Question

8 + If Problem la (10 points) Find the inverse Laplace transform of: Fo(s) 12 the ROC is defined as: -12 <Re(s) <0 Identify terms as right sided or left sided. S+12 Re(s) < 0 Re(s) >-12 Х X -12 0 Problem 1b (2 points) Circle one: The function f(t) Is: causal anti-causal not causal Explain why:

Answer 1

Question (a)

12 F, (s) = s +12 8 + - S

The Inverse Laplace Transform is

fo(t) = (-'{F, (s)}

8) fo(t) = (-11 12 (s + 12 + S

fo(t) = 12.4-1(+12}+8.1-18

Using

  

1 L{e-at u(t)} ROC: Re(s) > -a sta

L{u(-t)} = ROC: Re(s) < 0 S

So

1 L{e-12* u(t)}= ROC: Re(s) > -12 s + 12

L{u(-t)} = ROC: Re(s) < 0 S

We get

fo(t) = 12.e-12* u(t) +8.u(-t)

So the Inverse is

fo(t) = 12.e-12* u(t) +8.u(-t)

The right sided sequence is

12. e -12t ut)

The left sided sequence is

8.u(-t)

Question (b)

The function is not causal as the present value depends on the future inputs

For example, the output at time t = -2

fo(-2) = 12.e-12x-2u(-2) +8.u(2)

So the output at time t = -2 depends on the input at t = 2. So the function is not causal