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 odtuclass 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 z + 1 = j − 3¯z, (i) find |z|2 and (ii) plot z on the complex plane.
(b) (5 pts) Given z = rejθ and z2 = 25j, find z in polar form.
(c) (5 pts) Find the magnitude and angle of .
(d) (5 pts) Write z in polar form where z = je−jπ/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].
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4. (20 pts) Determine whether the following signals are periodic and if periodic find the fundamental period.
(a) (5 pts)
(b) (5 pts)
(c) (5 pts)
(d) (5 pts) x(t) = jej2t
5. (20 pts) Given the signal in Figure 1, check whether the signal is even or odd. If it is neither even nor odd, then find the even (Ev{x(t)}) and odd (Odd{x(t)}) decompositions of the signal and draw these parts.
6. (15 pts) Given the x(t) signal in Figure 3,
(a) (5 pts) Express x(t) in terms of the unit step function.
(b) (10 pts) Find and draw .
x(t)

Figure 3: t vs. x(t).
2