an important feature about linear viscoelastic measurements, i.e. the central role played by the relaxation function and the compliance. These terms can be used to describe the response of a material to any deformation history. If these can be modelled in terms of the chemistry of the system the complete linear rheological response of our material can be obtained. 

A note of caution should be sounded here. Whilst the curves shown in Figure 6.5 are characteristic of many charged dispersions it should be recalled that once we apply a sinusoid to a non-linear system the response need not be a sinusoid. As the strain is increased into the nonlinear region, the waveform passing through the sample becomes progressively distorted. The instrumental analysis in this case involves... 

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