1) Determine the autocorrelation function RXX(t) for the continuous-time random process X(t) that...

Question

1) Determine the autocorrelation function RXX(t) for the continuous-time random process X(t) that has the power spectrum (show all steps for your derivation XXXXXXXXXXcos rect 2 2 XX X X K W W p? ? ? ? ?? ? = ? ?? ? ? ?? ? L where K > 0 is a real constant and WX is the spectral extent of the process. 2) Write a Matlab code to produce the estimated power spectrum from the autocorrelation function obtained in step (1) for N=5 normalized frequencies. Note: You may use the FFT function to calculate the Fourier transform. See Example XXXXXXXXXXPlot on the same figure the estimated power spectrum on top of the given power spectrum 4) Repeat steps (2) and (3) for N=10 normalized frequencies. 5) Plot on the same figure the estimated power spectrum of step (4) on top of the given power spectrum 6) Compare your results
	
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1) Determine the autocorrelation function R (t) for the continuous-time random 
XX
process X(t) that has the power spectrum (show all steps for your derivation) 
?? ??
p? ?
2
L () ? = Kcos rect 
XX ?? ??
22 WW
??XX??
where 
K > 0 is a real constant and W is the spectral extent of the process. 
X
 
2) Write a Matlab code to produce the estimated power spectrum from the 
autocorrelation function obtained in step (1) for N=5 normalized frequencies. 
Note: You may use the FFT function to calculate the Fourier transform. See Example 
7.5-5 
 
3) Plot on the same figure the estimated power spectrum on top of the given power 
spectrum   
  
4)  Repeat steps (2) and (3) for N=10 normalized frequencies.  
 
5) Plot on the same figure the estimated power spectrum of step (4) on top of the 
given power spectrum   
 
6) Compare your results   
 
Points will be given for report format, correct program code, comparison, full clear 
complete plots, use of the cover sheet and submission on time 
Submit hardcopy of your work in the class on May 4 , 2013.

Robert · Accepted Answer

Solution: Given the continuous time random process X(t) that has a power spectrum as 
   ( )      
 (
  
   
)     (
 
   
)  
Where K>0 is a real constant and    is the spectral extent of the process. 
The autocorrelation function    ( ) is obtained by taking the inverse Fourier transform of 
the power spectral density 
   ( )  
 
  
∫    ( )

   ( )  
 
  
∫*     (
  
   
)     (
 
   
)+

   ( )  
 
  
∫ *     (
  
   
)+

   ( )  
 
  
∫ * 
 
  
     
  
  
   +
   ( )  
 
  
∫ [ 
 (
 
  
  )
        
  (
 
  
  )
]

   ( )  
 
  
{
    *(
 
  
  )  +
(
 
  
  )
 
       
 
 
    *(
 
  
  )  +
(
 
  
  )
} 
   ( )  
   
  
*    (   )     (     )     (     )+
clear 
N = 5; 
k = 1; 
Ts = 1; 
wx = pi/Ts; 
if rem(N,2) == 0 
    w = -wx : 2*wx/N : wx-2*wx/N; 
else 
    w = -wx+wx/N : 2*wx/N : wx; 
end 
  
tau = -0.5*N:(0.5*N-1); 
  
sinc1 = zeros(1,

1) Determine the autocorrelation function RXX(t) for the continuous-time random process X(t) that has the power spectrum (show all steps for your derivation XXXXXXXXXXcos rect 2 2 XX X X K W W p? ? ?...

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