1. (16 pts) Solve the following, showing your solution in detail.
(a)
(4 pts) Given a complex number in Cartesian coordinate system, z = x + jy and 2z − 9 = 4j − z¯,
i. find |z|2 and
ii. find and plot z on the complex plane.
(b)
(4 pts) Given z3 = −27j, find z in polar form (z = rejθ).
√
(c)
(4 pts) Find the magnitude and angle of z =
(1+j)( 3−j)
(√
3+j)
(d)
(4 pts) Write z in polar form where z = −(1 + j)8ejπ/2.
2.
(12 pts) Calculate power P and energy E of the given signals and determine whether they are Power signals, Energy signals or
neither of them.
(a)
(6 pts) x[n] = nu[n]
(b)
(6 pts) x(t) = e−2tu(t)
3.
(10 pts) Given the x(t) signal in Figure 1, draw the signal y(t) = 21 x(−31 t + 2).
x(t)
3
2
1
−3
−2
−1
1
2
3 t
−1
−2
−3
Figure 1: t vs. x(t).
4. (15 pts) Given the x[n] signal in Figure 2,
(a) (10 pts) Draw x[−2n] + x[n − 2].
(b) (5 pts) Express x[−2n] + x[n − 2] in terms of the unit impulse function.
1
x[n]
3
2
1
−7 −6 −5 −4 −3 −2 −1
n
1 2 3
−1
−2
−3
Figure 2: n vs. x[n].
5. (8 pts) Determine whether the following signals are periodic and if periodic find the fundamental period.
(a)
(4 pts) x(t) =
ej3t
−j
(b)
(4 pts) x[n] =
1
sin[
7π
n] + 4 cos[
3π
n −
π
]
2
8
4
2
6. (15 pts) Consider the signal in Figure 1.
(a) (5 pts) Show that the signal is neither even nor odd.
(b) (10 pts) Find the even and odd decompositions of the signal and draw these parts.
7. (12 pts) Given the x(t) signal in Figure 3,
(a) (5 pts) Express x(t) in terms of the unit step function.
(b) (7 pts) Find and draw dxdt(t) .
x(t)
3
2
1
t
−3−2−1 1 2 3 4 5 6 7 8 910
−1
−2
−3
Figure 3: t vs. x(t).
8. (12 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[n] = x[2n − 2]
(b) (6 pts) y(t) = tx( 2t − 1)
2
