Time Domain Representation of Continuous and Discrete Signals
97
P.1.24 Let f 1 (t) = 3e
2t/3
+ 3 and f 2 (t) = cos(πt). Create a script fi le that returns the energy and
power of f 1 (t) and f 2 (t).
P.1.25 Repeat P.1.24 for the following discrete sequences: f 1 (n) = 3e
2n/3
+ 3 and f 2 (n) =
cos(0.1πn), over the range −10 ≤ n ≤ 10.
P.1.26 Evaluate by hand which of the following discrete sequences are periodic, and if
periodic, evaluate its period.
a. cos(0.2n)
b. cos(0.2πn)
c. sin(2πn/3)
d. sin(9.1n)
P.1.27 Verify the following equalities:
a. Imp[( )]
n ϭ
Ϫ
ϭϪϱ
ϩϱ
(
)
n k
k
∑
b. u n
[( )] ϭ
Ϫ
ϭ
ϩϱ
(
)
n k
k 0
∑
c. r n
k n k
k
( )
(
)
ϭ
Ϫ
ϭ
ϩϱ
0
∑
P.1.28 Evaluate the following integrals:
a.
(
) (
)
2 3
1
t
t
dt
+
∫
Ϫ
ϭ
Ϫϱ
ϱ
b.
(
) (( / )
)
2 3 3 2 1
t
t
d t
+
∫
Ϫ
ϭ
Ϫϱ
ϱ
c.
sin( ) (( / ) ( / ))
wt
t
dt
3 2
2 3
Ϫ
ϭ
Ϫϱ
ϱ
∫
d.
e
t
d t
t
(
) (
( / ))
Ϫ ϩ
Ϫϱ
ϱ
Ϫ
ϭ
3 1 1
2 3
∫
P.1.29 Analyze and draw a fl ow chart of Example 1.15 and indicate in each average approximation the fi lter used as well as the effect of the fi lter.
P.1.30 Evaluate the fi rst and second derivative of the following expressions:
a. f 1 (t) = u(t) + 7u(t − 5) − 2u(t − 7)
b. f 2 (t) = tu(t) + e
3t u(t − 1) + 3δ(t − 3)
c. f 3 (t) = r(t)u(t) − r(t − 1)u(t − 1)
P.1.31 Let H(s) =
s + 3
_______________
s
3
+ s
2
+ 5s + 10
be the transfer function of a given analog system. Write
a MATLAB program that returns
a. The Bode plot of H(s), magnitude and phase
b. The impulse as well as the step responses
c. The zero/pole diagram
d. The system differential equation
P.1.32 Let
H z
z
z
z
z
z
z
( )
.
.
.
.
.
.
.
ϭ
ϩ
Ϫ
ϩ
ϩ
Ϫ
ϩ
Ϫ
Ϫ
0 32 0 43
0 85
0 2
1 0 802
0 42
0 62
1
2
3
1
2
3
−
−
−
−
CRC_47760_CH001.indd 97
CRC_47760_CH001.indd 97
7/25/2008 4:16:31 PM
7/25/2008 4:16:31 PM