Question

2. Let y(t)(e')u(t) represent the output of a causal, linear and time-invariant continuous-time system with unit impulse response h[nu(t) for some input signal z(t). Find r(t) Hint: Use the Laplace transform of y(t) and h(t) to first find the Laplace transform of r(t), and then find r(t) using inverse Laplace transform. 25 points

Answer 1

Given that y (t) = (1 - e^- t) u (t) here y (t) is output and h (t) is impulse response and x (t) is input we know by convolution theorem y (t) = x (t) times h (t) but convolution in Laplace domain is Y (s) = X (s) H (s) Rightarrow (1) we know that y (t) = (1 - e^- t) u (t) = u (t) - e^- t u (t) applying Laplace transform we get Y (s) = 1/s - 1/s + 1 = s + 1 - s/s (s + 1) = 1/s (s + 1) therefore Y (s) = 1/s (s + 1) WE KNOW THAT H (T) = E^- T U (T) APPLYING Laplace transform we get therefore H (s) = 1/s + 1 substituting Y (s) and H (s) in equation (1) Y (s) = X (s) H (s) 1
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