Tut5 mohit kumar - This is question bank for Laplace and Z transforms PDF

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This is question bank for Laplace and Z transforms...

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Indian Institute of Technology Roorkee Department of Electronics & Computer Engineering Signals & Systems (ECN 203)

Tutorial Sheet No: 5

1. Determine the Laplace transform and the associated region of convergence and pole-zero plot for each of the following functions of time:  1, 0≤t ≤1 (a) x(t) = e−2t u(t) + e−3t u(t) (g) x(t) = 0 , elsewhere −4t − (b) x(t) = e u(t) + e 5t sin(5t)u(t)  t, 0≤t ≤1 (c) x(t) = e2t u(−t) + e3t u(−t) (h) x(t) = 2 − t, 1 ≤t ≤2 (d) x(t) = te−2|t | (e) x(t) = |t| e−2|t| (f) x(t) = |t| e2t u(−t)

(i) δ (t) + u(t) (j) δ (3t) + u(3t)

2. Determine the function of time, x(t), for each of the following Laplace transforms and their associated region of convergence: (a) (b) (c) (d)

1 , ℜe{s} > 0 s2 +9 1 , ℜe{s} < 0 s2 +9 s+1 , ℜe{s} < −1 (s+1)2 +9 s+2 , −4 < ℜe{s} < −3 s2 +7s+12

(e)

s+1 , s2 +5s+6

(f)

( s+ 1 ) 2 , s2 −s+1

ℜe{s} > 21

(g)

s2 −s+1 , ( s+ 1 ) 2

ℜe{s} > −1

−3 < ℜe{s} < −2

3. An absolutely integrable signal x(t) is known to have a pole at s=2. Answer the following questions: (a) Could x(t) be of finite duration? (b) Could x(t) be left sided? (c) Could x(t) be right sided? (d) Could x(t) be two sided? 4. Suppose the following facts are given about the signal x(t)with Laplace transform X (s): (a) x(t) is real and even. (b) X (s) has four poles and no zeros in the finite s-plane. (c) X (s) has a pole at x(t) = (1/2)e jπ/4 . (d)

R∞

x(t)dt = 4.

−∞

Determine X (s) and its ROC. 5. Let x(t) be a signal that has a rational Laplace transform with exactly two poles, located at s = −1 and s = −3. If g(t) = e2t x(t) and G( jω ) converges, determine weather x(t) is left sided, right sided, or two sided.

6. Consider a continuous-time LTI system for which the input x(t) and output y(t) are related by the differential equation d 2 y(t) dy(t) − 2y(t) = x(t). − dt dt 2 Let X (s) and Y (s) denote Laplace transforms of x(t) and y(t), respectively, and let H(s) denote the Laplace transform of h(t), the system impulse response. (a) Determine H(s) as a ratio of two polynomials in s. Sketch the pole-zero pattern of H(s). (b) Determine h(t) for each of the following cases: i. The system is stable. ii. The system is casual. iii. The system is neither casual nor stable. s+1 . s2 +2s+2 = e−|t| , −∞ < t

7. The system function of a causal LTI system is H(s) = Determine and sketch y(t ) when the input is x(t )

< ∞.

8. We are given the following five facts about a real signal x(t) with Laplace transform X (s): 4) e2t x(t) is not absolutely integrable.

1) X (s) has exactly two poles. 2) X (s) has no zeros in the finite s-plane. 3) X (s) has a pole at s = −1 + j .

5) X (0) = 8.

Determine X (s) and its region of convergence. 9. Consider a signal related to two signals x1 (t) and x2 (t) by x(t) = x1 (t − 2) ∗ x2 (−t + 3) L

where x1 (t) = e−2t u(t) and x2 (t) = e−3t u(t). Given that e−at ←→ properties of the Laplace transform to determine Y (s).

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1 s+a ,

ℜe{s} > a, use...