1. (20 pts) Solve the following, showing your solution in detail.
(a) (5 pts) Given z = x + yj and 3z + 4 = 2j z, (i) nd jzj2 and (ii) plot z on the complex plane.
(b) (5 pts) Given z = rej and z3 = 64j, nd z in polar form.
p
(c) (5 pts) Find the magnitude and angle of z = (1 j)(1+ 3j) .
1+j
(d) (5 pts) Write z in polar form where z = jej =2.
2. (10 pts) Given the x(t) signal in Figure 1, draw the signal y(t) = x( 12 t + 1).
x(t)
3
2
1
t
4
3
2
1
1
2
3
4
1
2
3
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]
4
3
2
1
n
1
1
2
3
4
5
6
7
8
1
2
3
4
Figure 2: n vs. x[n].
1
4. (16 pts) Determine whether the following signals are periodic and if periodic nd the fundamental period.
(a)
(4 pts) x[n] = 3 cos[
13
n] + 5 sin[
7
n
2
]
3
10
3
(b)
(4 pts) x[n] = 5 sin[3n
]
4
(c)
(4 pts) x(t) = 2 cos(3 t
2
)
5
(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 nd the even (Evfx[n]g) and odd (Oddfx[n]g) decompositions of the signal and draw these parts.
6. (24 pts) Analyze whether the following systems have these properties: memory, stability, causality, linearity, invertibility, time-invariance. 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]
1
(d)
(6 pts) y[n] =
kP
x[n k]
=1
2
