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					EE422-Spring 2002                                                                                                     Quiz 3
Dr. Dickerson


1. A binary symmetric channel has an error probability of PE. The probability of transmitting 1 is
Q and the probability of transmitting 0 is 1 – Q. If the receiver detects the incoming digit as 0
what is the probability that the corresponding transmitted digit was a 1?

This question is looking for the probability that a 1 was sent given that a 0 was detected. This is
opposite of how we usually think of conditional probabilities: Pr(1 transmitted/0 detected).
However, we can still apply Bayes rule (theorem) to solve for it:
                     Prd etected transmitted  0 1 Prtransmitted 1
Prtrans det 1 0  
                                       Prdetected  0 
                                                            PE Q
         
             Prtransmitted 1 Prd etected transmitted  0 1  Prtransmitted  0  Prd etected transmitted  0 0 
                     PE Q
         
             QPE  1  Q 1  PE 
                       PE Q
         
             1  PE   Q 1  2 PE 

2. Two random processes x(t) and y(t) are:
 x  t   A cos 0t    and y t   B cos  n0t  n 
 where n is an integer not equal to 1 and A and B are constants. is a uniformly distributed RV
 in the range (0, 2). Show that the two processes are incoherent.

Solution:
Definition of Incoherent:
Rxy    0
Starting from the definition of cross correlation:
                   ______________________________
Rxy    AB cos 0t    cos  n0  t     n 


          AB 
                _____________________________ ____________________________ 
                                                                                                
              cos 0t  n0  t      n  1   cos  n0  t     0t   n  1  
           2                                                                                    
                                                                                                
Calculating the statistical averages gives:
_____________________________
                                                           1 2
cos 0t  n0  t      n  1                          cos 0t  n0  t      n  1  d
                                                          2 0
        0
Similarly
_____________________________
cos  0t  n0  t      n  1   0 and the cross correlation is Rxy    0 (incoherent)
EE422-Spring 2002                                                                           Quiz 3
Dr. Dickerson

3. The input to a linear system is a deterministic sinusoid ( s t   Asin 0t    ) and a white
Gaussian noise process with rms value N/2.
a. Find the input signal to noise ratio
b. Find the output signal to noise ratio of the filter with the following impulse response:
h  t   e j0t e . Use the attached integral and Fourier transform tables to calculate the result as
                  a t


far as you are able. Do not be alarmed if the result is not in the table. Show all work.

Solution:
                                               A2
a. Input power of a sinusoid is Ps               (from 421 and previously this semester)
                                               2
Noise power is:
  ____
                             N
  n   Sn  f df  
    i
      2
                                 df  
                           2
The input SNR is zero since any number over infinity is zero.

b. The Output signal to noise ratio is calculated as in Lecture 14:
Output is y(t)=so(t)+no(t) (superposition of linear systems)
b. Output Signal Power:
Calculate output signal using convolution

                                             
  so  t   si  t   h  t   A H  f o  sin 0t    ph  H  f o 
                                                                            
                   2
                     A
      so  t           H  f0 
       2                          2

                       2
From the Fourier Transform table given at the end of the quiz:
  a t                                           2a
e                                                            a0
                                             a 2
                                               2

and the complex exponential in the front is a frequency shift, so
                                             2a
h  t   e j0 t e            H    2
                   a t

                                        a    0 
                                                      2


Substituting into the above equation gives:
                             2
                 A2 2    2 A2
        s t  
            2
            o           2
                 2 a      a
Output Noise Power Spectral Density
                                                        2
                                          2a                N
Sn0    H   Sn   
                       2

                                  a 2    0 
                                                    2
                                                            2
Power of the random process is:
                 1                  1 
Pn  Rn  0       Sno   d   0 Sno   d
  o     o       2
EE422-Spring 2002                               Quiz 3
Dr. Dickerson

                       2 A2
SNRout                 a2
                                       2
                
           N            2a
           2                             d
                0 a     0 
                   2               2

				
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