Time Domain Representation of Continuous and Discrete Signals 99
P.1.35 The Cauchy window is defi ned by the following equation:wnM
MnM()
()
2(^22) 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:
wn
nM
M
() exp.
052
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:fn
n(^1) n
2
03 64
03 64
()
sin[. ( )]
.( )
fn
n(^2) n
2
02 64
02 64
()
sin[. ( )]
.( )
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 func-
tion, 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. There-
fore, 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 byfn
n(^1) n
2
03 64
03 64
()
sin[. ( )]
.( )
fn
n(^2) n
2
02 64
02 64
()
sin[. ( )]
.( )
P.1.40 Consider the analog time functions
f 1 (t) = [5sin(3πt)/(3πt)]^2
f 2 (t) = [3sin(2πt)/(2πt)]^2