lect13

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							                      Today’s Schedule
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        • Reading: Lathi 11.2-11.4 (also 7.2),
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11100   • Mini-Lecture 1:
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 1101   • Mini-Lecture 2:
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           – Random processes and linear systems
        • Activity: First Project Group Meeting
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         Lecture 13         Dickerson EE422        1
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            Autocorrelation of Random
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                     Process
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        • Correlation or second moment of a real
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          random process with itself at two times:
                        
11100                _________________     
00100
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                            1   2   
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        Lecture 13                  Dickerson EE422                   2
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        • Usually, t1=t and t2=t+t so that t1- t2 =t
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11100   • A random process is Wide-Sense
                         ________
          Stationary if: x  t   constant
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                                  Rx  t1 , t2   Rx t 
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        • Properties              _________
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                        Rx  0   x 2  t   second moment
                        Rx t   Rx  t 
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                        Rx  0   Rx t 
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        Lecture 13                Dickerson EE422              3
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        • Definition                       X      2
                                                       
                          S x    lim 
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                                             T
                                                       
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        • Relationship to Time Autocorrelation
                                        
                             Rx t   S x  
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                                              
                            1
           Px  Rx  0        sx   d  2 sx  f  df
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                           2 
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        Lecture 13             Dickerson EE422                 4
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        •
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11100   •   Px(f) > 0
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        •   When x(t) is real, Px(-f)= Px(f)
                
                    Px  f  df  P  total normalized power
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        •   
                                                              ____
                                       
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                                                                2

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            Px  0    Rx t  dt
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        Lecture 13                    Dickerson EE422                        5
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        • General expression for the PSD of a
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          Digital Signal:            F f  
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                          Ps  f            R  k  e j kT
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                                                                  s
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            – F(f) is the Fourier Transform of the Pulse
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              Shape f(t)
            – Ts is the sampling interval
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            – R(k) is the autocorrelation of the data:
                                 I
                      R  k     an ak  n  Pi
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        Lecture 13               Dickerson EE422                      6
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                                 I
                      R  k     an ak  n  Pi
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          (n+k)th symbol positions
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          product
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        Lecture 13                   Dickerson EE422   7
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 1111            Digital PSD Discussion
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                                  F f 
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                     Ps  f  
                                    Ts
                                                
                                                k 
                                                        R  k  e j kTs
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01010   • PSD only depends on the
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             – Pulse shape (f(t))
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        Lecture 13                Dickerson EE422                          8
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        •
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        •     Find the PSD:
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        1.    Find the spectrum of pulse:
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                 f  t       F  f   Tb
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        Lecture 13                Dickerson EE422           9
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0000       • For k = 0: there are 2 possibilities, an=A or an=0:
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          R  0     an an i Pi  A2 P 1  02 P  0   A2 / 2 (if P 1  0.5)
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           k     an ank i Pi  A2 P  an  1 and ank  1  0  A  P  an  0 and ank  1
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                  A  0  P  an  1 and an  k  1  02  P  an  0 and an  k  0   A2 / 4
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            Lecture 13                       Dickerson EE422                                  10
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 1111   Putting it all together
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                                                  2
                  A Tb  sin  f Tb                      
                                                               j 2 kf Tb 
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         P f                                       1   e
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                                                                        
                   4   f Tb 
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                                                       k              
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                       A Tb  sin  f Tb 
                              2
                                                           1          
                                                          1    f  
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                        4   f Tb                        Tb         
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                                                                                1
             – Poisson Sum Formula k e                                                 f n T 
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             – sin  f Tb
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        Lecture 13                         Dickerson EE422                                       11
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        • Find the PSD for BiPolar NRZ Signaling
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        2. Find Autocorrelation of signal
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        Lecture 13     Dickerson EE422        12
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        1. Find the spectrum of pulse:
                                  t               sin  fTb
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                     f  t       F  f   Tb
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                                  Tb                 fTb
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        • For k = 0: an=A or an= -A:
                                 2
                       R  0     an an i Pi  A2 P 1    A  P  0   A2
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        • For k >0: an=-A or A and an+k=-A or A:
                                                                                        
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          R  k     an an  k i Pi  A2  A   A     A  A    A 
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                                                                                                 0
                                                                                     2       1
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        Lecture 13                         Dickerson EE422                                            13
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0000    • A random process is said to be a white noise
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10100     process if the PSD is constant over all
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                                          R t   0  t 
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                                    2                   2
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        Lecture 13             Dickerson EE422                   14
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                 y t   h t   x t   Y  f   H  f  X  f 
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        • What is the autocorrelation and PSD for y(t)
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          when x(t) is known?
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                        Linear Network             Y(f)
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              Px(f)
        Lecture 13               Dickerson EE422                        15
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        • IF a WSS random process x(t) is applied to
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          a LTI system with impulse response h(t),
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          the output autocorrelation is:
                         
             Ry t     h  1 h  2  Rx t  2  1  d 1d 2
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                       h  t   h t   Rx t 
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 1111   • And the output PSD is:
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                     Py  f   H  f  Px  f 
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        Lecture 13              Dickerson EE422                     16
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 1111                  Next Time
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        • Reading: Lathi 11.2-11.4 (also 7.2),
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11100   • Mini-Lecture 1:
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         Lecture 13         Dickerson EE422      17