1. (20 pts) Solve the following, showing your solution in detail.
(a) (5 pts) Given z = x + yj and z + 1 = j 3z, (i) nd jzj2 and (ii) plot z on the complex plane.
(b) (5 pts) Given z = rej and z2 = 25j, nd z in polar form.
p
j)
.
(c) (5 pts) Find the magnitude and angle of z =
(1+j)(1
3
1
j
(d)
(5 pts) Write z in polar form where z = je
j =2.
2.
(10 pts) Given the x(t) signal in Figure 1, draw the signal y(t) =
1 x(2t
2).
2
x(t)
2
1
t
1
2
3
4
5
6
7
8
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]
6
5
4
3
2
1
n
2
1
1
2
3
4
5
6
7
8
9
10 11 12
1
2
3
4
Figure 2: n vs. x[n].
1
4. (20 pts) Determine whether the following signals are periodic and if periodic nd the fundamental period.
(a)
(5 pts) x[n] = 7 sin[
5
n
2
] + 2 cos[
2
n]
3
3
8
(b)
(5 pts) x[n] = 3 cos[5n
3
]
4
(c)
(5 pts) x(t) = 4 sin(5 t
3
)
5
(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 nd the even (Evfx(t)g) and odd (Oddfx(t)g) 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 dxdt(t) .
x(t)
2
1
t
1
2
3
4
1
2
Figure 3: t vs. x(t).
2
