Colorado State University, Ft. Collins
Spring 2016
ECE 312: Linear Systems Analysis II (Signal and Systems)
Homework 6
Assigned on: 04/19/2016, Due by: 05/05/2016
6.1
Consider a discrete-time periodic function xn with DTFT
2
4
X   

X
k  
0
2k 
 2k  

   

4 
 4  
 2k 
The values of X 0 
 are
 4 
0

 2k  6
X0

 
 4  0
6
k 0
k 1
k2
k 3
Determine xn .
6.2
Consider two length-4 discrete time signals
xn  e
j 2n
4
, n  0,1,2,3
and
hn  2 n   n  2, n  0,1,2,3
(a) Find X k , the DFT of xn .
(b) Find H k  , the DFT of hn .
(c) Find Y k  , the DFT of yn  xn  hn.
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(d) Find yn  xn  hn
6.3
The DFT of the analog signal f t   7 cos100t cos40t  is to be computed.
(a) What is the minimum sampling frequency to avoid aliasing?
(b) If a sampling frequency of S  300 rad/s is used, how many samples must be
taken to give a frequency resolution of 1 rad/s?
6.4
Let xt  be a continuous-time signal whose Fourier transform is X   . The follow
two signals are obtained by sampling xt  at sampling frequency  s .
x1 t  
x2 t  
where Ts 
2

 xnT  t  nT 
s
n  

s

 x nT
n  
s

Ts  
T 
  t  nTs  s 
2 
2
. Assume that X    0 for all  
s
no aliasing involved.
s
2
, or in other words, there is
a) Obtain the Fourier transforms of the following signals x1 t  , x2 t  ,
x1 t   x2 t  .
b) Construct two discrete-time sequences by x1 n  xnTs  ,
T 

x 2 n  x nTs  s  . Obtain the discrete time Fourier transforms of the
2

following discrete-time signals, X 1   DTFT x1 n,
X 2   DTFT x2 n. Compute X 1   X 2  and check whether it
has any direct connection with X 1    X 2  .
6.5
2
Let xn , n  0,1,2,, N  1 , be an N-point discrete-time signal whose discrete
Fourier transform is X k  , k  0,1,2,, N  1.
a) Obtain the N-point DFT of the flipped sequence x1 m,
m  0,1,2,  , N  1
x1 m  xN  1  n
Explain its connection with X k  .
b) Obtain the 2N-point DFT of the following zero-padded sequences
x3 m , m  0,1,2,,2 N  1
 xm
x2 m  
0
 x m  N 
x3 m   1
0
m  N 1
m  N 1
m  N 1
m  N 1
Explain their connections with X k  and X 1 k 
c) Obtain the 2N-point DFT of x2 m  x3 m .
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