Unit impulse

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... 

Eq.(2) describes an impulse with the area of 1 [1-3]. Fig. 1 (left) shows such an unit impulse S(x) and an example for an impulse response g(x) at the output of the system. 

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. 

Unit impulse fucnction Unit step function Example for an input signal 5(x) f s(x)f u( ),... 

Unit impulse response Unit step response responses for input example... 

The two types of turbines—axial-flow and radial-inflow turbines—can be divided further into impulse or reaction type units. Impulse turbines take their entire enthalpy drop through the nozzles, while the reaction turbine takes a partial drop through both the nozzles and the impeller blades. 

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). 

A sampler (i.e. an A/D eonverter) is represented by a switeh symbol as shown in Figure 7.5. It is possible to reeonstruet f t) approximately from f t) by the use of a hold deviee, the most eommon of whieh is the zero-order hold (D/A eonverter) as shown in Figure 7.6. From Figure 7.6 it ean be seen that a zero-order hold eonverts a series of impulses into a series of pulses of width T. Flenee a unit impulse at time t is eonverted into a pulse of width T, whieh may be ereated by a positive unit step at time t, followed by a negative unit step at time (t — T), i.e. delayed by T. 

The response-factor approach is based on a method in which the response factors represent the transfer functions of the wall due to unit impulse excitations. The real excitation is approximated by a superposition of such impulses (mostly of triangular shape), and the real response is determined by the superposition of the impulse responses (see Figs. 11.33 and 11.34). ... 

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... 

The differential distribution is the response to a unit impulse. Setting A = gives the expected result. Equation (15.19). 

In testing process systems, standard input disturbances such as the unit-step change, unit pulse, unit impulse, unit ramp, sinusoidal, and various randomised changes can be employed. 

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 ... 

Here, h(t) characterize the behaviour of the system, and is called the response function, or the impulse response, because it is identical to the response to a unit impulse excitation. 

The calorimeter response to a unit impulse must therefore be determined. This may be achieved by solving the Fourier equation [Eq. (23)] for a theoretical model of a heat-flow calorimeter and for this particular heat evolution. 

The problem is apparently simple and may be expressed in the following way knowing g(t), the thermogram, and h(t), the calorimeter response to a unit impulse, solve Eqs. (20) or (35) and determine/(<), the thermokinetics of the phenomenon taking place in the calorimeter. However, the digital information which is used in the computer does not allow the continuous integration of Eq. (35). Both functions g(t) and h(t) are indeed stored and manipulated as series of discrete steps (samples). For a computer s convenience, Eq. (35) must therefore be written... 

As a approaches zero, f(x) becomes oo at x = 0, when it is called a "unit impulse", or a delta function. Its transform g(k) = 1. Of equal importance is the Fourier transform of the Gaussian function... 

To answer this question, away from the context of PF, consider a characteristic function / ( ) that, at t = b, is suddenly increased from 0 to 1/C, where C is a relatively small, but nonzero, interval of time, and is then suddenly reduced to 0 at t = b + C, as illustrated in Figure 13.6. The shaded area of C(l/C) represents a unit amount of a pulse disturbance of a constant value (1/C) for a short period of time (C). As C - 0 for unit pulse, the height of the pulse increases, and its width decreases. The limit of this behavior is indicated by the vertical line with an arrow (meaning goes to infinity ) and defines a mathematical expression for an instantaneous (C - 0) unit pulse, called the Dirac delta function (or unit impulse function) ... 

Let us now define an infinite sequence of unit impulses or Dirac delta functions whose strengths are all equal to unity. One unit impulse occurs at every sampling time. We will call this series of unit impulses, shown in Fig. 18.4, the function /, . 

F. UNIT IMPULSE FUNCTION. By definition, the z transformation of an impulse-sampled function is... 

If/( is a unit impulse, putting it through an impulse sampler should give an /, , that is still just a unit impulse 5( . But Eq. (18.4) says that... 

But if/( ) must be equal to just, the term /,q in the equation above must be equal to 1 and all the other termsniust be equal to zero. Therefore the z transformation of the unit impulse is unity. 

Defining in this way permits us to use transfer functions in the z domain [Eq. (18.57)] just as we use transfer functions in the Laplace domain. G,, is the z transform of the impulse-sampled response of the process to a unit impulse function <5( . In z-transforming functions, we used the notation =... 

The hold must convert an impulse/( of area or strength /( rj at time t = nTj into a square pulse (not an impulse) of height and width. See Fig. 18.10. Let the unit impulse response of the hold be defined as /j,. If the hold is to do what we want it to do (i.e., convert an impulse into a step up and then a step down after Tj minutes), its unit impulse response must be... 

The term in parentheses is the Laplace transformation of the impulse-sampled response of the total combined process to a unit impulse input. We will call this (GiG,)S, in the Laplace domain and (Gi G2)(i) in the z domain. 

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