Sheet (5) (1) Sketch the pole-zero plot and region of convergence for the following signal ๐ฅ(๐ก) = ๐ ๐ข(๐ก) (2) Starting with the definition of La place transform , find the La place transform of the following signal : ๐ฅ(๐ก) = ๐ ๐ข(๐ก) (3) Using the time-shifting property , find the La place transform of the following signal ๐ฅ(๐ก) = ๐ข(๐ก) − ๐ข(๐ก − 1) (4) Using the complex-frequency-shifting property, find and sketch the inverse Laplace transform of the following 1 1 ๐(๐ ) = + (๐ + ๐4) + 3 (๐ − ๐4) + 3 (5) Using the time-scaling property , find the La place transform of these signals a. ๐ฅ(๐ก) = ๐ฟ(4๐ก) b. ๐ฅ(๐ก) = ๐ข(4๐ก) (6) Using the differentiation property , find the La place transform of these signals a. ๐ฅ(๐ก) = (๐ข(๐ก)) b. ๐ฅ(๐ก) = (4 sin(10๐๐ก) ๐ข(๐ก)) (7) Using the initial and final value theorems, find the initial and final values (if possible) of the signals whose Laplace transforms are these functions a. ๐(๐ ) = b. ๐(๐ ) = ( ) (8) Find the inverse Laplace transforms of these functions a. ๐(๐ ) = b. ๐(๐ ) = c. ๐(๐ ) = ( ) (9) Using the Laplace transform, solve these differential equations for t ≥ 0. a. ๐ฅฬ (๐ก) + 10๐ฅ(๐ก) = ๐ข(๐ก) , ๐ฅ(0) = 1 b. ๐ฅฬ (๐ก) − 2๐ฅฬ (๐ก) + 4๐ฅ(๐ก) = ๐ข(๐ก) , ๐ฅ(0) = 0 ๐๐๐ ๐ฅฬ (0) = 4 (10) Write the differential equations describing these systems and find and sketch the indicated responses : a. ๐ฅ(๐ก) = ๐ข(๐ก) , ๐ฆ(๐ก) ๐๐ ๐กโ๐ ๐๐๐ ๐๐๐๐ ๐ , ๐ฆ(0) = 0 b. ๐ฃ(0) = 10 , ๐ฃ(๐ก) ๐๐ ๐กโ๐ ๐๐๐ ๐๐๐๐ ๐ (11) Find the response ๐ฆ(๐ก) of the system whose impulse response โ(๐ก) to the excitation ๐ฅ(๐ก) where โ(๐ก) = ๐ ๐ข(๐ก) , ๐ฅ(๐ก) = 3๐ ๐ข(๐ก) − 12๐ ๐ข(−๐ก) (12) Using the integral definition find the the unilateral Laplace transform of these time Functions : a. ๐(๐ก) = ๐ ๐ข(๐ก) b. ๐(๐ก) = sin(๐ ๐ก) ๐ข(๐ก) (13) Using the unilateral la place transform properties find the unilateral Laplace transforms of the following functions : a. ๐(๐ก) = 5 sin 2๐(๐ก − 1) ๐ข(๐ก − 1) b. ๐(๐ก) = 2 cos(10๐๐ก) cos(100๐๐ก) ๐ข(๐ก) c. ๐(๐ก) = ๐ข(๐ก − 2) d. ๐(๐ก) = ∫ ๐ข(๐) ๐๐ (14) Given ๐ ๐ข(๐ก) ↔ ๐บ(๐ ) , find the inverse Laplace transforms of : a. ๐บ b. ๐บ(๐ − 2) + ๐บ(๐ + 2) c. ( )