18.03 Lecture Notes - Lecture 24: Derivative, Piecewise, Step Response
Document Summary
Unit impulse and step responses: generalized derivative, rest initial conditions, first order unit step/impulse response, second order unit step/impulse response. 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 f(t) = t + 2 u(t) , f"(t) = 1 + 2 delta(t) f"_r(t) = 1 f"_s(t) = 2 delta(t)
