3. Consider the time function x(t) shown in the figure below 12 (1) X 10 00...

 

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3. Consider the time function x(t) shown in the figure below 12 (1) X 10 00 6 st 4 2 2 x(t) = at², t≤s 3 mom 4 5 Time t (seconds) x(t) = 10, t > 5 6 9 10 Suppose this signal is sampled every T = 0.5 seconds, beginning at t = 0, to obtain the discrete- time signal x = x(KT). Find the z-transform X(z) of x using the following steps = (a) Express the signal x(t) in terms of the unit step u(t), unit ramp r(t) tu(t), and the unit acceleration a(t) = t² u(t). (b) Use the answer in (a) to obtain a closed-form expression for .xk (c) Compute the z-transform X(z) of x using the transform tables in the text, and any other properties of the z-transform that you might need (we listed many of these in class). (d) Use X(z) and the Final Value Thm to compute lim. Does this result agree with what you expect from x(t)? If not, there is a good chance that you solved (c) incorrectly! 3. Consider the time function x(t) shown in the figure below 12 (1) X 10 00 6 st 4 2 2 x(t) = at², t≤s 3 mom 4 5 Time t (seconds) x(t) = 10, t > 5 6 9 10 Suppose this signal is sampled every T = 0.5 seconds, beginning at t = 0, to obtain the discrete- time signal x = x(KT). Find the z-transform X(z) of x using the following steps = (a) Express the signal x(t) in terms of the unit step u(t), unit ramp r(t) tu(t), and the unit acceleration a(t) = t² u(t). (b) Use the answer in (a) to obtain a closed-form expression for .xk (c) Compute the z-transform X(z) of x using the transform tables in the text, and any other properties of the z-transform that you might need (we listed many of these in class). (d) Use X(z) and the Final Value Thm to compute lim. Does this result agree with what you expect from x(t)? If not, there is a good chance that you solved (c) incorrectly!

Answer rating: 100% (QA)

a Expressing xt in terms of unit functions Given that xt t2 for 0 t 5 xt 10 for t 5 ... View the full answer