Time Domain Representation of Continuous and Discrete Signals
99
P.1.35 The Cauchy window is defi ned by the following equation:
w n
M
M
n M
( )
(
)
ϭ
ϩ
Ϫ
2
2
2
2
Write a program that returns the plots of w(n) versus n, for n = 31 and α = 0.5, 1,
3, and 5, and discuss the effect of α.
P.1.36 The Gaussian window is defi ned by the following equation:
w n
n M
M
( ) exp
.
ϭ
Ϫ
Ϫ
0 5
2
2
Write a program that returns the plots of w(n) versus n, for n = 31, M = 0.5, and
α = 1, 3, and 5, and discuss the effect of α.
P.1.37 Consider the following discrete time sequences:
f n
n
n
1
2
0 3
64
0 3
64
( )
sin[ . (
)]
. (
)
ϭ
Ϫ
Ϫ
f n
n
n
2
2
0 2
64
0 2
64
( )
sin[ . (
)]
. (
)
ϭ
Ϫ
Ϫ
Write a program that returns
a. The plots of f 1 (n) versus n and f 2 (n) versus n, over 0 ≤ n ≤ 128 using the sinc function, and appropriately label the axis.
b. The plots of the expanded signals f 1 (n) and f 2 (n) by a factor L, by inserting L − 1
zeros between each one of the samples of f 1 (n) and f 2 (n), for L = 2, 3, and 4. Therefore, each signal f 1 (n) and f 2 (n) is expanded by a factor of 2, 3, and 4.
c. The plots of the down-sampled signals f 1 (n) and f 2 (n) by a factor of M = 2, 3, and 4.
P.1.38 Truncate or limit the length of each of the functions f 1 (n) and f 2 (n) defi ned in P.1.37
by using the following windows:
a. Parabolic
b. Cauchy
c. Gaussian
defi ned in P.1.34/36 over a length of N = 31.
P.1.39 Obtain the multiplex signal mult[f 1 (n), f 2 (n)]; for the sequences f 1 (n) and f 2 (n) given by
f n
n
n
1
2
0 3
64
0 3
64
( )
sin[ . (
)]
. (
)
ϭ
Ϫ
Ϫ
f n
n
n
2
2
0 2
64
0 2
64
( )
sin[ . (
)]
. (
)
ϭ
Ϫ
Ϫ
P.1.40 Consider the analog time functions
f 1 (t) = [5sin(3πt)/(3πt)]
2
f 2 (t) = [3sin(2πt)/(2πt)]
2
CRC_47760_CH001.indd 99
CRC_47760_CH001.indd 99
7/25/2008 4:16:31 PM
7/25/2008 4:16:31 PM
99
P.1.35 The Cauchy window is defi ned by the following equation:
w n
M
M
n M
( )
(
)
ϭ
ϩ
Ϫ
2
2
2
2
Write a program that returns the plots of w(n) versus n, for n = 31 and α = 0.5, 1,
3, and 5, and discuss the effect of α.
P.1.36 The Gaussian window is defi ned by the following equation:
w n
n M
M
( ) exp
.
ϭ
Ϫ
Ϫ
0 5
2
2
Write a program that returns the plots of w(n) versus n, for n = 31, M = 0.5, and
α = 1, 3, and 5, and discuss the effect of α.
P.1.37 Consider the following discrete time sequences:
f n
n
n
1
2
0 3
64
0 3
64
( )
sin[ . (
)]
. (
)
ϭ
Ϫ
Ϫ
f n
n
n
2
2
0 2
64
0 2
64
( )
sin[ . (
)]
. (
)
ϭ
Ϫ
Ϫ
Write a program that returns
a. The plots of f 1 (n) versus n and f 2 (n) versus n, over 0 ≤ n ≤ 128 using the sinc function, and appropriately label the axis.
b. The plots of the expanded signals f 1 (n) and f 2 (n) by a factor L, by inserting L − 1
zeros between each one of the samples of f 1 (n) and f 2 (n), for L = 2, 3, and 4. Therefore, each signal f 1 (n) and f 2 (n) is expanded by a factor of 2, 3, and 4.
c. The plots of the down-sampled signals f 1 (n) and f 2 (n) by a factor of M = 2, 3, and 4.
P.1.38 Truncate or limit the length of each of the functions f 1 (n) and f 2 (n) defi ned in P.1.37
by using the following windows:
a. Parabolic
b. Cauchy
c. Gaussian
defi ned in P.1.34/36 over a length of N = 31.
P.1.39 Obtain the multiplex signal mult[f 1 (n), f 2 (n)]; for the sequences f 1 (n) and f 2 (n) given by
f n
n
n
1
2
0 3
64
0 3
64
( )
sin[ . (
)]
. (
)
ϭ
Ϫ
Ϫ
f n
n
n
2
2
0 2
64
0 2
64
( )
sin[ . (
)]
. (
)
ϭ
Ϫ
Ϫ
P.1.40 Consider the analog time functions
f 1 (t) = [5sin(3πt)/(3πt)]
2
f 2 (t) = [3sin(2πt)/(2πt)]
2
CRC_47760_CH001.indd 99
CRC_47760_CH001.indd 99
7/25/2008 4:16:31 PM
7/25/2008 4:16:31 PM