Description

• Grouping: You are allowed to work in pairs.
• Submission: We provide a latex template for your solutions. Use that template and create a hw2.tar.gz file that includes hw2.tex and all other related files. Tar.gz file should not contain any directories and should create a hw2.pdf file with the following commands, otherwise you will get zero; tar xvzf hw2.tar.gz pdflatex hw2.tex
Submit hw2.tar.gz to the COW page of the course.

−4
(a) (5 pts) Find the differential equation which represents this system.
(b) (15 pts) Find the output y(t), when the input x(t) = (e−t + e−2t)u(t). Assume that the system is initially at rest.
2. (20 pts) Evaluate the following convolutions.
(a) (10 pts) Given x[n] = δ[n−1]−3δ[n−2]+δ[n−3] and h[n] = δ[n+1]+2δ[n]−3δ[n−1], compute and draw y[n] = x[n]∗h[n]. (b) (10 pts) Given x(t) = u(t) + u(t − 1) and h(t) = e−2t cos(t)u(t), calculate
3. (20 pts) Evaluate the following convolutions.
(a) (10 pts) Given h(t) = e−3tu(t) and x(t) = e−tu(t), find y(t) = x(t) ∗ h(t).
(b) (10 pts) Given h(t) = etu(t) and x(t) = u(t − 1) − u(t − 2), find y(t) = x(t) ∗ h(t).
4. (20 pts) Solve the following homogeneous difference equations with the specified initial conditions.
(a) (10 pts) y[n] − 15y[n − 1] + 26y[n − 2] = 0,y[0] = 10 and y[1] = 42.
(b) (10 pts) y[n] − 3y[n − 1] + y[n − 2] = 0,y[0] = 1 and y[1] = 2.
5. (20 pts) Consider a continuous LTI system represented by the following differential equation, which is initially at rest:

(a) (12 pts) Find the impulse response of this system.
(b) (8 pts) Using the impulse response you found in (a), analyze if this system is i. causal,
ii. memoryless,
iii. stable,
iv. invertible.
1