Question

Problem 7.1 (10 points) Express the unilateral z-transforms of the following functions as rational functions. Find also the ROC. You may use tables. (a) xl[n]-1-0.2)" (b) x2[n] (0.3)" +2(-5) -0.2n Problem 7.2 (10 points) Express the unilateral z-transforms of the following functions as rational functions. Find also the ROC. You may use tables. (a) xl[n] = 3e-j02" (b) x2[n]- 5cos(5n) (c) x3[n] = e-0.gn sin(0.7n) Problem 7.3 (10 points) The signals given are sampled every 0.3 s, beginning att-0. Find the unilateral z- transforms of the sampled functions, with each transform expressed as a rational function. (a) x1(t) = 3e" + 2e' (b) x2(t)-3e°3/" (c) x3(t) -5e"cos(31) -0.3jt Problem 7.4 (10 points (a) Use the z-transform to evaluate the following series: ? x1 (0.5)" (ii) x2->(0.5)" (b) Use the z-transform to evaluate the series: x3-?(0.3)" cos(0.2n) (Hint: Compute the z-transform of the expression under the sum, and see if you can use specific values of the z-transform to evaluate the series.)

Answer 1

"(a) x(n) = (-0.2)^n X(Z) = (-0.2)^2 X(z) = sigma^infinity_n = 0 (-0.2)^n z^-n = sigma^infinity_n = 0 (-0.2z^-1)^n = 1 + (0.2z^-1) + (-0.2z^-1)^2 + - - - - infinity = 1/1 - (-0.2z^-1) X(Z) = 1/1 + 0.2z^-1
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