Consider the following problems related to the convolution integral.

(a) The impulse response of a LTI system is h(t) = e^−2tu(t) and the system input is a pulse x(t) = u(t) − u(t − 3). Find the output of the system y(t) by means of the convolution integral graphically and by means of the Laplace transform.

(b) It is known that the impulse response of an analog averager is h₁(t) = u(t) − u(t − 1), consider the input to the averager x₁(t) = u(t) − u(t − 1), determine graphically as well as by means of the Laplace transform the corresponding output of the averager y₁(t) = [h₁∗x₁](t). Is y₁(t) smoother than the input signal x₁(t)? provide an argument for your answer.

(c) Suppose we cascade 3 analog averagers each with the same impulse response h_i(t) = u(t) − u(t − 1), i = 1, 2, 3, determine the transfer function of this system. If the duration of the input to the first averager is M sec., what would be the duration of the output of the 3th averager?