Sheet (5)
(1) Sketch the pole-zero plot and region of convergence for the following signal
𝑥(𝑡) = 𝑒
𝑢(𝑡)
(2) Starting with the definition of La place transform , find the La place transform of the
following signal :
𝑥(𝑡) = 𝑒 𝑢(𝑡)
(3) Using the time-shifting property , find the La place transform of the following signal
𝑥(𝑡) = 𝑢(𝑡) − 𝑢(𝑡 − 1)
(4) Using the complex-frequency-shifting property, find and sketch the inverse Laplace
transform of the following
1
1
𝑋(𝑠) =
+
(𝑠 + 𝑗4) + 3 (𝑠 − 𝑗4) + 3
(5) Using the time-scaling property , find the La place transform of these signals
a. 𝑥(𝑡) = 𝛿(4𝑡)
b. 𝑥(𝑡) = 𝑢(4𝑡)
(6) Using the differentiation property , find the La place transform of these signals
a. 𝑥(𝑡) =
(𝑢(𝑡))
b. 𝑥(𝑡) =
(4 sin(10𝜋𝑡) 𝑢(𝑡))
(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. 𝑋(𝑠) =
b. 𝑋(𝑠) =
(
)
(8) Find the inverse Laplace transforms of these functions
a. 𝑋(𝑠) =
b. 𝑋(𝑠) =
c. 𝑋(𝑠) =
(
)
(9) Using the Laplace transform, solve these differential equations for t ≥ 0.
a. 𝑥̇ (𝑡) + 10𝑥(𝑡) = 𝑢(𝑡) , 𝑥(0) = 1
b. 𝑥̈ (𝑡) − 2𝑥̇ (𝑡) + 4𝑥(𝑡) = 𝑢(𝑡) , 𝑥(0) = 0 𝑎𝑛𝑑 𝑥̇ (0) = 4
(10) Write the differential equations describing these systems and find and sketch the
indicated responses :
a. 𝑥(𝑡) = 𝑢(𝑡) , 𝑦(𝑡) 𝑖𝑠 𝑡ℎ𝑒 𝑟𝑒𝑠𝑝𝑜𝑛𝑠𝑒 , 𝑦(0) = 0
b. 𝑣(0) = 10 , 𝑣(𝑡) 𝑖𝑠 𝑡ℎ𝑒 𝑟𝑒𝑠𝑝𝑜𝑛𝑠𝑒
(11) Find the response 𝑦(𝑡) of the system whose impulse response ℎ(𝑡) to the excitation
𝑥(𝑡) where ℎ(𝑡) = 𝑒 𝑢(𝑡) , 𝑥(𝑡) = 3𝑒 𝑢(𝑡) − 12𝑒 𝑢(−𝑡)
(12) Using the integral definition find the the unilateral Laplace transform of these time
Functions :
a. 𝑔(𝑡) = 𝑒 𝑢(𝑡)
b. 𝑔(𝑡) = sin(𝜔 𝑡) 𝑢(𝑡)
(13) Using the unilateral la place transform properties find the unilateral Laplace transforms
of the following functions :
a. 𝑔(𝑡) = 5 sin 2𝜋(𝑡 − 1) 𝑢(𝑡 − 1)
b. 𝑔(𝑡) = 2 cos(10𝜋𝑡) cos(100𝜋𝑡) 𝑢(𝑡)
c. 𝑔(𝑡) =
𝑢(𝑡 − 2)
d. 𝑔(𝑡) = ∫ 𝑢(𝜏) 𝑑𝜏
(14) Given 𝑒
𝑢(𝑡) ↔ 𝐺(𝑠) , find the inverse Laplace transforms of :
a. 𝐺
b. 𝐺(𝑠 − 2) + 𝐺(𝑠 + 2)
c.
( )