In convolution problems the impulse response h(t)of the system and the input x(t)are given and one is interested in finding the output of the system y(t). The so-called €œdeconvolution€ problem consists in giving two of x(t), h(t) and y(t) to find the other. For instance, the output y(t) and the impulse response h(t) of the system and one wants to find the input. Consider the following cases:

(a) Suppose the impulse response of the system is h(t) = e^ˆ’tcos(t) u(t) and the output has a Laplace transform

what is the input x(t)?

(b) The output of a LTI system is y₁(t) = r(t) ˆ’ 2r(t ˆ’ 1) + r(t ˆ’ 2), where r(t)is the ramp signal. Determine the impulse response h₁(t)of the system if it is known that the input is x₁(t) = u(t) ˆ’ u(t ˆ’ 1).

Y (s) = s((s+ 1)? + 1)