Delta function response

Delta function response - Over most of the wavelengths of interest, optical and infrared detectors produce one photoelectron for every detected photon, which provides a one-to-one correspondence between detected photons and photoelectrons. This means that the detector response is exactly linear to the intensity incident on the detector - an attribute that allows astronomers to precisely remove sky background and electronic bias to accurately measure the intensity of the astronomical object. 

Figure 28. Geometry for computing the delta function response of the resist. (Reproduced with permission from Ref. 43)...

This equation is derived on the assumption that the dielectric responds to E in a linear fashion. Linearity means that the total response to several stimuli is the sum of the rehouses to each stimulus acting on its owil It should be noted that equation (29) involves da/dr, the derivative of the st iunction response. It is usually referred to as the pulse or delta-function response because... 

Figure 4.14 (a) Energy levels for a spin 3/2 nucleus in the presence of a magnetic field, (b) The resultant NMR absorption spectrum in a single peak at the frequency Vq. (c) Dipolar interactions broaden the resonance symmetrically so that it is not the delta-function response depicted in (b). 

The function g(x) is named impulse response of the system, because it is the response to an unit pulse 5(x) applied at =0 [2]. This unit impulse 5(x), also called Dirac impulse or delta-function, is defined as... 

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. 

Example 15.4 The differential distribution can be defined as the outlet response of a system 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... 

Curve fitting using a delta function for the pulse input for a TAP reactor should be limited to the latter % part of the response curve for curves of FWHM < 3 times pulse width, while for curves with FWHM > 4 times pulse width, it is a fair approximation fijr most of the curve. The assumption of a zero concentration at the reactor outlet is not good evrai for a pumping speed of 1,500 Is and broad response curves with FWHM > 1000 ms. 

Now consider foe external force acting on the system to be composed of a series of instantaneous impacts, each of which can be expressed mathematically by a delta function. The response of the system can then be represented by a function G(f ) The differential equation to be solved then takes on the form... 

When rh delta function, an instantaneous heat deposition, and the resultant voltage response of the transducer is simply the instrument response function, T (t). Although heat depositions cannot be time resolved in this regime, their magnitude and consequently the enthalpy... 

In signal-response terms, cAo = 8(6), the Dirac input, and = Caao the normahzed response, C(0). For the Dirac delta function,... 

When emitted light from several species (j) and some scattered light are superimposed, the true sum response to a delta-function exciting pulse is... 

Since all tracer entered the system at the same time, t = 0, the response gives the distribution or range of residence times the tracer has spent in the system. Thus, by definition, eqn. (8) is the RTD of the tracer because the tracer behaves identically to the process fluid, it is also the system RTD. This was depicted previously in Fig. 3. Furthermore, eqn. (8) is general in that it shows that the inverse of a system transfer function is equal to the RTD of that system. To create a pulse of tracer which approximates to a dirac delta function may be difficult to achieve in practice, but the simplicity of the test and ease of interpreting results is a strong incentive for using impulse response testing methods. 

The curve which describes the concentration-time function of tracer in the exit stream of any vessel in response to an idealized instantaneous or pulse tracer injection is called the C curve. Such an input is often called a delta-function input. As with the F curve, dimensionless coordinates are chosen. Concentrations are measured in terms of the initial concentration of injected tracer if evenly distributed throughout the... 

Fig. 6. Typical downstream response to an upstream delta function input in the dimensionless form shown here this response is called the C curve (L13).

A gated deuterium lamp which has a full width at half-maximum (FWHM) ofabout2nsanddecay time of 1 ns has been used. The decay curves are deconvolved by numerical convolution technique with the assumption that the delta-pulse response is a single exponential function. A programme is used that varies the lifetime until the sum oi the squares or tne deviations between the observed and the calculated decay curves is a minimum (Fig. 11.5). If t0 = unquenched fluorescence lifetime and t = lifetime of quenched... 

The variable to be perturbed is either the potential, E, or the Current density, j. The response on a potential perturbation, logically, will be the resulting current j (t), but it is additionally useful to measure the integral of j(t), i.e. the charge q (0 that has passed the interface. As a counterpart, a technique is known where the perturbation is a current pulse of infinitely small time duration (delta function) comprising a certain amount of charge the coulostatic impulse technique. 

The sensor s response characteristics can often be approximated by delta functions (Finlayson and Hordley 2001b). Even though each sensor responds to a range of wavelengths, in practice they are either close enough to delta functions or they can be sharpened (Barnard et al. 2001 Finlayson and Funt 1996 Finlayson et al. 1994a,b) such that this assumption is approximately correct. Assuming that the sensor s response characteristics can be described by delta functions, we have... 

Suppose that we have two different illuminants. Each illuminant defines a local coordinate system inside the three- dimensional space of receptors as shown in Figure 3.23. A diagonal transform, i.e. a simple scaling of each color channel, is not sufficient to align the coordinate systems defined by the two illuminants. A simple scaling of the color channels can only be used if the response functions of the sensor are sufficiently narrow band, i.e. they can be approximated by a delta function. 

In Chapter 3 we have seen that sensors that have very narrow band-response functions simplify the functions which relate the energy measured by the sensor to the geometry of the object, the reflectance of the object patch, and the irradiance falling onto the patch. If the response function is very narrow, it can be approximated by a delta function. In this case, the energy measured by a sensor I at sensor position X/ is given by... 

For all of the following algorithms, we will assume that the response functions of the sensors are very narrow-band, i.e. they can be approximated by delta functions. Let A., with i r,g, h) be the wavelengths to which the sensors respond. We will now denote the sensor coordinates by (x, y). The intensity measured by the sensor at position (x, y) is then given by... 

If one assumes a linear relationship between the response of the sensor and pixel colors, i.e. Ci (x, y) = I, (x. y). and one also assumes that the sensor s response characteristic is similar to delta functions, then the light illuminating the scene simply scales the product of the geometry term G and the reflectance Rt of the object. 

A completely different model is given by Richards and Parks (1971). It is based on modified von Kries coefficients. If we assume that the sensor response functions have the shape of delta functions, then it is possible to transform a given color of a patch taken under one illuminant to the color of the patch when viewed under a different illuminant by multiplying the colors using three constant factors for the three channels. These factors are known as von Kries coefficients, as described in Section 4.6. The von Kries coefficients are defined as... 

Solution This solution illustrates a possible definition of the delta function as the limit of an ordinary function. Disturb the reactor with a rectangular tracer pulse of duration At and height A/t so that A units of tracer are injected. The input signal is Cm =0,t < 0 Ct = A/At, 0 < t < At Cin = 0, and t > At. The outlet response is found from the dynamic model of a CSTR, Equation (14.2). The result is... 

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