Question

Can someone please help derive these convolutions. they are from a signals and systems book. show with steps.

eat-1 u(t) eat -ehi u t eat u(t) *e" u(t) = teat u(t) ebteat (a - b)2 teu(t) u(t)

Answer 1

solution:

Continuous Time convolution y (t) = integral ^infinity _ T = -infinity x (T) . n (t - T) dT = x (t) * h (t) e ^at u (t) * u (t) = (e ^at - 1/ a) u (t) \r\n Here two conditions y (t) = { 0: if t < 0 (multiplication = 0) integral ^t _ 0 e ^at (1) dt: if t > 0 = integral ^t _ 0 e ^at dt = e ^at/a] ^t _ 0 = e ^at/a - 1/a = e ^at -1/a e ^at u (t) * u (t) = e ^at - 1/a \r\n e ^at u (t) * e ^bt u (t) = [e ^at - e ^bt/a - b] u (t) , a noteequalto b x (t) = e ^at u (t) h (t) = e ^bt u (t) Rightarrow h (t - T) = e ^b (t - T) u (t) y (t) = { integral ^t _ 0 e ^at e ^h (t - T) d T u (t): t > 0 0: t < 0 = e ^bt integral ^t _ 0