EED-201: Assignment 4 (Laplace transform)
1. Find the inverse laplace transform of:
i) 𝑋(s) = s2+2s+5
(s+3)(s+5)2 , Re(s) > -3 ii) 𝑋(s) = 2𝑠+1
s+2 , Re(s) > -2
iii) 𝑋(s) = 𝑠3+2s2+6
𝑠2+3𝑠 , Re(s) > 0
2. Find the laplace transform of the following x(t):
i) x(t) = cos(𝜔0𝑡 + 𝜙) 𝑢(𝑡)
ii) x(t)= 𝑒−𝑎𝑡 𝑢(𝑡)− 𝑒𝑎𝑡𝑢(−𝑡)
iii) x(t)= 𝑠𝑔𝑛(𝑡)
3. The step response of a continuous-time LTI system is given by 1− 𝑒−𝑡𝑢(𝑡). For a certain
unknown input x(t), the output y(t) is observed to be (2 − 3𝑒−𝑡 + 𝑒−3𝑡)𝑢(𝑡). Find the input
x(t).
4. Consider two right-sided signals x(t) and y(t) related through the differential
equations: 𝑑𝑥(t)
dt = −2𝑦(𝑡)+ 𝛿(𝑡)
and 𝑑𝑦(t)
dt = 2𝑥(𝑡)
Determine X(s) and Y(s) along with their regions of convergence.
5. A causal LTI system S has the block diagram representation shown in the figure below.
Determine a differential equation relating the input x(t) to the output y(t) of this system.
6. The system function of a causal LTI system is given by:
𝐻(𝑠)=𝑠+1
𝑠2+2𝑠+2
Determine and sketch the response y(t) when the input is: 𝑥(𝑡)= 𝑒−|𝑡|, −∞ < 𝑡 < ∞.
