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 odtuclass page of the course.
• Late Submission: Not allowed.

1. (12 pts) Analyze whether the following systems have these properties: memory, stability, causality, linearity, invertibility, timeinvariance. Provide your answer in detail.

(b) (6 pts) y(t) = tx(2t + 3)
2. (13 pts) Consider an LTI system given by the following block diagram:

−5
(a) (3 pts) Find the differential equation which represents this system.
(b) (10 pts) Find the output y(t), when the input x(t) = (e−t + e−3t)u(t). Assume that the system is initially at rest.
3. (15 pts) Evaluate the following convolutions.
(a) (10 pts) Given x[n] = 2δ[n] + δ[n + 1] and h[n] = δ[n− 1] + 2δ[n + 1], compute and draw y[n] = x[n] ∗h[n].
(b) (5 pts) Given x(t) = u(t− 1) + u(t + 1) and h(t) = e−t sin(t)u(t), calculate
4. (20 pts) Evaluate the following convolutions.
(a) (10 pts) Given h(t) = e−2tu(t) and x(t) = e−tu(t), find y(t) = x(t) ∗h(t).
(b) (10 pts) Given h(t) = e3tu(t) and x(t) = u(t) −u(t− 1), find y(t) = x(t) ∗h(t).
5. (20 pts) Solve the following homogeneous difference and differential equations with the specified initial conditions.
(a) (10 pts) 2y[n + 2] − 3y[n + 1] + y[n] = 0, y[0] = 1 and y[1] = 0.
(b) (10 pts) y(3)(t) − 3y00(t) + 4y0(t) − 2y(t) = 0, y00(0) = 2, y0(0) = 1 and y(0) = 3.
6. (20 pts) Consider the following discrete time LTI system which is initially at rest:

where
(a) (10 pts) Find h0[n].
(b) (5 pts) Find the overall impulse response, h[n], of this system.
(c) (5 pts) Find the difference equation which represents the relationship between the input x[n] and the output y[n].
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