unit_impulse

unit_impulse - Unit Impulse, Unit Step, and Impulse...

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Unit Impulse, Unit Step, and Impulse Response Revisited Do not distribute these notes 83 Prof. R.T. M’Closkey, UCLA
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The Unit Impulse Function The unit impulse function is a signal defined on ( -∞ , ) . It is of fundamental importance for linear systems. The idea behind the unit impulse is that it has a short duration for which it is non-zero compared to the time constants of the system on which it is acting . Thus, the duration for which the impulse is non-zero will depend on the application. In fact, the unit impulse should be considered not as a single function but rather a “set" of functions, from which any member will suit our needs. Any element from the set will be denoted δ . One distinguishing feature is that it has area equal to one: Z -∞ δ ( t ) dt = 1 . This means that if the duration for which the value of the unit impulse is non-zero is reduced, the peak value of the impulse must correspondingly go up. The following functions are all realizations of the unit impulse function: δ = ( 1 Δ t [0 , Δ] 0 otherwise (34) δ = ( ce - ct t 0 ( c > 0) 0 t < 0 (35) δ = ( c 2 te - ct t 0 ( c > 0) 0 t < 0 (36) δ = ( 1 2 c 3 t 2 e - ct t 0 ( c > 0) 0 t < 0 (37) The last three realizations can be useful when computing solutions to ODE’s because the signals are of exponential form which make determining particular solutions somewhat easier than other representations of the unit impulse. Furthermore, the last case is also continuously differentiable which is a useful property when the time derivative of the input appears in the ODE. Perhaps the most important point to remember is that the shape of the unit impulse does not matter, only its duration in relation to the system under test –this will become clear in some examples. Do not distribute these notes 84 Prof. R.T. M’Closkey, UCLA
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Although many engineering texts represent the unit impulse as a function of zero duration and infinite magnitude (but with, remarkably, unit area), in a practical sense it is a just function which is non-zero for only a “short" duration in a neighborhood of t = 0 . Modal testing instruments such as impact hammers make use of this fact. See below for plots of these functions when c = 100 and Δ = 1 / 100 . The graphical representation of the idealized unit impulse is shown on the right. ï 0.02 ï 0.01 0 0.01 0.02 0.03 0.04 0.05 0 20 40 60 80 100 120 140 160 180 200 sec impulse amplitude square pulse 200e ï 200t 400 2 te ï 400t 0.5 u 500 3 t 2 e ï 500t ï 1 ï 0.8 ï 0.6 ï 0.4 ï 0.2 0 0.2 0.4 0.6 0.8 1 ï 0.5 0 0.5 1 1.5 time magnitude t =0 t =0 + impulse only takes on non-zero values in this interval several realization of the unit impulse typical unit impulse representation The notation “ 0 - and “ 0 + " are also introduced and denote the interval of time during which the impulse has non-zero values. Here’s an important property of the unit impulse, Z -∞ g ( c + τ ) δ ( τ ) = Z 0 + 0 - g ( c + τ ) δ ( τ ) = g ( c ) Z 0 + 0 - δ ( τ ) = g ( c ) .
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unit_impulse - Unit Impulse, Unit Step, and Impulse...

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