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MIT18_03S10_c24

# MIT18_03S10_c24 - 18.03 Class 24 April 5 2010 Unit impulse...

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18.03 Class 24, April 5 , 20 10 Unit impulse and step responses 1. Generalized derivative 2. Rest initial conditions 3. First order unit step/impulse response 4. Second order unit step/impulse response [1] Generalized derivative The unit step and delta functions help deal with events happening on a time scale which is very short relative to our interest. u(t) can be thought of as a function which, except for t very near zero, is given by u(t) = 0 for t < 0 and u(t) = 1 for t > 0 . delta(t) can be thought of as a function which is zero except for t very near zero, and has area under its graph equal to 1 . u'(t) = delta(t) . A generalized function is by definition a sum g(t) = g_r(t) + g_s(t) , where its *regular part* g_r(t) is piecewise smooth, and its *singular part* g_s(t) is a linear combination of shifted delta functions. Any regular f(t) function has a "generalized derivative" f'(t) which is a generalized function: f'(t) = f'_r(t) + f'_s(t) . The regular part is the ordinary derivative of f(t) (except at the break points, where it is undefined). The singular part is a sum of delta functions, one for each break in the graph: (f(a+)-f(a-)) delta(t-a) There's no separate notation for the generalized derivative to distinguish it from the ordinary derivative, and we will just write f'(t) or dot-x (t). For example, if

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MIT18_03S10_c24 - 18.03 Class 24 April 5 2010 Unit impulse...

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