ENGR 451 Problem Set #1 1.1 Find and plot()H?and()H?(, the magnitude and phase of the DTFT of the...

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ENGR 451 Problem Set #1 1.1 Find and plot()H?and()H?(, the magnitude and phase of the DTFT of the pulse,[]hn, shown below. 01201[]hnnMake sure your plot of()H?(is appropriately phase wrapped. 1.2 Find and plot()H?and()H?(, the magnitude and phase of the DTFT of the sequence,[][ 1] [ 3]hnnndd= XXXXXXXXXXMake sure your plot of()H?(is appropriately phase wrapped. 1.3 Given a system with impulse response, [] [] [ 1] [ 2]hn n nnddd=--+-, find the response of the system when 23[] cosxnnp=. 1.4 A system is defined by the linear constant coefficient differential equation12[][ 1] [ 1]yn yn xn--= -. Find the output,[]yn, when the input14[][]nxnun=. 1.5 a) Given the system below, find()H?. b) Find[]ynwhen2[] cosxnnp=. c) Find[]ynwhen4[] cosxnnp=. Delay by 2+Delay by[][]xnyn2-1.6 Consider a system with input12[] [][ 1]xn nndd=- -, and output is14[][][]nynn und=- +, find[]hn[]xn[]hn[]yn1.7 For the following system, assume that[]xnhas the spectrum shown below a) Make a fully labeled sketch of()X?,()Q?,()R?and()Y?. b) Find[]ynin terms of[]xn. 1()H??cos(2)n?[]xn[]qn[]rn[]yn×cos(2)n?×2p-2p01?pp/20p/2p()X?
	
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Problem Set #1  
1.1  
Find and plot H()? and( H()? , the magnitude and phase of the DTFT of the pulse,hn [] , shown below.

hn []
1
0
n
0 1 2
 
Make sure your plot of( H()? is appropriately phase wrapped. 
1.2  
Find and plot H()? and( H()?, the magnitude and phase of the DTFT of the 
sequence,hn [] =-d[n +1]-d[n -3] . Make sure your plot of( H()? is appropriately phase wrapped. 
1.3  
Given a system with impulse response, hn []=d[n]--dd [n 1]+ [n-2], find the response of the system 
2p
when x[]nn=cos . 
3
1.4  
1
A system is defined by the linear constant coefficient differential equationyn []- yn [ -= 1] x[n-1]. Find 
2
n
1
the output, y[] n , when the inputx[]nu= []n .  
4
1.5  
a) Given the system below, findH()? . 
p
b) Find y[] n whenx[]nn=cos . 
2
p
c) Find y[] n whenx[]nn=cos . 
4
 
Delay by 2
x[] n yn []
+
Delay by-2
 
1.6  
n
1 1
Consider a system with inputxn []=-dd []n [n-1] , and output isyn [] =-d[]n + u[]n , findhn [] 
2 4
 
x[] n hn [] yn []1.7  
For the following system, assume thatx[] n has the spectrum shown below 
a) Make a fully labeled sketch ofX()? ,Q()? ,R()? andY()? . 
b) Find y[] n in terms ofx[] n . 
 
H()?
qn [] rn []
1
x[] n yn []
××
?
-p 2p 2
cos(n? 2) cos(n? 2)

?
X()
1
?
0
pp/2 0p/2p

Robert · Accepted Answer

Sol: (1.1)  
              DTFT of pulse  h n is, 
                                            j n
n
H h n e 



   
From the figure, DTFT of pulse  h n is,
   
       
 
     
     
2
0
2
2
0 1 2
1
1 cos sin cos 2 sin 2
1 cos cos 2 sin sin 2
j n
n
j j
j j
H h n e
H h h e h e
H e e
H j j
H j

 
 



    
    


 
 

  
  
    
    

 
Now taking a magnitude of  H  ,
     
     
     
   
   
2 2
2 22
2 22 2
2 2
1 cos cos 2 sin sin 2
1 cos 2cos 1 sin 2sin cos
cos 1 2cos sin 1 2cos
1 2cos cos sin
1 2cos
H
H
H
H
H
    
     
    
   
 
    
     
   
  
 
 
And phase,
 
 
1
1
2
1
sin sin 2
tan
1 cos cos 2
sin 2sin cos
tan
1 cos 2cos 1
sin 1 2cos
tan
cos 1 2cos
 

 
  

 
 

 



 
    
 
     
 
  
  
 
                                                                    1
sin
tan
cos



     
 
                                                   
 1tan tan 
 
 
 
 
Now plot the magnitude and phase,
                                                  Magnitude plot
                                                            Phase plot 
Sol: (1.2)  
              DTFT of pulse  h n is, 
                                      j n
n
H h n e 



   
Given that 
                                        1 3h n n n       
Now taking a DTFT of  h n ,
 
     
     
3
cos sin cos3 sin 3
cos cos3 sin sin 3
j jH e e
H j j
H j
 
    
    
  
    
     
 
Now taking a magnitude of  H  ,
     
     
       
   
    
    
 
2 2
2
2 2 2 2
2 2 2 2
2
cos cos3 sin sin 3
cos cos 3 2cos cos3 sin sin 3 2sin sin 3
cos sin cos 3 sin 3 2 cos cos3 sin sin 3
1 1 2 cos 4
2 1 cos 4
2 1 2cos 2 1
2cos 2
H
H
H
H
H
H
H
    
        
        
 
 
 
 
     
     
     
  
 
  

 
And phase,
 
 
 
 
1
1
sin sin 3
tan
cos cos 2
sin sin 3
tan
cos cos 2
 

 
 

 


  
  
  
 
  
 
 
                                
 1tan tan 
 
 
 
 
       Now plot the magnitude and phase,
                                       Magnitude plot
                                              Phase plot
Sol: (1.3)    Given that, 
          Impulse response        1 2h n n n n        
    On taking the DTFT of  h n , 
                                             21 j jH e e       
                         Input             
2
cos
3
x n n

  
 On taking the DTFT of  x n ,
   
 
2
cos
3
2 2
3 3
X DTFT x n DTFT n
X


 
     
 
     
 
    
       
    
 
We know that, 
                                              
 
 
Y
H
X



  
Or 
                                               Y H X    
Now substitute the value of     and H X  in above equation,

ENGR 451 Problem Set #1 1.1 Find and plot()H?and()H?(, the magnitude and phase of the DTFT of the pulse,[]hn, shown below. 01201[]hnnMake sure your plot of()H?(is appropriately phase wrapped. 1.2 Find...

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