9. find the following convolution - - i) [u(t)u(t 1)] u(t − 1) ii) e atu(t) e¯btu(t) iii) (t - - 1)u(t 1) u(t) vi) tu(t)u(t)

7. A CT function x(t) is non-zero over a rahge of its argument from -2 to 3. It is convolved with another function y(t) which is non-zero over a range of its argument from 0 to 3. What is the non- zero range of the convolution of these two functions? Explain your answe. x(t) -2 0 y(t) 3

For e(t) = -3t (a) Express E*(s) as a series. (b) Express E*(s) in closed form. (c) Express E*(s) as a series which is different from that in part (a).

5. Express x(t) i. in terms of elementary signals r(t) x(t) 1 -2 ii. In terms of elementary Signals r(t) & 8(t) x(t) 2 + time -3 -2 -1 2 3 -2 !

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Solve for x(t) in the system shown in the figure below if f(t) is a unit step

8. Given below is the signal m(t). Decompose this signal in terms of the unit step function u(t). -6 -5 -4 -2 -1 m(t) 2 1 0 -1 0 1 2 3 4 t Homework 2 5

Required information Consider the CTFT pair. u (t + 5) – u (t – 7) → A sinc (bf) eicf The numerical values of the literal constants are A = b = and c=

Consider the function given below. Determine whether the given function is causal and BIBO-stable. x(t) = 4 rect (t/2)   Please show work

3. Given x₁(t) = u(t) - u(t - 4) and x₂(t) = e-tu(t - 2), compute the following convo- lutions and sketch the resulting signals: (a) x₁(t) * x₂(t) (b) x2(t) * 8(t − 1)