The impulse function

4 Common time domain input functions 3.4.1 The impulse function 

The Laplaee transform of an impulse funetion is equal to the area of the funetion. The impulse funetion whose area is unity is ealled a unit impulse 8 t). 

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The (unit) impulse function is called the Dirac (or simply delta) function in mathematics.1 If we suddenly dump a bucket of water into a bigger tank, the impulse function is how we describe the action mathematically. We can consider the impulse function as the unit rectangular function in Eq. (2-20) as T shrinks to zero while the height 1/T goes to infinity ... 

The impulse function is rarely defined in the conventional sense, but rather via its important property in an integral ... 

The Laplace transform of the impulse function is obtained easily by taking the limit of the unit rectangular function transform (2-20) with the use of L Hospital s rule ... 

In mathematics, the unit rectangular function is defined with a height of 1/2T and a width of 2T from -T to T. We simply begin at t = 0 in control problems. Furthermore, the impulse function is the time derivative of the unit step function. 

The impulse function is an infinitely high spike that has zero width and an area of one (see Fig. 9.1a). It is a function that cannot occur in any real system, but it is a useful mathematical function that will be used in several spots in this book. 

Examining a given concentration profile in greater detail, we may construct the model illustrated in Figure 2.3, which approximates the bell shape by many hypothetical impulse functions of different heights located Ax distance increments apart. Simple diffusion is a linear process that fits the two criteria discussed earlier (Chap. 1). According to the superposition principle, the impulse functions may be treated completely independently if there is no solute-solute interaction (e.g., dimerization). The future of the total profile will be described accurately by a summation of the behavior of each independent impulse. 

The impulse is a special case of the pulse function in which to - 0 but the area A/ under the impulse function remains constant and finite (Figure 7.26). Thus, for the impulse function ... 

The mismatch of surpassing 100% absorption of the active metabolite is probably due to method constraints in combination with the immediate release data, as the deconvolution method requires data from a formulation with zero order absorption for the impulse function, e.g. an oral solution (oral bolus input) the immediate release formulation only provides an approximation to the required properties. 

Figure 11 Some well characterized Fourier pairs. The white spectrum and the impulse function (a), the boxcar and sine functions (b), the triangular and sintf functions (c), and the Gaussian pair (d)...

Consider an impulse function. The impulse function is also known as a Dirac delta function and is represented by S(t). The function has a magnitude oo and an area equal to unity at time t = 0. The Laplace transform of an impulse function is obtained by taking the limit of a pulse function of unit area ast 0. Thus, the area of pulse function HT = 1. The Laplace transform is given by... 

This may seem a rather trivial result, but it shows that a signal can be fully described by a set of appropriately scaled impulse responses. The time invariance property of LTl systems tells us that a shifted input will give a shifted response. For the impulse function, the following is therefore true ... 

The most important control element in the glottal source is of course the rate at which the cycles occur. In Equation 11.30, this is determined by the positions oiN and N2, in Equation 11.32 by Tq, Te and 7b and in Equation 11.31 this is determined by the impulse function u n. In the last case, the glottal volume velocity function can be thought of as a low pass filtering of an impulse stream. From these expressions and empirical measurements, it is known that this low pass filter creates a roll off of about -12dB per octave. It is this, combined with the radiation effect, that gives all speech spectra their characteristic spectral slope. 

It is useful to note that the impulse function arises from the rate of change of a step function since... 

Thus, the derivative of the step function also defines the impulse function. This is seen to be also true for any response function the time derivative of the step response produces the impulse response. 

The Relationship Between Time and Frequency Domain Laplace and Fourier Transform As long as this is true, Fourier transform (F(V(t)), F(I(t)) - settled situation, a static case) or Laplace transform (L(V(t)), L(I(t)) - transition situation, a dynamic case) can be used for the conversion between time and frequency domain, where z(t) is the impulse function characterizing the complex impedance Z(j(o) in time domain. 

If we use a tunable source of pure color (such as a dye laser), then the intensity can be modeled as /(A) = d(X — Aq) with d( ) as the impulse function. The collection of chromaticity coordinates x, y] that will be generated by varying Aq gives the CIE chromaticity triangle as shown in Figure 13.6. 

As can be seen from the sketch, as s —> 0, the rectangular pulse Kit - to) gets taller and thinner, but the limit does not exist. Even though the limit does not exist in the usual sense, one can still derive useful properties of the impulse function. For example. 

A convenient method for providing feedforward compensation that does not interfere with either norm reset or antireset windup involves the use of an impulse relay and a summer as shown in Figure 9.8. In pneumatics these fimctions are sometimes combined into a single device. The impulse function... 

It can be shown by equation (16.39) that P(f) will have a slight offset from set point. The impulse function shovm in Figure 16.9 after the predictor eliminates this. 

Note that the impulse function time constants are the same as the reset time constants of the loops to which the decouplers are coimected. 

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