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# lect13 by ajizai

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```									                      Today’s Schedule
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• Reading: Lathi 11.2-11.4 (also 7.2),
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10100
11100   • Mini-Lecture 1:
00100
0100       – Power spectral densities
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0111
1101   • Mini-Lecture 2:
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– Random processes and linear systems
• Activity: First Project Group Meeting
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1111
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10000   • Project Discussion and Questions
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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
0100    Rx  t1 , t2   x t x t    x1 x2 f x  x1 , x2  dx1dx2
0011
1   2   
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Lecture 13                  Dickerson EE422                   2
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1111                Wide Sense Stationary
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• Usually, t1=t and t2=t+t so that t1- t2 =t
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0000
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10100
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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1111                Power Spectral Density
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0100                                     ________________
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• Definition                       X      2

S x    lim 
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T

10100                                T      T       
11100                                                 
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0011
• Relationship to Time Autocorrelation

Rx t   S x  
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01010
10010
10111   • Power of a Random Process
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                
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Px  Rx  0        sx   d  2 sx  f  df
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2 
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Lecture 13             Dickerson EE422                 4
10111
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1111                     Properties of PSD
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•
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0000        Px(f) is always real
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10100
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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10111   •   If x(t) is WSS,  Px  f  df              P  x  Rx  0 
2

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1010    •                 
Px  0    Rx t  dt
00011                     
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Lecture 13                    Dickerson EE422                        5
10111
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1111           General Expression PSD
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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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0100                                                 Ts   k 
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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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01010                           i 1
Lecture 13               Dickerson EE422                      6
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1111                Autocorrelation Term
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I
R  k     an ak  n  Pi
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11100                           i 1
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0111   • an and ak+n are the levels of the nth and
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(n+k)th symbol positions
10111
1101   • Pi is the probability of having the ith anan+k
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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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00100
Ps  f  
Ts

k 
R  k  e j kTs
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0011
0111
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01010   • PSD only depends on the
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– Pulse shape (f(t))
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1010         – Statistical properties of the data
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Lecture 13                Dickerson EE422                          8
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1111           Example: Unipolar NRZ
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•
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10100
11100   •     Square pulses of width Tb
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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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1111                 Evaluate Autocorrelation
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1010
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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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00100                    i 1
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0011
0111      • For k >0: there are 4 possibilities, an=0 orA and
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01010        an+k=0 or A:
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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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1101 R
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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
10111
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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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0011                                  
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4   f Tb                        Tb         
01010
10010
10111   • Simplify using:                                                            
1
– Poisson Sum Formula k e                                                 f n T 
1101                                                            j 2 kf Tb
1111                                                                                           b
1010                                                                         Tb   k 
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– sin  f Tb
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01010                 f Tb
Lecture 13                         Dickerson EE422                                       11
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1111                 Activity
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• Find the PSD for BiPolar NRZ Signaling
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11100   1. Find F.T. of pulse
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0011
2. Find Autocorrelation of signal
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Lecture 13     Dickerson EE422        12
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1111                                  Solution
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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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11100
 Tb                 fTb
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0111   2. Autocorrelation
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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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1101                           i 1
1111
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• For k >0: an=-A or A and an+k=-A or A:
                                             
4
R  k     an an  k i Pi  A2  A   A     A  A    A 
00011
0
2       1
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01010
Lecture 13                         Dickerson EE422                                            13
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1111                White Noise Process
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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
11100
00100     frequencies: P  f   N0               N
R t   0  t 
0100                      x                      x
0011
2                   2
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01010                          t                             f
Lecture 13             Dickerson EE422                   14
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1111                  Linear Systems
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1010
1100
0100    • Recall:
y t   h t   x t   Y  f   H  f  X  f 
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11100   • This is still valid if x and y are random processes,
00100
0100      x might be signal plus noise or just noise
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• What is the autocorrelation and PSD for y(t)
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when x(t) is known?
10111          x(t)                                y(t)
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Linear Network             Y(f)
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00011         Rx(t)                                Ry(t)
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01010
Px(f)
Lecture 13               Dickerson EE422                        15
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1111                   Key Theorem
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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
0011
0111
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 h  t   h t   Rx t 
01010
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1101
1111   • And the output PSD is:
1010

Py  f   H  f  Px  f 
00011                                       2
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01010
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),
1000
0000
00001
10100
11100   • Mini-Lecture 1:
00100
0100       – Equivalent Noise Bandwidth
0011
0111
1101   • Activity
01010
10010
10111   • Mini-Lecture 2:
1101
1111      – Quiz 2
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00011
10000   • Mini-Lecture 3:
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Lecture 13         Dickerson EE422      17

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