18.03 Lecture Notes - Weight Function, Hyperbola, Exponential Decay
Document Summary
Laplace transform iv: the pole diagram: another example, t-shift rule, poles, what the pole diagram of f(s) says about f(t) Other input signals? x" + 4x = cos(2t) rest initial conditions (no delta functions in this signal, so we know that x(0+) = 0 and x"(0+) = 0, but that will come out automatically. ) From our latest addition to the table, we find that x = u(t) 1/2 t sin(t) (resonance!) General fact: p(d) x = f(t) with rest initial conditions has laplace transformed equation p(s) x(s) = f(s) with solution. In the s-domain, the system response is obtained by multiplying by the transfer function! [2] another rule: let a > 0 and define f_a(t) = 0 if t < a. = f(t-a) if t > a (if f_s(t) contains c delta(t) , i also want f_a(t) to contain c delta(t-a) .
