Step function unit impulse

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( ),... 

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. 

Example 2.10 What is the time domain response C (t) in Eq. (2-27) if the change in inlet concentration is (a) a unit step function, and (b) an impulse function ... 

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... 

Inversion of the Laplace transformation gives the time function. If we have a transfer function the unit step function is C [G(,y s] and the impulse response is C" [G(,)]. 

FIGURE 4 Indiviclvial impulse-response functions for the carbon cycle (response R. of the atmospheric COt concentration to a 5-function COi input at time t = 0, left) and the variables global mean near-surface temperature (center) and mean sea-level rise (right) for the physical ocean-atmosphere sy stem (response to a step-function increase in the CO2 concentration to a constant level at time t = 0). The units for temperature response Rj and sea-level response R< refer to the amplitudes of the first empirical orthogonal functions (EOFs) of the response patterns of the respective variables and are essentially arbitrary (see text). Adopted from Hooss ft /., 2001. 

Passing the step function through an impulse sampler gives = Kun(t)I t)-, where /(,) is the sequenee of unit impulses defined in Eq. (14.3). Using the definition of z transformation [Eq. (14.22)J gives... 

The response from the test administration is deconvolved using the perturbed unit impulse response determined in step 3 to give the input functions ... 

The unit impulse function. Sit), is often called the Dirac delta function. It behaves in the manner shown in Fig. 9.8. The area under the curve is always unity, and as 6 becomes small, the height of the pulse tends to infinity. We can define this distribution in terms of unit step functions... 

Z(w) = 1/wC, and in the s-domain Z(s) = 1/sC. The Laplace transforms of some very important excitation waveforms are very simple for example, for a unit impulse it is 1, a unit step function 1/s, a ramp 1/s, etc. That is why the excitation with, for example, a unit impulse is of special interest examining the response of a system. In the extended immittance definition, calculations with some nonsinusoidal waveforms become very simple. Even so, Laplace transforms are beyond the scope of this book. 

A single pulse or a step function excitation is the basis of relaxation theory. Power dissipation and temperature rise may for instance impede the use of repeated waveforms, and single pulse excitation is necessary. A single pulse is a pulse waveform with repetition interval oo, it has a continuous frequency spectrum as opposed to a line spectrum. The unit impulse (delta function) waveform is often used as excitation waveform. It is obtained with the pulse width 0 and the pulse amplitude oo, keeping the product = 1. The frequency spectrum consists of equal contributions of all frequencies. In that respect, it is equal to white noise (see the following section). Also, the infinite amplitude of the unit pulse automatically brings the system into the nonlinear region. The unit impulse is a mathematical concept a practical pulse applied for the examination of a system response must have limited amplitude and a certain pulse width. 

Thus, F(9) is a unit step function shifted along the time axis by 0 and E(0) is a unit impulse function shifted along time axis by 0. Sketches of F(0) and E(0) for ideal PFR are shown in Figures 3.52 and 3.53, respectively. 

Figure 2.3. Depiction of unit step, time delay, rectangular, and impulse functions.

When identifying the hypothetical system S we need u. The weighting function found in Example 5.6 is substituted for the input of the hypothetical system. This input does not contain an impulse or a unit step component, and hence we set DC = 0 and US = 0. The response of the hypothetical system equals the "Observed" response. The program is the one used in Example 5.6, only tha data lines are changed as follows ... 

In the dynamic optimization method [234-236], Eq. (2.9) is taken as a mathematical model of the calorimeter, and thus appropriate zero initial conditions are assumed. This method assumes the existence of one input function T(t) and one output function P(t). The impulse response H(t) is determined as a derivative with respect to time of the response of the calorimetric system to a unit step. As a criterion of accordance between the measured temperature change T(t) and the estimated course of temperature x(t), the integral of the square of the difference between these two courses is taken ... 

A discrete-time impulse or step response model can be developed from a transfer function model or a linear differential (or difference) equation model. For example, consider a first-order transfer function with t = 1 min and K=l, and a unit step input change. The first-order difference equation corresponding to Eq. 7-34 with y(0) = 0 and Ar = 0.2 is... 

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