3.
9 Problems
1. Find the Laplace transform X(s) of x(t) and determine its region of
convergence
If 𝑥(𝑡) = 2[𝛿(𝑡 + 1) + 𝛿(𝑡 − 1)]
2. Find the Laplace transform of 𝑦(𝑡) = 𝑠𝑖𝑛(2𝜋𝑡)𝑢(𝑡) − 𝑠𝑖𝑛(2𝜋(𝑡 − 1))𝑢(𝑡 −
1)) and its region of convergence.
3. Find the Laplace transform of the reflection of the unit-step signal (i.e., u(−t))
and its region of convergence. Then use the result together with the Laplace
transform of u(t) to see if you can obtain the Laplace transform of a constant
or u(t)+u(−t) (assume u(0) =0.5 so there is no discontinuity at t =0).
4. The input to an LTI system is 𝑥(𝑡) = 𝑢(𝑡) − 2𝑢(𝑡 − 1) + 𝑢(𝑡 − 2)
If the Laplace transform of the output is given by
(𝑠 + 2)(1 − 𝑒 −𝑠 )2 )
𝑌(𝑠) =
𝑠 2 (𝑠 + 1)2
determine the transfer function of the system.
5. Find the inverse Laplace transform of
𝑠2 − 3
𝑋(𝑠) =
(𝑠 + 1)(𝑠 + 2)
6. Find the inverse Laplace transform of
3𝑠 − 4
𝑋 (𝑠) =
𝑠 (𝑠 + 1)(𝑠 + 2)
7. Find the poles and zeros of
1 − 𝑠𝑒 −𝑠
𝑋(𝑠) =
𝑠(𝑠 + 2)
Find the inverse Laplace transform x(t)
8. The impulse response of an LTI system is ℎ(𝑡) = 𝑒 −2𝑡 𝑢(𝑡) and the system
input is a pulse 𝑥(𝑡) = 𝑢(𝑡) − 𝑢(𝑡 − 3). Find the output of the system y(t) by
means of the convolution integral graphically and by means of the Laplace
transform.
9. Suppose that the transfer function of a LTI system is
𝑠
𝐻(𝑠) =
𝑠2 +𝑠+ 1
Find the unit-step response s(t) of the system, and then use it to find the
response due to the following inputs:
63
𝑥1(𝑡) = 𝑢(𝑡) − 𝑢(𝑡 − 1)
𝑥2(𝑡) = 𝛿(𝑡) − 𝛿(𝑡 − 1)
10. Suppose 𝑥(𝑡) = 𝑢(𝑡) − 𝑢(𝑡 − 1). is the input of an LTI system with a
transfer function 𝐻(𝑠) = 1/(𝑠 2 + 4𝜋 2 ).
Find and plot the poles and zeros of 𝑌(𝑠) = 𝐿[𝑦(𝑡)] = 𝐻(𝑠)𝑋(𝑠) where y(t)
is the output of the system.
64
