Description

• Grouping: You are allowed to work in pairs.
• Submission: We provide a latex template for your solutions. Use that template and create a hw1.tar.gz file that includes hw1.tex and all other related files. Tar.gz file should not contain any directories and should create a hw1.pdf file with the following commands, otherwise you will get zero; tar xvzf hw1.tar.gz pdflatex hw1.tex
Submit hw1.tar.gz to the COW page of the course.
• Late Submission: Not allowed.

1. (20 pts) Solve the following, showing your solution in detail.
(a) (5 pts) Given z = x + yj and 3z + 4 = 2j − z¯, (i) find |z|2 and (ii) plot z on the complex plane.
(b) (5 pts) Given z = rejθ and z3 = 64j, find z in polar form.
(c) (5 pts) Find the magnitude and angle of .
(d) (5 pts) Write z in polar form where z = −jejπ/2. 2. (10 pts) Given the x(t) signal in Figure 1, draw the signal
x(t)

Figure 1: t vs. x(t).
3. (15 pts) Given the x[n] signal in Figure 2,
(a) (10 pts) Draw x[−n] + x[2n + 1].
(b) (5 pts) Express x[−n] + x[2n + 1] in terms of the unit impulse function.
x[n]

Figure 2: n vs. x[n].
1
4. (16 pts) Determine whether the following signals are periodic and if periodic find the fundamental period.
(a) (4 pts)
(b) (4 pts)
(c) (4 pts)
(d) (4 pts) x(t) = −jej5t
5. (15 pts) Given the signal in Figure 2, check whether the signal is even or odd. If it is neither even nor odd, then find the even (Ev{x[n]}) and odd (Odd{x[n]}) decompositions of the signal and draw these parts.
6. (24 pts) Analyze whether the following systems have these properties: memory, stability, causality, linearity, invertibility, timeinvariance. Provide your answer in detail.
(a) (6 pts) y(t) = x(2t − 3)
(b) (6 pts) y(t) = tx(t) (c) (6 pts) y[n] = x[2n − 3]
∞
(d) (6 pts) y[n] = P x[n − k]
k=1
2